A&A 450, 1037-1050 (2006)
DOI: 10.1051/0004-6361:20054626

Are 44Ti-producing supernovae exceptional?[*]

L.-S. The 1 - D. D. Clayton1 - R. Diehl2 - D. H. Hartmann1 - A. F. Iyudin2,3 - M. D. Leising1 - B. S. Meyer1 - Y. Motizuki4 - V. Schönfelder2

1 - Department of Physics and Astronomy, Clemson University, Clemson, SC 29634-0978, USA
2 - Max-Planck-Institut für Extraterrestrische Physik, Postfach 1312, 85741 Garching, Germany
3 - Skobeltsyn Institute of Nuclear Physics, Moscow State University, Vorob'evy Gory, 119992 Moscow, Russia
4 - Cyclotron Center, RIKEN, Hirosawa 2-1, Wako 351-0198, Japan

Received 1 December 2005 / Accepted 15 December 2005

According to standard models supernovae produce radioactive 44Ti, which should be visible in gamma-rays following decay to 44Ca for a few centuries. 44Ti production is believed to be the source of cosmic 44Ca, whose abundance is well established. Yet, gamma-ray telescopes have not seen the expected young remnants of core collapse events. The 44Ti mean life of $\tau$ $\simeq$ 89 y and the Galactic supernova rate of $\simeq$3/100 y imply $\simeq$several detectable 44Ti gamma-ray sources, but only one is clearly seen, the 340-year-old Cas A SNR. Furthermore, supernovae which produce much 44Ti are expected to occur primarily in the inner part of the Galaxy, where young massive stars are most abundant. Because the Galaxy is transparent to gamma-rays, this should be the dominant location of expected gamma-ray sources. Yet the Cas A SNR as the only one source is located far from the inner Galaxy (at longitude 112$^\circ $). We evaluate the surprising absence of detectable supernovae from the past three centuries. We discuss whether our understanding of SN explosions, their 44Ti yields, their spatial distributions, and statistical arguments can be stretched so that this apparent disagreement may be accommodated within reasonable expectations, or if we have to revise some or all of the above aspects to bring expectations in agreement with the observations. We conclude that either core collapse supernovae have been improbably rare in the Galaxy during the past few centuries, or 44Ti-producing supernovae are atypical supernovae. We also present a new argument based on 44Ca/40Ca ratios in mainstream SiC stardust grains that may cast doubt on massive-He-cap type I supernovae as the source of most galactic 44Ca.

Key words: ISM: abundances - Galaxy: abundances - gamma rays: observations - ISM: supernova remnants - supernovae: general - dust, extinction

1 Introduction

Supernovae are the agents that drive the evolution of gaseous regions of galaxies. As end points of the evolution of massive stars that have formed out of the interstellar gas, their explosions eject matter enriched with freshly formed isotopes and stir interstellar gas. However, the explosions themselves are still not understood (Janka et al. 2003; Burrows 2000). Parametric descriptions are used to describe the core collapses ("cc-SN'', supernovae of types II and Ib/c) (Woosley & Weaver 1995; Thielemann et al. 1996) as well as thermonuclear explosions of white dwarfs (supernovae of type Ia; Nomoto et al. 1997). A prominent issue in astrophysics is whether the supernova explosion itself is a well-regulated, robust physical process, or if intrinsic variability over a wider range of physical conditions are rather common.

Supernova homogeneity by type can be studied in different ways. One approach is to analyze the rate of supernovae of a specific type in different environments and over different time scales. In this work we do this by asking if the current rate of supernovae in our Galaxy which produce radioactive 44Ti is in line with expectations from other observables and from supernova theory.

44Ti decay offers a unique window to the study of supernova rates. Specific aspects of this window are:

Gamma rays penetrate the entire galactic disk with little extinction.
44Ti gamma-rays reflect the current rate of supernovae, with the 44Ti mean decay time scale of $\tau = 89$ years; this present-day snapshot which has not yet fed back into chemical evolution can be directly related to the observable current population of massive stars.
Most 44Ti is co-produced with 56Ni in relatively frequent core collapse supernovae (cc-SN). The radioactive energy of 56Ni is responsible for well-observed supernova light.
Nucleosynthesis of 44Ti is primarily from $\alpha$-rich freeze-out of nuclear statistical equilibrium and secondarily from silicon burning.
The origin of abundant cosmic 44Ca occurs mainly through 44Ti  nucleosynthesis.
44Ti traces have been found in pre-solar grains which have been attributed to condensation within core-collapse supernovae.
In this paper we estimate what the gamma-ray sky of 44Ti sources would be expected to look like by adopting an average 44Ti source model having a characteristic source event recurrence rate, 44Ti yield per event, and spatial distribution. We compare this to the present-day gamma-ray survey and find apparent and serious conflicts. Then we analyze whether deviations from these average expectations can occur from the known or expected variability of models and parameters involved. We use a Monte Carlo simulation of the expected sky image within reasonable distributions of parameters for that purpose. This leads us to discuss each of the relevant parameters, which may explain an anomalous 44Ti sky; these are, specifically:

2 44Ti and supernovae in the Galaxy

Supernova rates in the Galaxy can be inferred from different observables. But observational incompleteness and bias requires that several assumptions are an essential part of such inferences. We compare here the observed 44Ti sky with expectations for the occurrences of young 44Ti emitting supernova remnants, as they result from astrophysical assessments of supernova characteristics for our Galaxy. We choose a time interval unit of 100 years for comparison of chemical history with supernova event rates. The extrapolation of the past history and yields of 44Ti-ejecting supernova events should give a production rate which can be compared with recent supernova rates, as no other source of 44Ca has yet been identified.

Have the 44Ca-producing supernovae been typical; is all of 44Ca produced through radioactive 44Ti; or do exceptional events contribute most of the 44Ti? Gamma-ray surveys for 44Ti sources, and presolar supernova grains provide ways to address this question.

In the following section we will then examine each of the critical assumptions in more detail.

For a 44Ti sky reference, we adopt the result from COMPTEL's survey in the 1.157 MeV band, which is the most complete survey to date (Dupraz et al. 1997; Iyudin et al. 1999). In this survey, one object has been clearly detected (340-year-old Cas A at a distance of 3.4 kpc), candidates at lower significance have been discussed (most prominently GRO J0852-4642 in the Vela region, Iyudin et al. 1999), and a weak signal from the Per OB2 association (Dupraz et al. 1997). Apparently, no bright young 44Ti emitting supernova remnants are found in the inner region of the Galaxy (see Fig. 1).

What do we expect the Galaxy to look like in 44Ti emission?

2.1 44Ti from supernovae

Supernova observations directly demonstrate that these events are at the origin of 44Ti production: the 1.157 MeV $\gamma $-ray line following 44Ti decay has been detected in the 340-year old Galactic supernova remnant Cas A (Iyudin et al. 1994; Vink et al. 2001). Furthermore, SN1987A's late light curve, observed in unique detail over more than 15 years, appears powered by a similar amount of 44Ti (0.2- $2.0\times 10^{-4}$ $M_{\odot}$), from modeling of radioactive energy deposition and photon transport in the SNR (Fransson & Kozma 2002; Woosley et al. 1989). Gamma-ray detection and proof of this interpretation is still lacking, but INTEGRAL's recent observations may prove sufficiently sensitive.

\end{figure} Figure 1: Maximum-entropy map of the Galactic plane (within latitude $\pm $30$^\circ $) in the 44Ti energy window (1.066-1.246 MeV) for the combination of CGRO observations from 0.1 to 617.1. Two crosses mark the positions of Cas A and GRO J0852-4642. This figure is adopted from Iyudin (1999).
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Presolar grains have been identified in meteoritic samples through their unusual isotopic abundance patterns, and hold rich isotopic abundance detail for characterizing their condensation environments (e.g. Clayton & Nittler 2004; Zinner 1998). SiC grains of the X-type are attributed to core-collapse supernovae from their large excesses in characteristic isotopes (Nittler et al. 1996). 44Ti-produced overabundance of 44Ca is found in these SiC X grains, indicating presence of 44Ti  at their time of condensation. Measured 44Ca/40Ca ratios are large (Nittler et al. 1996; Clayton et al. 1997a; Hoppe et al. 2000). This proves their supernova origin, on one hand, and likewise it proves that dust-producing supernovae may eject 44Ti in significant amounts. Their attribution to core collapse supernovae rather than supernovae of type Ia (Nittler et al. 1996) is plausible, if we believe that core collapse supernovae probably dominate 44Ti  production and because there is no direct evidence of dust condensation in thermonuclear supernovae (Dwek 1998). It is likely that all of the observed X grains sample both different condensation environments and different production events. Therefore, their 44Ca abundances cannot be interpreted in absolute terms as a measure of the mass of 44Ti ejected per supernova.

The most plausible cosmic environment for production of 44Ti  is the $\alpha$-rich freeze-out from high-temperature burning near Nuclear Statistical Equilibrium (e.g. Arnett 1996; Woosley et al. 1973). The required high values for the entropy are found in core-collapse supernovae. Therefore the simplest plausible assumption is that core-collapse supernovae are responsible for any substantial sources of 44Ti.

It is likewise plausible that traditional SNIa do not add significantly to nucleosynthesis at mass A = 44, specifically no 44Ti, since their NSE freeze-out conditions will not be favorable for 44Ti production (Timmes et al. 1995). We consider the symbiotic-star scenario as a rare subclass of SNIa, even though their 44Ti yields may be large (Woosley & Weaver 1994); so they indeed would be rare outliers, rather than typical 44Ti producing supernovae. We will discuss uncertainties in each of these sites in more detail below (see Sect. 3.5).

2.2 Ca from supernovae

Measurements of cosmic isotopic abundances can be converted into isotope production rates; however, for all long-lived isotopes models of chemical evolution have to be applied (e.g. Matteucci 2003; Timmes et al. 1995; Clayton 1988; Pagel 1997). How can we use these for predicting the shortlived 44Ti source appearance in gamma-rays? Its lifetime is too short for mean chemical evolution arguments.

Over the time scale of chemical evolution of the Galaxy, the cumulative and averaging effects of different explosion types would integrate to a smooth pattern of "standard abundances'', as they are observed throughout the universe. Evolutionary pathways for individual elements can be associated with the evolution of metallicity and the rates of different supernova types. The abundance of 44Ca in solar-system matter thus can be translated through models of chemical evolution into a current average production rate of $\simeq$ $3\times 10^{-4}$ $M_{\odot}$ of 44Ti per 100 y (Leising & Share 1990, see below).

Chemical evolution calculations (Timmes et al. 1995) using computed ejecta masses (Woosley & Weaver 1995) were shown to account reasonably for most solar abundances, including specifically 40Ca, but failing by a factor of three for 44Ca; this fact is also evident from the ratio of 44Ti to Fe in cc-SN models. But over short time scales where only a limited number of source events contribute, great variability among sources would present a significant difference of the present-day picture from the average. Therefore we may compare the expectations from long-term averaged 44Ti  production to the present-day 44Ti source record imaged in gamma-rays, allowing for the short-term fluctuations with Monte-Carlo realizations of the sources.

The solar-system 44Ca abundance is rooted plausibly in 44Ti nucleosynthesis (Clayton 1982; Woosley et al. 1973), and is a result of the integrated Galactic nucleosynthesis prior to formation of the solar system. Given a time-dependence of the production rate, it can be normalized to the rate required to give precisely the measured solar abundance 4.5 Gyr ago, which also fixes the long-term average production rate today. Models of galactic chemical evolution, constrained by a number of observables, provide us with that time dependence, albeit subject to a number of assumptions and parameter choices. Adjusting the 44Ti yield in the aforementioned calculation (Timmes et al. 1995) upward by a factor of three to achieve the solar abundance, we infer a production rate of 44Ca of about $3.6\times 10^{-6}$ $M_{\odot}$  yr-1. Other considerations using different chemical evolution models (see Appendix C) lead from the 44Ca abundance to a current 44Ti production rate: p(44Ti) =  $5.5\times 10^{-6}$ $M_{\odot}$  yr-1, with full uncertainty range 1.2- $12\times 10^{-6}$ $M_{\odot}$  yr-1 (see Appendix C).

2.3 Supernova rates

Direct supernova rate measures have been made through correlations between supernova activity and other tracers of the massive star content of a galaxy. van den Bergh (1991) finds ( $2.62~\pm 0.8$) h1002 SN century-1. For h100= 0.75, the rate is $1.5 \pm 0.8$ SN century-1. This rate is based on a combined study of galactic supernova remnants, historical SNe, and novae in M 31 and M 33. Cappellaro et al. (1993) refer to this rate as the best estimate. van den Bergh & Tammann (1991) find SNR = 4.0 SN century-1. The authors review supernova rates in external galaxies and derive a specific supernova frequency, in units of 1 SNu = one SN per century per 1010 $L_\odot$(B), for various galaxy types. If one assumes that the Galaxy is intermediate between types Sab-Sb and types Sbc-Sd, the specific rate is  3  h1002 SNu. For a Galactic blue-band luminosity of L(B) =  $2.3\times 10^{10}$ $L_\odot$(B) (their Table 11) and h100 = 0.75 we infer SNR = 4.0 SN century-1. Their review paper also discusses estimates from internal tracers in the Milky Way: from radio supernova remnant (RSNR) statistics they infer SNR =  $3.3 \pm 2.0$ SN century-1. From the historic record of nearby (<a few kpc) supernovae in the past millennium they find SNR = 5.8 $\pm~ 2.4$ SN century-1. The large extinction corrections in the galactic plane make this small sample highly incomplete, which results in large uncertainties in extrapolations to the full galactic disk. The authors also review efforts based on the pulsar birth rate, but extensive observational selection effects in combination with the strong and poorly understood evolution of luminosity and beaming geometry (see Lyne & Graham-Smith 1998; and Lorimer & Kramer 2004) renders this method impractical for estimating the galactic SNR. Continuing the studies of van den Bergh & Tammann, Cappellaro et al. (1993) find SNR =  $1.4 \pm 0.9$ SN century-1, when scaling to external galaxies of similar type. The sample is obtained from surveys carried out at the Asiago and Sternberg Observatories. The authors provide an extensive discussion of the uncertainties of this method, which can exceed 200% for some late type galaxies. More recently, van den Bergh & McClure (1994) find SNR = (2.4-2.7) h752 events per century. This estimate is based on re-evaluation of the extra-galactic SN rates obtained from Evans' 1980-1988 observations. This method depends on a somewhat uncertain type of the Galaxy and the value of its blue-band luminosity, while the uncertainty due to the Hubble constant is now very small. Given the error analysis in the paper, the rate is uncertain by at least 30%.

van den Bergh & McClure (1994) in studying the supernova rates of local spiral galaxies of types Sab-Sd of Evans' observations estimated that 80%-90% of supernova in that galaxies are of types Ibc and II. Recently Cappellaro (2003) combining five SN searches to include 137 SNe in 9346 galaxies estimates the SN type ratios in the Galaxy to be Ia:Ib/c:II = 0.22:0.11:0.67. From these observations, one infers a ratio of core-collapse to thermonuclear supernova of R = (II+Ibc)/Ia = 3.5. However, note that the Galactic historical record in the last millennium shown in Table B.1 contains only two type Ia SNR out of six SNRs. An often used alternative distribution over types is Ia:Ib/c:II = 0.1:0.15:0.75 (Hatano et al. 1997; Dawson & Johnson 1994; Hartmann et al. 1993; Tammann et al. 1994), which implies a three times higher cc-SN fraction. In this work, we adopt this set of parameters, and note that the small 44Ti yield of type Ia renders our results insensitive to this ratio.

Over the last millennium, the historic record contains six events (see Appendix B.4), which implies a rather low rate at face value. However Galactic extinction at visible wavelengths and embedded supernovae will lead to large occultation bias, and with extinction models plus Monte Carlo simulations this historic record can be assessed to approximately agree with extragalactic rate determinations (see Sect. 3 and Appendix A.2). An often cited rate of galactic cc-SN of three per century is consistent both with astronomical arguments (van den Bergh & McClure 1990) and with the rate inferred by Timmes et al. (1995) from their chemical evolution model that produces solar abundances successfully. We adopt a supernova recurrence rate of 30 years as a baseline for our 44Ti sky expectations.

\end{figure} Figure 2: The expected Ti sky of 10 simulated galaxies of model A where the supernova recurrence time is taken to be 30 years and the supernovae ratio of Ia:Ib:II = 0.10:0.15:0.75. Simulating a 105 galaxy sky, a gamma-ray detector with a detection limit of $10^{-5}~\rm cm^{-2}~s^{-1}$ would have a probability of detecting 0, 1, and 2 44Ti sources of 0.0017, 0.012, and 0.037, respectively. A slightly better instrument than the $10^{-5}\rm ~cm^{-2}~s^{-1}$ detection limit would detect several 44Ti sources.
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2.4 Supernova locations

The 1.157 MeV $\gamma $-ray line following 44Ti decay has been detected in the 340-year old Galactic supernova remnant Cas A (Iyudin et al. 1994; Vink et al. 2001). COMPTEL's survey (Dupraz et al. 1997; Iyudin et al. 1999) has resulted in other candidate sources, such as the so-called Vela junior SNR (Iyudin et al. 1998). INTEGRAL's inner-Galaxy survey has been studied with IBIS Imager data, which did not reveal a new source in this region (Renaud et al. 2004). The difficulties of MeV observations have thus not led to convincing new 44Ti rich supernova remnants, especially in the inner Galaxy region (l = 0$^\circ $ $\pm $ 30$^\circ $) where observations are deepest. Yet, Cas A seems an established 44Ti detection in COMPTEL (Schönfelder et al. 2000) and Beppo-Sax (Vink et al. 2001) and INTEGRAL/IBIS (Vink 2005) measurements, while OSSE (The et al. 1996) and RXTE (Rothschild & Lingenfelter 2003) measurements of Cas A were not sufficiently sensitive. There exist no well-understood supernova remnants other than Cas A where the 44Ti production issue can be tested. It is apparently the only supernova whose yield, age, and nearness makes 44Ti visible in gammas. Naturally we ask ourselves if Cas A is a typical supernova or an anomalous case of a high-44Ti yield supernova?

Current supernova models predict an amount of 44Ti  which is of the same order than what these observations suggest, though generally slightly less. Can we take this as a satisfactory confirmation of our understanding of core collapse supernova 44Ti production? These two identified core collapse events and their association with 44Ti production appear to be in line with models which attribute 44Ti production to the more frequent standard core collapse events but not to standard thermonuclear supernovae. Is this correct?

If true, the location of 44Ti sources should match the locations of young massive stars which have rather short lifetimes. There is substantial evidence that massive star formation occurs in spiral arms and predominantly in the inner Galaxy (Kennicutt 1998; Elmegreen et al. 2003; Scoville et al. 2001). Massive stars can be observed directly in the infrared (e.g. Maeder & Conti 1994), though extinction corrections are large in regions of dense clouds. Possibly an even better (though more indirect) massive-star census can be derived from 26Al decay $\gamma $-rays (Prantzos & Diehl 1996; Knödlseder 2000; Diehl et al. 2005; Diehl et al. 1995). 26Al is understood to originate predominantly from massive stars and $\gamma $-rays easily penetrate even dark clouds in star forming regions. So, do we see the 44Ti sources in regions where we expect them to occur? Or do other factors which are not yet understood conspire to make 44Ti  ejection a phenomenon of core collapse events occurring in special regions and environments?

2.5 Estimating the 44Ti sky appearance

If we want to estimate how the 44Ti sky should appear in a gamma-ray survey, we need to follow a statistical approach, due to the rare occurrence of supernovae. We therefore apply a Monte Carlo approach of sampling plausible probability distributions for supernova rates, their 44Ti yields, and their Galactic distribution, thus calculating a large statistical sample of possible appearances of the 44Ti sky.

\end{figure} Figure 3: Cumulative 44Ti gamma-ray line flux distribution of supernovae with $f_{\gamma }>$1$\times $10-12 cm-2 s-1 according to our Model A (see Appendix A.2) with supernovae type ratio of Ia:Ib:II = 0.1:0.15:0.75 ( left). Expected average number of supernovae per sample galaxy with their 44Ti gamma-line fluxes above detection limits, for a supernova recurrence time of 30 yr, irrespective of their position in the sky ( right). The detection limit of COMPTEL instrument is shown as a dotted line.
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Randomly selected sample results from 105 such realizations of an expected 44Ti sky are shown in Fig. 2, to be compared with the observed 1.157 MeV COMPTEL image shown in Fig. 1 (Iyudin et al. 1999). We illustrate how typical these expected images would be, by showing the distribution of source brightnesses over a much larger number of Monte Carlo samples (Fig. 3, left): a 44Ti flux above a representative limit of 10-5 ph cm-2 s-1 occurs in 19% of our 44Ti gamma-line flux distribution. The expected number of 44Ti point sources per galaxy lies above the gamma-ray survey sensitivity limit, i.e., we do expect typically 5-6 positive detections of 44Ti sources (see Fig. 3 right).

\end{figure} Figure 4: The longitude distribution of supernovae for four 44Ti gamma-ray line flux bandwidths in Model A of Appendix A.2 with supernovae type ratio of Ia:Ib:II = 0.1:0.15:0.75. The equal-bin width is 20 degrees in longitude. Only the $f_{\gamma }<1\times 10^{-6}$ cm-2 s-1 distribution is normalized and multiplied by a factor of 0.298.
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It is immediately evident from Figs. 2 and 3 that expectations based on such seemingly plausible assumptions look very different than the observed 44Ti sky: Fig. 2 shows $\sim$4-7 observable 44Ti sources in an area of the Galaxy that contained none (see Fig. 1) above the observable flux limit used for comparison. The brightest source in that realization (large-filled circles) has $I_\gamma$ $\sim$ 10-4 cm-2 s-1, which would have been seen as a $\sim$10 $\sigma $ source by COMPTEL, and would have been detected already in INTEGRAL's inner-Galaxy survey (Vink 2004). The majority of our calculated samples lead to this same type of conflict (see Fig. 3). The probability of having no sources within the central galaxy is small for current surveys, as illustrated in Fig. 4 (for model A): for a survey down to $I_\gamma$ $\sim$ 10-5 cm-2 s-1, $\sim$12% of supernovae appear within longitudes $0\pm60$$^\circ $, whereas only 72% of supernovae is within that volume. An interesting feature, however, is that the longitude distribution for $f_{\gamma}<1\times10^{-5}$ cm-2 s-1 is dominated by sources from type Ia (the bulge) and farther away SNe, while the distribution for $f_{\gamma}>1\times 10^{-5}$ cm-2 s-1 is dominated by sources from type Ib and type II (disk) and nearby SNe. This feature also can be seen in Fig. 3, where around $f_{\gamma}\simeq1\times10^{-5}$ cm-2 s-1the distribution changes its slope.

It is clear that something is wrong with at least one of these assumptions:

In the following we examine these questions, and seek possible explanations for the apparent conflict.

3 Resolving the 44Ti sky conflict

3.1 Gamma-ray observations

The 44Ti gamma-ray sky can be studied in lines from the primary decay to 44Sc at 67.9 and 78.4 keV and from the following decay to 44Ca at 1.157 MeV. In this latter line, the COMPTEL imaging telescope had reported the pioneering detection of 44Ti from Cas A (Iyudin et al. 1994), clearly showing a point source image at 1.157 MeV as well as the line in a spectrum from this source. This detection had initially created some controversy because other gamma-ray instruments apparently did not see it (Rothschild et al. 1998; The et al. 1996). We now believe that this is due to the high initially-reported COMPTEL gamma-ray flux value, reduced later with better statistical accuracy of the measurement (Dupraz et al. 1997; Iyudin et al. 1999). The independent detections with BeppoSax (Vink et al. 2001) and with INTEGRAL/IBIS (Vink 2005) in both lower-energy lines (Beppo-SAX) and in the 67.9 keV line (IBIS) now consolidate the 44Ti detection from Cas A, but also suggest that indeed the 44Ti gamma-ray flux of Cas A is in a range between 0.8 and $3.5\times 10^{-5}$ ph cm-2 s-1; an "average'' of ( $2.6\pm0.4\pm0.5$$\times $ 10-5 ph cm-2 s-1 has been derived (Vink 2005). An upper limit from INTEGRAL/SPI reported from first studies is consistent with this flux value, and may suggest that the 44Ti line is broader than 1000 km s-1 (Vink 2005).

COMPTEL's sky survey allowed for mapping of the plane of the Galaxy in the 44Ti line (Dupraz et al. 1996; Iyudin et al. 1999). Secondary features in these COMPTEL 44Ti maps kept the discussion about statistical significances and systematic uncertainties alive (see Schönfelder et al. 2000, for a comparison of Cas A to RX J0852, a promising second source candidate, Iyudin et al. 1998). The COMPTEL point source detection algorithm (de Boer et al. 1992) has been tested with simulations over the full sky: likelihood statistics has been verified to reproduce the expected number of artificial sources for a full sky survey, as the noise level is approached. The problem is that in the range of all the 44Ti gamma-ray lines, all gamma-ray telescopes suffer from a large background from local radioactivity induced by cosmic-ray bombardment of the instruments (Weidenspointner et al. 2002; Gehrels & Michelson 1999). Determination of this background is crucial. For an imaging instrument, this can be done rather well by interpolation of imaging signatures from adjacent energies. But furthermore, for COMPTEL the 1.157 MeV line of 44Ti is not far above its lower energy threshold, and in fact imaging selections strongly affect the sensitivity of the instrument up to $\simeq$1.5 MeV. Nevertheless, imaging analysis in adjacent energy bands should all experience similar problems, and therefore differences between images in the 44Ti band and in neighboring energy bands can be attributed to 44Ti rather than continuum sources or instrumental artifacts, once they are confirmed to be point-like sources (instrumental background lines would in general spread over data space and hence lead to extended or large-scale artifacts in the image, hence impact on the flux measurement rather than on point source detection). Detailed comparisons of results for different energy bands, data subsets, and selections have led to the more cautious report about the Vela-region source (Schönfelder et al. 2000), so that still no convincing second 44Ti source clearly above the noise level is claimed. The INTEGRAL limits for 44Ti lines from the Vela region source are now close to the reported COMPTEL flux value for this candidate source (Renaud 2004; von Kienlin et al. 2004) and we therefore consider only one single 44Ti source (Cas A) as being detected down to flux levels of 10-5 ph cm-2 s-1.

For the low-energy lines measured by other instruments, in addition to the instrumental background the underlying continuum (see The et al. 1999 for Cas A) presents a major source of uncertainty (see Vink & Laming 2003; Vink 2005; Vink et al. 2001). This becomes even more of a problem for recombination line features from 44Sc at 4.1 keV, which have been studied with ASCA (Iyudin et al. 2005; Tsunemi et al. 2000), yet without clear detections (though tantalizing hints have been discussed for the Vela-region source).

For our quantitative comparisons of the 44Ti supernova rate with the historical record of supernovae in the last millennium (see Appendix B.2), we make use of the COMPTEL survey. We avoid the regions where reduced exposure might eventually lead to increased artifact levels, and concentrate on the inner Galaxy where exposure for the COMPTEL sky survey is deep and homogeneous. We also avoid using data after the second re-boost of the satellite, whereafter the activation level of 22Na had increased substantially (Weidenspointner et al. 2002). This leaves us with a dataset covering the sky range of $\vert l\vert\leq 90^\circ$, $\vert b\vert\leq 30^\circ$ using 7 years of the 9-year sky survey. In these gamma-ray data, the 44Ti source sensitivity should be rather well-behaved and useful for our study. Note that the first inner-Galaxy survey from INTEGRAL (longitudes of $\pm $20$^\circ $ around the GC) (Renaud 2004) is consistent with our dataset in that also there no source is found to flux levels of 10-5 ph cm-2 s-1 (Renaud 2005).

3.2 Galactic supernova rate

Many determinations of the galactic star formation rate (SFR) and supernova rate (SNR) have been made in past decades (see Diehl et al. 2005 for a compilation of estimates, and Stahler & Palla 2005 on general astrophysical aspects of star formation). In this work we are concerned with the 44Ti from supernovae, so we need the formation rate of massive stars (above, say, 10 $M_{\odot}$). The conversion between SNR and SFR is a sensitive function of the Initial Mass Function (IMF), and values given in the literature thus vary depending on the author's chosen IMF. For a generic transformation equation, we use the calibration from McKee & Williams (1997): SFR = 196 SNR, where the SFR is measured in $M_{\odot}$ yr-1 and the SNR in events per year. A star formation rate of 4 $M_{\odot}$ yr-1 thus corresponds to a supernova rate of two events per century. These rates only include core collapse supernovae (type II and Ibc), but not SNIa. Generally these supernova rates are averages over time scales much longer than 44Ti decay.

Many papers discuss the star formation (rate) history (SFH) in relative terms, or the star formation rate surface density ($M_{\odot}$ yr-1 kpc-2) in the solar neighborhood and its radial dependence. None of these papers is useful (for our purpose) without an absolute calibration, based on a model of the galactic distribution of star formation.

Generically, the SFR is obtained from a tracer that can be corrected for observational selection effects and is understood well enough so that possible evolutionary effects can be taken into account. One either deals with a class of residual objects, such as pulsars or supernova remnants, or with reprocessed light, such as free-free, H-alpha, or IR emission that follows from the ionization and heating of interstellar gas and its dust content in the vicinity of the hot and luminous stars. One must be careful to include time-dependent effects, as the afterglow of an instantaneous starburst behaves differently than the steady state output from a region with continuous star formation. Here, we are concerned with an average star formation rate.

To set the stage, let us collect SFR values from the literature. Smith et al. (1978) concluded SFR = 5.3 $M_{\odot}$ yr-1, while Talbot (1980) finds a very low value SFR = 0.8 $M_{\odot}$ yr-1, and Guesten & Mezger (1982) report a very high value of SFR = 13.0 $M_{\odot}$ yr-1. Later papers appear to converge on a median value: Turner (1984) finds SFR = 3.0 $M_{\odot}$ yr-1, and Mezger (1987) finds 5.1 $M_{\odot}$ yr-1. Measurements in the past decade confirm this moderate rate: McKee (1989) derives SFR = 3.6 $M_{\odot}$ yr-1 from the analysis of thermal radio emission (free-free) from HII regions around massive stars. This emission is directly proportional to the production rate of ionizing photons, which in turn is directly proportional to the SFR. He pointed out that this method is very sensitive to the slope of the high-mass IMF. We also note that the method depends on stellar model atmospheres in conjunction with models for massive stars, which change with treatments of mass loss, rotation, and convection. This paper also briefly discusses the use of the far-IR luminosity, due to warm dust heated by the absorption of photons from massive stars. The author uses the measured IR luminosity of the Galaxy of $4.7 \times 10^9$ $L_\odot$ (from Mezger 1987) to derive SFR = 2.4 $M_{\odot}$ yr-1.

We must convert between star formation rate (SFR) and supernova rate (SNR) (see above). McKee & Williams (1997) promote the value SFR = 4.0 $M_{\odot}$ yr-1, and based on the Scalo's IMF (Scalo 1986) convert this rate into a total number rate of 7.9 stars per year. They assume that all stars above 8 $M_{\odot}$ become supernovae, corresponding to a supernova fraction of $2.6\times 10^{-3}$. The mean stellar mass is $\langle m\rangle = 0.51$ $M_{\odot}$. The corresponding cc-SN rate is 2 per century. This value is also supported by a completely independent method, based on the production of radioactive 26Al in cc-SNe, which can be traced though its gamma-ray line at 1.809 MeV. Timmes et al. (1997) use this method and conclude SFR = ($5~\pm~4$$M_{\odot}$ yr-1, utilizing the 26Al line flux derived from COMPTEL. The steady-state mass of 26Al obtained in their work is in the range 0.7-2.8 $M_{\odot}$, consistent with the value presented in the recent study by Diehl et al. (2005). Based on the Salpeter IMF in the range 0.1-40 $M_{\odot}$ and the 26Al yields from Woosley & Weaver (1995) [which do not include contributions from the Wolf-Rayet wind phase] the authors derive the above quoted SFR and an associated cc-supernova rate of $3.4~\pm2.8$ per century. They neglect hydrostatically produced 26Al that is injected into the ISM in massive star winds, which causes their SFR to be overestimated. The large uncertainty is mostly due to the steady-state mass of 26Al inferred from the COMPTEL flux. INTEGRAL data presented by Diehl et al. (2005) have significantly reduced the error in this key quantity, and with wind yields included the latter study finds SNR =  $1.9 \pm 1.1$ supernovae per century, which corresponds to SFR =  $3.8 \pm 2.2$ $M_{\odot}$ yr-1, similar to the value given in McKee & Williams (1997). This SFR is very similar to the one obtained for M 51 (Calzetti et al. 2005), and thus places the Galaxy in the group of quiescently star-forming galaxies.

The most recent paper addressing this issue is by Reed (2005), who does not derive the SFR, but states that the galactic supernova rate is estimated as probably not less than 1 nor more than 2 per century. Using the conversion factors from McKee & Williams (1997), one infers that the SFR is in the range 2-4 $M_{\odot}$ yr-1. Reed uses a sample of a little over 400 O3-B2 dwarfs within 1.5 kpc of the Sun to determine the birthrate of stars more massive than 10 $M_{\odot}$. The galaxy wide rate is derived from this local measurement by extrapolation based on models for the spatial distribution of stars, a model for galactic extinction (to accomplish corrections for stellar magnitudes), and a model of stellar life times. Reed emphasizes various sources of errors, such as lacking spectral classifications of some bright OB stars, the (poorly known) inhomogeneous spatial structure of extinction as well as stellar density, and non-unique connection between mass and spectral type. Finally, Reed also draws attention to the fact that one would have to include B3 dwarfs as well, if the lower mass limit for supernovae is 8 $M_{\odot}$ and not 10 $M_{\odot}$ (see Heger et al. (2003) for comments on this mass limit). The OB-star catalog of the author was used to perform a modified $V/V_{\rm max}$ test to obtain a present-day star count as a function of absolute V-band magnitude. From the stellar life times, and the assumption of steady state, the local birthrate follows. A double exponential model (in galactocentric radius and scale height above the plane) of the spatial distribution of these stars (which includes an inner hole of radius R =4.25 kpc) ultimately leads to a total birthrate of 1.14 OB stars per century. Variations in the size of the hole change this number significantly, which leads the author to finally claim a rate of 1-2 supernovae per century.

In the context of our interpretation of the 44Ti observations, we would argue that Supernova rates between one and three cc-SNe per century are consistent with the large set of studies reviewed above. To solve the 44Ti Sky conflict with a choice of the SNR (or SFR), an extremely low rate outside this range would of course explain the absence of 44Ti gamma-ray line sources in the sky. However, chemical evolution arguments for 44Ca would then require correspondingly higher 44Ti yields which are not supported by explosive nucleosynthesis studies (as discussed below) and which in any case would lead to higher fluxes from supernova remnants and thus again to a source count that exceeds the observed count of $\sim$1. The observational constraints are on the product of rate and yield. The natural solution to the problem may be the very rare events with extremely high yields, which is discussed below. In that case it is of course totally unclear what to use for the spatial distribution of these events, and "unusual'' positions of a gamma-ray line source on the sky (such as that of Cas A) would be hard to interpret.

But perhaps we have overlooked another option. We know that the galactic star formation process is strongly correlated in space and time (Elmegreen et al. 2003). Could it be that the Galaxy just had a brief hiatus in its SFR? It would not take too much of a pause (say a few centuries) to explain the absence of bright 44Ti sources if the past few centuries were very untypical with respect to the SFR (or SNR). This possibility is included in our Monte Carlo simulations, which show that this is not a likely solution when one simply considers Poisson fluctuations. This solution is thus not acceptable, unless one can point out a physical cause of the hiatus in the recent SFR.

3.3 Galactic supernovae locations

Likewise, we may wonder about the possibility of very large spatial fluctuations. Could it be that the recent Galaxy exhibits an average star formation, rate-wise, but a lopsided distribution in space. If the opposite side of the Galaxy currently forms stars, and regions in the solar sector are relatively inactive ( $l=90{-}270\mbox{$^\circ$ }$), we expect to detect fewer gamma-ray line sources (because of their somewhat larger average distances). But at the same time the supernovae would also suffer from enhanced extinction, and matching the historic SN record would require a much higher rate. We have simulated the effects of a lopsided profile with a von Mises Distribution, the analog of a Gaussian distribution for circular data (Mardia 1975; Fisher 1996). This function allows us to change from an axis-symmetric galactic distribution to a one-parameter distribution (measured by a parameter k, where k=0 corresponds to the uniform circular distribution) in a chosen direction which we choose it to be the longitude $l=0\mbox{$^\circ$ }$ direction. We re-simulated the 105 Galaxies sample as described in Sect. 2.5, and find that the detection probabilities (stated in the caption of Fig. 2) change to 0.0042, 0.023, and 0.062 for parameter k=0.5 and 0.0068, 0.035, and 0.087 for parameter k=1.0, respectively. The probability curves for 44Ti $\gamma $-line source detection (see Fig. B.4) are shifted towards higher rates, as expected, but the overall likelihood of these models decreases. The constraint from the historic record demands new rates that are even larger, as the extinction correction affects the results more strongly than the D-2 distance effect for the flux. The combination of these two constraints make lopsided models less acceptable than axis-symmetric ones.

A lopsided model would also make Cas A even more special, regarding its unexpected location on the sky. To alleviate this problem we simulated lopsided star-forming galaxies in which the solar sector was the more active. The detection probabilities of 0, 1, and 2 44Ti sources of this model are 0.0005, 0.003, and 0.012 for parameter k=1.0, respectively. This shows that the number of the most probable 44Ti $\gamma $-line source detection in the Galaxy is >2 detections and the model is less probable than the model used for Fig. 2 for consistency with the observed 44Ti sky.

Is it reasonable at all to consider one-sided star forming galaxies? That major merger events should be able to tidally induce a lopsided starburst activity is perhaps obvious, but the Galaxy is not undergoing such an event. However, Rudnick et al. (2000) have shown that even minor mergers (Ibata et al. 1995) generically termed "weak interactions'' may lead to a boost in the star formation rate correlated with their lopsidedness. Another mechanism for the creation of non-symmetric star formation patterns is the interaction between odd numbers of spiral density waves, as it may be at work in M 51 (Henry et al. 2003). We do not advocate such an asymmetry for our Galaxy, but just wanted to consider this real possibility as one of the potential fixes for the 44Ti sky problem. Our simulations indicate that even such an extreme solution does not work, as the various combined constraints operate against each other. While a lopsided Galaxy helps on the gamma-ray source count side, the historic record is harder to explain if recent supernovae are located preferentially on the far-side.

3.4 Galactic extinction map

Analyzing the consistency of the supernova rates derived from the Galactic historical record and the COMPTEL's gamma-ray map (Appendix B.4), the consistency would improve if the SN rate from historical record is smaller. This could be realized if the true Galactic extinction map is lower than the visual extinction map we used here. Recently there are two published Galactic extinction maps that are useful for the type of study of this paper. The optical reddening model of Mendez & van Altena (1998), which is based on Galactic dust distribution model, makes use of the same optical sky surveys implemented by Hakkila et al. (1997) in addition to some other restricted surveys. However this extinction map is reliable only for solar neighborhood within 6 kpc. Another large scale three-dimensional model of Galactic extinction based on the Galactic dust distribution of Drimmel et al. (2003) has been shown to give a good agreement with the empirical extinction derived from NIR color-magnitude diagrams within 0.05 mag and furthermore it is reliable for a distance up to $\simeq$8 kpc. This extinction model gives a larger magnitude of extinction than that of Hakkila et al. (1997) for longitude $\vert l\vert \leq 1.5$$^\circ $and for most pointing directions from the Sun for distance larger than 6 kpc. For distance less than 5 kpc, the extinction of this map is smaller than that of Hakkila et al. (1997) which could give a better agreement between the SN rates of the historical record and of the COMPTEL's gamma-ray map.

3.5 The SN model

3.5.1 Lifetime of 44Ti

Although the lifetime of 44Ti measured in laboratories had exhibited a large uncertainty since its first measurement in 1965, a compilation of five recent experiments performed after 1998 gives an averaged lifetime of $87 \pm 1$ yr, where the quoted error is of statistical and of one standard deviation (see, e.g., Fig. 5 of Hashimoto et al. 2001 and also Görres et al. 1998). Apparently, this small uncertainty in the measured 44Ti lifetime does not affect the discrepancy discussed here.

It is noted that the above-mentioned lifetime measured in laboratories is for neutral atoms. Since 44Ti is a pure orbital-electron-capture decay isotope, its lifetime depends on the electronic environment in the evolutional course of a supernova remnant. For example, a fully-ionized 44Ti is stable, and the lifetime of 44Ti in the Hydrogen-like ionization state becomes longer by a factor of 2.25 than that of the neutral 44Ti (see Motizuki & Kumagai 2004). Let us briefly consider the effect of 44Ti ionization on our problem.

In young supernova remnants, the reverse shock propagates inward through the ejecta and the resulting increase in temperature and density may lead to highly ionized ejecta material through thermal collisions with free electrons. A high-degree of ionization may then result in a longer lifetime of 44Ti, which would significantly alter the inferred 44Ti mass. In fact, H-like and He-like Fe ions have been observed in Cas A (see, e.g., Hwang et al. 2004). Because the electron binding energies of Ti are smaller than those of Fe, it is easier to ionize Ti than Fe. Accordingly, 44Ti atoms in Cas A may be expected to be in such high ionization states at least in part if they are accompanied by the highly ionized Fe (this is expected because 44Ti is synthesized at the same location as where 56Ni is also produced in the innermost region of a supernova).

Since the present-day radioactivity was entirely affected by the history of various ionization stages and their duration time for which the 44Ti has experienced through the evolution, detailed discussion requires numerical simulations as was done by Mochizuki et al. (1999) and Mochizuki (2001). However, we can get a rough idea of the ionization effect on the radioactivity by using the result of simple linear analysis, i.e., Eq. (7) of Motizuki & Kumagai (2004):

 \begin{displaymath}\Delta F_{\gamma}/F_{\gamma} = (t/\tau - 1) \Delta \tau/\tau,
\end{displaymath} (1)

where we have replaced the radioactivity A and the decay rate $\lambda$appeared in Eq. (7) of Motizuki & Kumagai (2004) with the $\gamma $-ray flux $F_{\gamma}$ and the 44Ti lifetime $\tau$, respectively. In Eq. (1), t is the age of a SNR, $\Delta F_{\gamma}$ is the change of the flux by ionization, $\Delta \tau$ is that of the lifetime.

Note that $\Delta \tau$ is always positive because the ionization always increases its lifetime. As was pointed out by the above authors, the sign of $\Delta F_{\gamma}$ is then determined by that of the term in the parenthesis in the right-hand side of Eq. (1). This means that the flux is enhanced by the ionization when a SNR is older than the 44Ti lifetime, and that the flux is reduced when it is younger.

Our concern here is whether the effect of ionization on the lifetime of 44Ti in SNRs can reduce the disagreement between the observed 44Ti Galactic map and the model's map or not. From the above arguments, we can easily understand that the discrepancy may be diminished if the $\gamma $-line fluxes in Fig. 2 could be smaller which may be realized if most of the $\gamma $-ray detected SNRs in Fig. 2 are younger than the 44Ti lifetime.

To get a rough idea, we performed a calculation in which all parameters are the same as employed for Fig. 2 except 1) the fluxes are multiplied by a factor 0.5 for SNRs with ages less than 100 y, and 2) the fluxes are multiplied by a factor 2 for SNRs with ages between 200 and 400 y. The selection of 200-400 years old SNRs as enhanced targets here is because the effect of the ionization due to the reverse shock is considered to be distinguished for these ages and the further inclusion of the enhanced-flux effect on SNRs older than 400 y only makes the discrepancy larger (see Mochizuki et al. 1999 for details). Simulating a 105 galaxy sky, we found that a $\gamma $-ray detector with a detection limit of $1 \times 10^{-5}$ ph cm-2 s-1would have a probability of detecting 0, 1, 2 44Ti sources of 0.0012, 0.008, and 0.026, respectively. Therefore, from this simple analysis it is suggested that the disagreement cannot be compensated by the ionization effect; in effect it becomes worse in our simple calculations above than the reference calculations of Fig. 2.

A more precise estimate requires the knowledge of the temperature and the density evolution of a supernova remnant, and the distribution of 44Ti in it. However, any detailed calculations taking into account the retardation of 44Ti decay due to ionization will not alter the situation better: in any case, the older SNRs whose fluxes may be enhanced always dominate in number the younger SNRs whose fluxes may be decreased.

3.5.2 Nucleosynthesis reaction rates

Estimates of yields of 44Ti from nucleosynthesis in supernovae depend crucially on key nuclear reaction rates, and uncertainties in these rates limit our ability to constrain the supernova rate. The et al. (1998) studied the sensitivity of 44Ti yields in alpha-rich freezeouts to uncertainties in nuclear reaction rates. They did this by computing the alpha-rich freezeout with reference values for the reaction rates and then comparing these results with ones from calculations with individual rates varied upwards and downwards by a factor of 100 from their reference values. The results were that the production of 44Ti was most sensitive to the rates for the following reactions: 44Ti $(\alpha,
{\rm p})^{47}$V, $\alpha(2\alpha,\gamma)^{12}$C, 44Ti $(\alpha,\gamma)^{48}$Cr, and 45V $({\rm p},\gamma)^{46}$Cr for matter with equal numbers of neutrons and protons ($\eta = 0$). For neutron excess $\eta$ greater than zero, the importance of the reaction 45V $({\rm p},\gamma)^{46}$Cr drops, but other reactions become more important. In particular, these reactions are 12C $(\alpha,\gamma)^{16}$O, 40Ca $(\alpha,\gamma)^{44}$Ti, 27Al $(\alpha,{\rm n})^{30}$P, and 30Si $(\alpha,{\rm n})^{33}$S.

For our purposes, the relevant question is how much the 44Ti may vary from current supernova models given these uncertainties. Motivated by the work of The et al. (1998), Sonzogni et al. (2000) measured the cross section for the 44Ti $(\alpha,
{\rm p})^{47}$V reaction at the astrophysically relevant energies. They found that the experimental cross section for this reaction was a factor of two larger than in the rate compilation of Thielemann et al. (1987) used in the The et al. (1998) calculations. From this result, Sonzogni et al. (2000) inferred a 25% reduction in the amount of 44Ti produced in alpha-rich freezeouts in supernovae. Other of the key reactions found by The et al. (1998) had similar sensitivities of 44Ti yield to reaction rates; therefore, if other experimental reaction rates are also a factor of $\sim$2 different from the theoretical calculations, we can expect similar ${\sim} 25\%$ effects on the 44Ti yield. From these results, we might thus conservatively expect the 44Ti yield to be uncertain by less than a factor of $\sim$2 due to reaction rate uncertainties. Such a conclusion is supported by the study of the reaction rate sensitivity of nucleosynthesis yields in core-collapse supernovae by Hoffman et al. (1999). These authors compared the yields from core-collapse supernova models using two different reaction rate libraries. For the 15 $M_{\odot}$ stellar model studied, the two calculations gave 44Ti yields that agreed to within 20%, in spite of the fact that many individual nuclear reaction rates differed by a factor of two or more between the two rate compilations.

On the other hand, Nassar et al. (2005) have recently measured the 40Ca( $\alpha,\gamma)^{44}$Ti reaction cross section in the energy range for nucleosynthesis in supernovae. In that energy range, the authors find that the reaction rate is 5-10 times larger than the previously used theoretical rate calculated from a statistical model (Rauscher et al. 2000). This large difference between the experimental rate and the theoretical rate may be due to the fact that the low level density in the 44Ti compound nucleus limits the applicability of the statistical model for theoretical predictions for the rate of this reaction. In any event, the larger rate increases the yield of 44Ti by a factor of ${\sim}2$ in the stellar models Nassar et al. (2005) explored. Such a large increase in the 40Ca $(\alpha,\gamma)^{44}$Ti reaction rate may allow normal core-collapse supernovae to account for the solar system's supply of 44Ca; however, this result would worsen the discrepancy between the observed Galactic 44Ti gamma-ray flux and our predictions.

3.5.3 The supernova explosion model

In core-collapse supernovae, 44Ti production occurs by the alpha-rich freezeout near the mass cut. The location of the mass cut in the star will then certainly affect the 44Ti yield. Also important is the question of whether the simple-minded notion of a mass cut at a single radial shell in the star even makes sense in more realistic models that account for large-scale fluid motions behind the stalled supernova shock prior to the explosion and for stellar rotation. These more realistic models suggest that the material ejected from near the mass cut will in fact be a mixture of parcels that arose from both inside and outside the mass cut. We can certainly expect variations in the entropies of those parcels, which, in turn will have attendant variations in the 44Ti yield (e.g., Pruet et al. 2005).

For our purposes, the important issue to consider is how much variation can we expect in the 44Ti yield from differences in the mass cut and multi-dimensional effects. One-dimensional models suggest that typically half or more of the 44Ti produced during the explosion might fall back on the remnant (e.g, Nassar et al. 2005). Similar results are possible for the multi-dimensional models. It is therefore quite conceivable that yields of 44Ti from supernovae of the same mass might vary by factors of $\gtrsim $2 simply due to variations in the location of the mass cut or multi-dimensional effects. Of course, if the supernova forms a black hole with mass greater than $\sim$2 solar masses, it will swallow up its innermost material and, thus, most or all of the 44Ti (and 56Ni) it produced. Such supernovae would be dim in both visible and gamma radiation. Perhaps the Galactic 44Ti map is indicating that supernovae over the last few hundred years have been predominantly of this type.

3.6 Supernova homogeneity

Although often taken for granted, homogeneity among supernovae of a type remains an open issue: for thermonuclear supernovae, light curves have been found to be fairly similar (Branch 1998). Their successful empirical relative adjustment through a light-curve-decline parameter apparently makes them "standard candles'' over the full range of cosmic evolution (this is the basis for the determination of cosmic expansion history, see e.g. Dahlén et al. 2004). The homogeneity of the r-process elemental abundance pattern in low-metallicity stars suggests that the r-process, which is commonly attributed to core collapse supernovae, also presents a fairly well-regulated nucleosynthesis environment (e.g. Thielemann et al. 2002). On the other hand, the 56Ni masses ejected in supernovae appear to scatter, within $\simeq$30% for SNIa (Benetti et al. 2005), and for core collapse events over a wide range from 0.01 to 1 $M_{\odot}$  (Woosley et al. 1995; Woosley & Weaver 1995; Thielemann et al. 1996), suggesting more variability in the core collapse nucleosynthesis than in thermonuclear explosions (Nomoto et al. 1997; Thielemann et al. 2004). Still, for supernovae of type Ia alternative model types are also discussed that would produce quite distinctly different nucleosynthesis products than central carbon ignition in Chandrasekhar-mass white dwarfs (Nomoto 1982; Livne & Arnett 1995; Woosley & Weaver 1994).

In summary: supernovae of both types are quite homogeneous in some of their characteristics, but anomalies suggest a deeper study of physical regulations and their observational impact.

3.6.1 2-3D effects in core collapse supernovae

Nagataki et al. (1997) in their explosive nucleosynthesis calculations of 2-D axisymmetric type II supernova found that materials engulfed by energetic shock waves along polar directions undergo higher temperatures (or higher entropy per baryon, Fryer & Heger 2000) to produce a higher amount of 44Ti  than in spherical explosions. Recently, Maeda & Nomoto (2003) also studied hydrodynamics and explosive nucleosynthesis in bipolar supernova/hypernova explosions. Their bipolar models produce a large amount (>10-4 $M_{\odot}$) of 44Ti  and at the same time eject a relatively small amount ($\sim$0.1-0.2 $M_{\odot}$) of 56Ni. These features of 2D supernova models inspire Prantzos (2004) to suggest that the "missing 44Ti'' problem (to be in concordance with SN1987A and Cas A observations and also to account for the 44Ca solar abundance) could be solved by avoiding the overproduction of 56Ni. More systematic studies are therefore required under variety of progenitor masses, explosion energies, metallicities, and other physical variables. As this 2D effect could elude the discrepancy between the 44Ti production in spherical supernova models and the amount observed from Cas A SNR in gamma-line fluxes and the amount deduced from 44Ca in solar abundance, the axisymmetric explosion also seems to be the natural consequences of rotation and magnetic field effects during pre-supernova phase (Yamada & Sato 1994; Müller & Hillebrandt 1981) and the neutrino-driven convection (Burrows et al. 1995) for the core-collapse supernovae. Furthermore, evidence to support the axisymmetric explosion have been inferred from the measurements of pulsar velocities (Hobbs et al. 2004; Burrows & Hayes 1996) and from the jet features observed in radio and X-ray images from several supernova remnants (Gaensler et al. 1998). With various degrees of Rayleigh-Taylor instabilities that may develop in core-collapse supernova (either in 2-D or 3-D simulations), we expect there would be a wide distribution of 44Ti production in supernova events as inferred indirectly by the pulsar velocity distribution. Thereby the task is to find 44Ti distribution production with supernovae synthesizing higher than typical amount of 44Ti produced in a spherical model (to explain the amount of 44Ca in solar abundance and Cas A SNR), but also produces small amount of 44Ti for the most-recent supernova so their gamma-line emissions are too weak to be detected. For this, we have to wait until the explosive 3-D nucleosynthesis can be performed within reasonable time.

3.6.2 Rare 44Ti rich events

The apparent deficit of 44Ti remnants from typical supernovae as the main source of 44Ca is explained if typical supernovae are not the main source. Some rare type of event with a proportionately higher 44Ti yield could be the major source and not leave detectable remnants today (e.g., Woosley & Pinto 1988). For example, these could be He-triggered sub-Chandrasekhar-mass thermonuclear supernovae (Woosley & Weaver 1994). If their recurrence time is several times the 44Ti lifetime, the Poisson probability of having none detectable now can be large. Woosley & Weaver (1994) find some models with up to one hundred times higher 44Ti yields than the typical supernova values discussed above. These need only occur now every one to two thousand years to provide the necessary 44Ca synthesis. Such a scenario, while potentially invisible to gamma-ray astronomy, would imply inhomogeneities among the relative abundances of 44Ca and other isotopes (see Sect. D.3). A galactic survey at 6.9 keV could test this scenario (Leising 2001). Another $\alpha$-rich freezeout nucleus, 59Cu, decays to 59Ni whose half-life is 75 000 years. The nearest remnants of these rare objects from the past 105 years could be easily detectable in the subsequent cobalt K$_{\alpha}$ X-rays.

Mainstream SiC grains, however, argue against rare producers of large amounts of 44Ti being responsible for most of Galactic 44Ca, as the 44Ca/40Ca ratio in mainstream SiC grains does not vary much from grain to grain; here sample size is much larger than for X grains, so a more representative sampling may be assumed (see Appendix D).

We perform a simple simulation to test the viability of the rare-event scenario. In this model, we take the model that produces Fig. B.4 (where the amount of 44Ti in supernovae are as produced by the supernova models explained in Sect. 2.5) but modify the 10% of core-collapse supernovae to produce 20$\times $the 44Ti of the supernova models. Therefore on average the amount of 44Ti in this model is about a factor 2.7 of the model shown in Fig. B.4 to account for the 44Ca solar abundance. This model gives a better agreement with the observed 44Ti sky than the model used to produce Fig. 2 where all supernovae produce 3$\times $ the amount of 44Ti of supernovae models. As shown in the caption of Fig. 2, the probability of this model detecting 0, 1, and 2 44Ti sources are 0.012, 0.053, and 0.12, respectively. This results show that the model is more probable than the model of Fig. 2 in explaining why COMPTEL and INTEGRAL only detect one 44Ti source (less than expected). For comparison, the supernovae recurrence rates of the peak of the probability curve of the 1.157 MeV $\gamma $-line fluxes that are consistent with the COMPTEL observed fluxes (see Appendix B.2 and the dotted curve in Fig. B.4) for the model of Fig. B.4, this model, and the model of Fig. 2 are 36 yr (Fig. B.4), 40 yr, and 58 yr, respectively. This result is encouraging that a set of supernovae models with a rare type supernova that produces most of the solar 44Ca gives a closer recurrence rate to the one implied by the historical supernova record ($\sim$17 yr) than the model with 3$\times $ amount of 44Ti of the calculated supernova models. Still we have no good explanation for why some small number of supernovae are so rich in 44Ti.

4 Conclusions

The observed distribution of pointlike sources of 44Ti in our Galaxy is inconsistent (at a probability near 10-3) with the combined current understanding of 44Ti production, specifically

assuming the last few centuries are representative of the steady-state 44Ti  production. The inconsistency is exacerbated if we further demand, as currently understood to be so, that the standard solar abundance of 44Ca originates from 44Ti-producing supernovae. The larger discrepancy is because the product of the above two quantities falls short by a factor of three of the requisite current 44Ca production rate. The disagreement persists for any combination of rate and yield whose product is the required value, unless the yield of 44Ti is very much higher and the rates very much lower, i.e., 44Ca is not made primarily by typical supernovae events, but by very rare ones. These might include He-triggered detonations of sub-Chandrasekhar SNe Ia, or rare variants of core collapses, perhaps those most departing from spherical symmetry. Evidence of He-cap SNIa as source of the 44Ca abundance might be identifiable in 44Ca/40Ca ratios greater than solar in some mainstream SiC grains.

A future survey with gamma-ray line sensitivity of 10 $^{-6}~ \rm cm^{-2}~ s^{-1}$ would be expected to detect $\geq$10 sources (Fig. 2), and so could rule out 44Ca production by frequent supernovae at confidence $5 \times 10^{-5}$. Regardless of specific assumptions, the Cas A supernova remnant as the brightest 44Ti remnant in the galaxy is a priori very unlikely. That the brightest SNR should be found in the outer galaxy or that it is over 300 years old are each improbable. It suggests that yields higher than suggested by many current calculations are possible, which, of course, makes the lack of other detectable remnants even more puzzling.

Part of this work was supported by NASA Grant NAG5-6892, NAG5-13565, NAG5-10764 to Clemson University and DOE's Scientific Discovery through Advanced Computing Program (grant DE-FC02-01ER41189). Research by DDC was supported by NASA's Origin of Solar Systems Program.



Online Material

Appendix A: Simulations of 44Ti skies

A.1 Monte carlo simulation details

In generating supernova events of our Monte Carlo simulations, we adopt the procedure shown by Higdon & Fowler (1987); Mahoney et al. (1992); Hartmann et al. (1993). A random number between 0 and 1000 is uniformly generated to represent a supernova age between zero and 1000 y ($\simeq$ $11.4\times\tau_{\rm Ti44}$) where $\tau_{\rm Ti44} = (87.7 \pm 1.7)$ yr (Ahmad et al. 1998; Görres et al. 1998; Norman et al. 1997) is the mean life of 44Ti. This range is large enough that a supernova age older than 1000 y does not contribute to the Galactic 44Ti flux.

Another random number is generated to choose the type of the supernova event. Then, several random numbers are generated to give us the location of the supernova according to its spatial distribution. A detail procedure in generating the locations of supernova from a disk and spheroid populations of type Ia events can be found in the paper of Higdon & Fowler (1987); Mahoney et al. (1992). The distance of the supernova location to the Sun can be calculated easily.

A random number is generated from a Gaussian distribution to give us the peak bolometric magnitude of the supernova following the distribution given in Sect. A.2. The apparent magnitude then can be calculated knowing the location, the distance, extinction magnitude (from Hakkilla et al.'s empirical model), and the peak bolometric magnitude of the supernova. We find in model A (Sect. A.2) the fraction of SNe in the last millennium brighter than apparent magnitude 0 is $\sim$11% (see Fig. B.3).

The amount of 44Ti of the supernova is given by a uniform distribution of 44Ti according to its type as described in Sect. 2.5, however note that the analysis in this appendix does not use the multiplying factor of 3 that is used in 2.5. $F_{\gamma}$, the 44Ti gamma-line flux at the Sun location, then can be determined knowing the location, the distance, the age, and the amount of 44Ti of the supernova:

\begin{displaymath}F_{\gamma} = 8.21\times10^{-3} \; M_4 \; \exp(-t/87.7~{\rm yr})/d_{\rm kpc}^2
\; \gamma \; {\rm cm}^{-2} \;\rm s^{-1}\end{displaymath}

where M4 is the 44Ti of the supernova in units of 10-4 $M_{\odot}$ and $d_{\rm kpc}$ is the distance (in kpc) of the supernova from the Sun. We find that in Model A the fraction of supernovae in the simulations with 44Ti $\gamma $-line fluxes larger than $1 \times 10^{-5}$ $\gamma \; \rm cm^{-2} \; s^{-1}$ and $3\times 10^{-5}$ $\gamma \; \rm cm^{-2} \; s^{-1}$ are $\sim$11% and $\sim$5%, respectively (Fig. 3). Its longitude distribution along the Galactic plane in some relevant 44Ti gamma-line flux bandwidths can be seen in Fig. 4, while for models B and C (Sect. A.2) are shown in Fig. A.1.

\end{figure} Figure A.1: Same as Fig. 4 but for models B (left) and C (right).
\end{figure} Figure A.2: The distribution of supernovae in model A in Galactic longitude and latitude.

We generate 1 million supernovae according to the above prescription and construct galaxies with numbers of supernovae between 1 and 300. For each galaxy with a certain ${\it n}$ number of supernovae, the 44Ti gamma-line flux distribution $F(f_{\gamma},n)$and the SN apparent magnitude distributed M(m,n) can be extracted.

For a galaxy with SN recurrence time of $T_{\rm rec}$ or average supernova rate, $\mu = 1000$/ $T_{\rm rec}$ (we only simulate supernova events in the last millennium as described above), the 44Ti gamma-line flux distribution, $F(f_{\gamma},\mu)$ can be obtained from $F(f_{\gamma},\mu)$ = $\sum_{n=0}^{\infty} \frac{e^{-\mu} \; \mu^n}{n!} \;

A.2 Supernova types and spatial distribution

Supernova events as the source of Galactic 44Ti can either be type Ia, Ib/c, or II SNe. For this study, we choose a generally accepted (though uncertain) value of the type ratio, Ia:Ib/c:II = 0.1:0.15:0.75 (Hatano et al. 1997; Dawson & Johnson 1994). Yoshii et al. (1996) and Tsujimoto et al. (1995) using their chemical evolution model find that the ratio of the total number of type Ia to type II SNe of 0.12 gives a good agreement with the observed solar abundance. This is consistent with the observed Ia frequency which is as low as 10% of the total SNe occurrence (van den Bergh & Tammann 1991). Monte Carlo representations of type Ia SNe are generated using a nova distribution template that traces the blue light distribution in M 31 (Higdon & Fowler 1987). These populations form an axisymmetric disc and a spherically symmetric bulge. The spheroid density distribution follows:

and the disc density distribution follows: The fraction of SNIa occurring in the spheroid is taken to be $\sim$1/7 of the total SNIa (Bahcall & Soneira 1980).

Supernovae of types Ib & II are associated with massive stars whose birth places are exponentially distributed in height above the plane with a scale length of $\sim$100 pc. Because the rate of core collapses is larger than the rate of thermonuclear supernovae, we study several type II distributions to ensure that our results do not depend significantly on this choice. We consider three cases:

model A: exponential disk with no supernova within 3 kpc of the Galactic center, with radial scale length of 5 kpc Hatano et al. (1997):
$n(\rho,z) \propto {\rm e}^{-\vert z\vert/\sigma_z} \; {\rm e}^{-\rho/\rho_{\rm d}}$
$\sigma_z$ = 0.100 kpc
$\rho_{\rm d}$ = 5.0 kpc;
model B: exponential disk with radial scale length of 3.5 kpc that produces an acceptable fit to the COMPTEL's 26Al $\gamma $-line map Diehl et al. (1995);
model C: Gaussian-ring disk at radial distance of 3.7 kpc and radial distance scale length of 1.27 kpc (Taylor & Cordes 1993).
The distribution of supernovae in model A in directional is shown in Fig. A.2 and in distance from the Sun is shown in Fig. A.3.

\end{figure} Figure A.3: The distribution of supernovae distance in model A.

Appendix B: Comparison of supernova rates from 44Ti gamma-rays and the historical record

In order to produce a consistent picture of the recent Galactic supernova rate with the best known 44Ti supernova yields, supernovae types and spatial distribution, and Galactic extinction model we utilize our Monte Carlo simulation method to compare two observables of the same phenomenon (Higdon & Fowler 1987; The et al. 1999; Mahoney et al. 1992; Hartmann et al. 1993). We use the COMPTEL gamma-ray maps and the six historical Galactic supernova record of the last millennium to constrain the range of Galactic supernova rates.

Similar to the above determination of 44Ti flux distribution from a simulated set of supernova events, other distribution such as the apparent magnitude distribution of supernovae for a galaxy with an average supernova rate $\mu$ can be constructed (Sect. B.3). From this distribution we can determine the probability of optical detection for a certain detection-limiting apparent magnitude, such as magnitude 0, for example. In this way, we can use the same underlying assumptions about supernova characteristics, and compare two different observables, i.e. 44Ti gamma-ray versus optical detections of the supernovae at the adopted rates. Varying the rates then so that they conform to the observational constraints, we obtain a handle on systematic differences in observed supernova rates through these two observational windows.

For the 44Ti gamma-ray observations, we utilize the COMPTEL survey discussed above (Sects. 2.1 and 3.1). For the optical observations, we utilize the historical record summarized in Table B.1.

Table B.1: Recent galactic supernova record.

For comparison of the COMPTEL gamma-ray map with the results of Monte Carlo simulations, due to our limited computational power, for the purpose to reduce systematic effects from regions of low exposure, and due to our limited ability to analyze the map, we perform two analysis. In the flux dataspace (Sect. B.2) we use the flux information exclusively, without using the observed location of the gamma-ray source. In the map dataspace analysis (Sect. B.1), we only use the COMPTEL map of inner galaxy ( $\vert l\vert\leq 90^\circ$, $\vert b\vert\leq 30^\circ$) which does not include the locations of the detected 44Ti $\gamma $-line from Cas A and GRO J0852-4642. In order to see how consistent the model with the Galactic supernova record, we perform the historical record analysis below.

B.1 Map dataspace analysis

In map (imaging) analysis, a simulated COMPTEL data set is produced by convolving the directions and the 44Ti $\gamma $-line fluxes of the Monte-Carlo-generated supernovae through the COMPTEL detector response. The probability of the consistency of the simulated data with the measured counts is carried out by calculating the likelihood of the simulated data plus the best-fit background model, and comparing this to the likelihood of the background model only (de Boer et al. 1992). The change in the likelihood gives the relative probability that the specific realization of the model is consistent with the measured COMPTEL data. This process is repeated for at least 104 galaxies to obtain the average relative probability of the model at a particular supernova rate as dashed lines in Figs. B.4, and B.5.

B.2 Flux dataspace analysis

In flux dataspace analysis, in order to include the detected sources, the locations of the 44Ti supernovae are not used, flux information is used exclusively. The flux dataspace consists of the two 44Ti detected fluxes from Cas A SNR and GRO J0852-4642 Iyudin et al. (1998) with other fluxes assumed to be zero (with which they are consistent.) For each Monte Carlo Galaxy, a $\chi^2$ value is calculated by comparing the two strongest fluxes with the fluxes measured from Cas A and GRO J0852-4642 and the other fluxes are compared with null fluxes. The probability of the model to be consistent with the COMPTEL fluxes is determined from the $\chi^2$ and using the number of degrees of freedom as the number of independent 10 $^\circ\times10 ^\circ$ image elements (fields of view) of the supernova positions in the model. This size of independent fields of view is obtained by calibrating the size to produce the results of the analysis of COMPTEL's map of the inner Galaxy. Averaging the probabilities of 106 Monte Carlo galaxies, we obtain the average probability shown in Figs. B.4 and B.5 as dotted lines.

B.3 Historical record analysis

\end{figure} Figure B.1: The distribution of the peak absolute magnitudes of supernova in the B-band of model A.

The historical supernova record covering the last millennium shows a total of only six Galactic SNe during that era (Table B.1). Of course, this small number is due to significant losses from extinction and incomplete monitoring of the sky, especially during the early centuries. In fitting this data, we count the fraction of galaxies that have six SNe brighter than magnitude 0. In this approach, we assume that historical SNe were detected if they were brighter than magnitude 0.

The peak absolute magnitudes of supernovae in the B-band are approximated by Gaussian distributions. For the mean values and the one standard deviation we adopt for type Ia MB = -19.4, $\sigma=0.2$ (Branch 1998), for type Ib MB = -18.2, $\sigma=0.3$ (Dawson & Johnson 1994), and for type II MB = -17.2, $\sigma=1.2$ (van den Bergh & Tammann 1991). Their distribution is shown in Fig. B.1 Observed magnitudes of simulated SNe are obtained by convolving absolute magnitudes through a Galactic extinction model of (Hakkila et al. 1997). In this extinction model, the total visual extinction from the Sun's location to the Galactic Center is 11.64 mag and the total visual extinction perpendicular to the Galactic plane at Sun's location is $\sim$0.1 mag (Fig. B.2). The cumulative distribution of supernovae having blue peak apparent magnitude brighter than mB in model A is shown in Fig. B.3.

\end{figure} Figure B.2: The Galactic visual extinction magnitude as a function of distance from the Sun to various direction according to empirical model of (Hakkila et al. 1997).

\end{figure} Figure B.3: The apparent magnitude of supernovae in model A according to empirical extinction model of Hakkila et al. (1997).

The average fraction of the model that is consistent with six events brighter than magnitude 0 is shown in Figs. B.4, and B.5 as solid lines.

B.4 The supernova rates: optical versus gamma-ray constraints

\end{figure} Figure B.4: Normalized relative probability of Galactic SNe rate calculated from three different analysis vs. SNe recurrence time for model  A. The solid line is the probability of the model in which the number of SNe with $m_V \leq 0$ is 6 in a 1000 year duration. The dashed-line is the probability inferred by maximum likelihood analysis of COMPTEL's 1.157 MeV $\gamma $-line inner galactic ($l\leq 90$, $b\leq 30$) image map. The dotted-line is the relative probability from chi-square test that the 1.157 MeV $\gamma $-line fluxes in the model are consistent with the COMPTEL observed fluxes. The labels of the curves show the optimal, 1$\sigma $ ranges, or the 1$\sigma $ lower limits of the SNe recurrence time. The boxes, circles, and diamonds on the curves show the 1$\sigma $, 90%, and 95% relative probabilities. The shaded area shows the range of the Galactic SNe recurrence time that is consistent with better than 1$\sigma $ probability with each of the historical record and the COMPTEL flux dataspace analysis.

Our Galactic supernova record analysis leads to a most probable Galactic supernova recurrence time of $\sim$17, $\sim$16, and $\sim$13 y based on models A, B, and C, respectively. This implies a larger rate than given by previous investigations based on the historical record. For example, Dawson & Johnson (1994) estimated $\sim$3 SNe per century as also obtained by Tammann et al. (1994). The rate is smaller than ours because Dawson & Johnson (1994) considered 7 observed SNe within the last 2000 yr and assumed that the historical record is 80% complete. A higher SNe rate of 5 SNe per century was obtained by Hatano et al. (1997) who include a population of "ultradim'' SNe in addition to 4 observed Galactic SNe having V<0 and 80% completeness within the last millenium.

The map and flux methods of estimating supernova rates based on $\gamma $-ray data appear to be significantly different, but they are statistically consistent with each other for a wide range of supernova rates. The imaging analysis gives a smaller rate than the flux analysis, because the data used in the map analysis do not include the Cas A and GRO J0852-4642 $\gamma $-line detections. However, the probability estimate from the flux analysis is a somewhat coarser estimate because the size of independent FOVs used in determining the probability is not known exactly. Ignoring the spatial information begs the question as to why the two brightest SN are in the outer Galaxy rather than in the inner Galaxy where we expected them.

\end{figure} Figure B.5: Same as Fig. B.4 but for models B ( left) and C ( right).

In the flux dataspace analysis, where we ignore the expected spatial distribution of 44Ti remnants and consider only the measured flux distribution, we find a most probable supernova rate that is more compatible with standard values (such as the rate inferred by the historical record), even for standard 44Ti yields. However, the COMPTEL map data indicate a lower SN rate ($\sim$1 SN/36 yr) than that suggested by the historical record ($\sim$1 SN/17 yr). However model A which does not have a supernova event in the inner 3 kpc radius has better agreement between the map analysis and the historical record analysis. This suggests that a model with less concentrated supernova near the Galactic center than the model used here could give a better agreement than we obtained here. Also, an extinction model with smaller extinction toward the Galactic center than Hakkilla et al.'s model could give a better agreement than what we present here.

Based on chemical evolution studies, Timmes et al. (1996) estimated that only $\sim$1/3 of the solar 44Ca abundance is accounted for. Models with a SN rate of $\sim$3 SNe per century and standard 44Ti yields fail to produce the solar 44Ca abundance. This rate, when confronted with the gamma-ray data (dashed line in Figs. B.4 and B.5) is too large: the COMPTEL gamma-ray data worsen an already serious problem. Timmes et al. (1996) suggest 3 possibilities:

increase the 44Ti yields by a factor of $\sim$3;
increase the supernova rate by a factor of $\sim$3;
there is another source of 44Ca in the Galaxy.
Our analysis in Figs. B.4, and B.5 shows that the first and second option are not compatible with COMPTEL's 44Ti $\gamma $-line map, which would be brighter by a factor of 3 or exhibit a larger number of 44Ti hot spots than actually observed. We are thus left with the third option, to seriously entertain the idea that there exists some rare type of supernova (i.e., detonation of helium white dwarf), not realized in recent centuries, that produces very large amounts of 44Ti.

Appendix C: Solar 44Ca constraints

We now wish to estimate the effects of choices of chemical evolution model parameters on the 44Ca constraint. We do this with analytic models (Clayton 1985). Leising & Share (1994) used that chemical evolution model to estimate the current 44Ca production rate by requiring the yield that gave exactly the solar abundance 4.55 Gyr ago. With model parameters k (infall contribution) and $\Delta$ (time delay) defining the shape of the galactic infall history, the current production is

\begin{displaymath}p(^{44}{\rm Ca})= X_\odot(^{44}{\rm Ca}) ~ \frac{k+1}{\Delta}...
\left(\frac{T_\odot+\Delta}{\Delta}\right)^{-k} \right]^{-1},
\end{displaymath} (C.1)

where $T_{\rm G}$ is the age of the Galaxy, $T_\odot =T_{\rm G}-4.55$ Gyr, and $M_{\rm g}$($T_{\rm G}$) is mass of interstellar gas participating in star formation now. In this formulation, the 44Ca production depends mainly on the infall parameter $k= [f(t)/M_{\rm G}(t)] (t + \Delta)$ and the gas mass, and weakly on $\Delta$ and the age of the Galaxy. The closed-box model, k=0, is ruled out by observations, and k=2-4 is favored from a number of considerations (Clayton et al. 1993). For the total 44Ca production rate, therefore, a low extreme value of p(44Ca) =  $1.1\times 10^{-6}$ $M_{\odot}$  yr-1 is obtained for k=1, $\Delta=1.0$ Gyr, $M_{\rm G}(T_{\rm G})=4\times 10^{9}$ $M_{\odot}$ , and $T_{\rm G}=13$ Gyr. An upper limit is estimated as p(44Ca) =  $1.2\times 10^{-5}$ $M_{\odot}$  yr-1 for k=4, $\Delta=0.1$ Gyr, $M_{\rm G}(T_{\rm G})=1\times 10^{10}$ $M_{\odot}$ , and $T_{\rm G}=10$ Gyr. Our favored value is p(44Ca) =  $5.5\times 10^{-6}$ $M_{\odot}$  yr-1, derived for k=2, $\Delta=0.1$ Gyr, $M_{\rm G}(T_{\rm G})=1\times 10^{10}$ $M_{\odot}$ , and $T_{\rm G}=12$ Gyr. As in previous such studies (e.g. Leising & Share 1994) we assume that this equals the 44Ti production rate. (However, we note the statement by Timmes et al. (1996) that one-half of 44Ca is made directly as 44Ca rather than as 44Ti.) In such models, the solar 44Ca abundance thus could be provided by, for example, 2.8 SNe per century at the present epoch ejecting on average $2\times 10^{-4}$ $M_{\odot}$  of 44Ti.

Appendix D: Presolar grains

Do the presolar SiC grains (Clayton & Nittler 2004) that condensed within the expanding supernova interior (Clayton et al. 1997a; Amari & Zinner 1997) shed light on the 44Ti  origin? Called "X grains'', each represents a sample of selected portions of a supernova interior, but "what supernova'' and "what portions of it'' are unidentified questions for each X grain. Deneault et al. (2003) present a physical condensation argument to identify the region within the supernovae; but their argument awaits digestion by the condensation-chemist community. The X grains certainly contain evidence of having condensed with abundant live 44Ti in the form of very large 44Ca/40Ca ratios (Nittler et al. 1996; Clayton et al. 1997a; Hoppe et al. 2000) that can be explained only through the action of live 44Ti. This proves their supernova origin, and was a predicted signature of supernova origin for presolar grains (Clayton 1975) from the beginning. If each grain contains sufficient Ca and Ti the SIMS analysis can yield the 44Ti/48Ti abundance ratio at the time of condensation (Fig. 2 of Clayton et al. (1997a) and Figs. 8 and 9 of Hoppe et al. 2000). Many X grains having 44Ti/48Ti = 0.1-0.6 are found. The production ratio required by the assumption that all 44Ca within solar abundances is the product of decay of 44Ti is P(44Ti)/P(48Ti) = 0.72; therefore, supernova X grains are found with ratios up to the required bulk production consistent with that assumption. Despite this it must be clearly understood that the measured production ratio P(44Ti)/P(48Ti) within material in a specific grain bears no simple relationship to the bulk production ratio in the supernova within which that grain condensed. This illustrates the problem of not knowing precisely what supernova material the grain's isotopes reflect. However, this problem seems capable of eventual solution (Deneault et al. 2003).

D.1 SNII X grains

The SiC X grains probably originated in core-collapse SNII. Within them the pure alpha-rich freezeout (Woosley et al. 1973, Sect. VII), which produces the bulk of 44Ti ejected from SNII, has production ratio near P(44Ti)/P(48Ti) = 1-2 (see Fig. 23 of The et al. 1998; Woosley et al. 1973, Sect. 3); so no X grain is pure alpha-rich freezeout material, although some of them must contain a significant fraction of their Ti from the alpha-rich freezeout. On the other hand, the production ratio during normal silicon burning is very much smaller, near P(44Ti)/P(48Ti) = 0.01 (Woosley et al. 1973, Fig. 19). Therefore, X grains having 44Ti/48Ti near 0.01 or less may contain no alpha-rich freezeout material at all, but instead condense from Si-burning ejecta rich in 28Si.

In their study of 99 SiC X grains from the Murchison meteorite, Hoppe et al. (2000) found that 25 contained enough Ti and Ca for isotopic analysis with the CAMECA IMS3f ion microprobe in Bern. Now that the new nano-SIMS ion microprobe is functional, we may hope for even more complete future surveys with its higher sensitivity. Of the 25 having enough Ti and Ca, 5 revealed large and easily resolved 44Ca excesses, corresponding to P(44Ti)/P(48Ti) > 0.01. Although the chemistry of condensation of SiC in supernovae has not been solved, initial studies (Deneault et al. 2003) of the location of C and Si suggest that no more than 20% of SiC X grains should be expected to contain P(44Ti)/P(48Ti) ratios in excess of those available during normal O and Si burning. That is, a majority of the X grains should contain no alpha-rich matter even if each supernova ejects such matter. If this be taken as so, the data suggest that the parent supernova population responsible for the 100s of X grains that have been studied did eject alpha-rich freezeout material in addition to its Si-burning matter. If ejection of alpha-rich freezeout matter were instead rare, we expect that the number of SiC grains having large P(44Ti)/P(48Ti) ratios would be significantly less than 20%. Without more detailed understanding of the origin of the X grains it seems plausible to take their evidence to suggest that all SNII eject alpha-rich-freezeout matter.

If the SiC X grains do originate in SNII, their frequent 44Ca excesses demonstrate that either the X grains all originated in a single nearly supernova that did eject 44Ti or originated in many presolar SNII that mostly ejected 44Ti (although not necessarily always). The properties of the "mainstream SiC grains'' enable us to argue against the possibility of a single supernova as origin of the X grains. The much more abundant mainstream grains, which are thought to have originated in presolar carbon stars, have been argued on the basis of their Si isotopes to have been the result of a great many C stars. Their Si isotopic compositions represent approximately the initial Si isotopic compositions from which the intermediate-mass stars formed, and their correlation between excess 29Si and 30Si is explained by galactic chemical evolution effects that have produced a wide range of Si isotopic compositions in a large number of C stars (Timmes & Clayton 1996; Clayton 2003). Because the mainstream grains comprise about 10-4 of all interstellar Si, much of which is quite old, and because the condensed SiC from the C-star phase is but a small fraction of all Si ejected from stars, it seems that the lifetime of SiC grains in the ISM is not short. If this be so, the lifetime of X SiC grains is also not short, suggesting a substantial number of contributing supernovae to the matter gathered into any cloud in the ISM.

D.2 SNIa X grains

For the question of 44Ti gamma-ray hotspots the issue arises whether the SiC X grains may instead have arisen from exploding SNIa. After all, the high expectation for seeing 44Ti supernova remnants shown in Fig. 1 came from the assumption that type II are the sources of 44Ti nucleosynthesis and therefore also of the solar 44Ca abundance. The absence of such sources could be explained if rare type I events produced much larger yields than are expected from type II. Do the supernova grains offer any guidance? Clayton et al. (1997b) showed that isotopically good fits to X grains might originate within He caps on exploding C,O white dwarfs. Huge production ratios, up to P(44Ti)/P(48Ti) = 100, occur in those He shell zones having peak T9>1 (see their Fig. 6). These are too great for existing observations of X grains, suggesting that SNIa are not their sources. However, Clayton et al. (1997b) showed that good fits to all isotopes require that the He caps undergo considerable post-explosive mixing prior to condensation of the X grains. Their Table 2 shows production ratios as small as 0.01 in an average of zones 1-8 (the coolest eight zones) and near 10 in an average over all sixteen zones. (The reader must note that much 48Ti production is listed by them under 48Cr, its radioactive progenitor.) The isotopic possibility must therefore be addressed that the 44Ti-rich X grains are from SNIa He caps rather than from SNII. The discovery of even a single X-type SiC grain containing an initial ratio 44Ti/48Ti = 5 or greater would demonstrate that the He-cap SNIa do exist and that they can condense SiC. This would amount to an existence proof for these rare 44Ti producers. But no such grain has yet been detected.

Clayton et al. (1997b) pointed out that if the SNIa are near the Chandrasekhar mass, the He cap can be no more than 0.01 $M_{\odot}$   so that less than 10-5 $M_{\odot}$ of 44Ti is ejected. Such events would be both rare and dim in 44Ti lines. However, a much larger He cap is involved if the white dwarf is sub-Chandrasekhar (larger) and the detonation begins in a massive He cap. Large amounts of 44Ti are ejected from such models (Woosley & Weaver 1994), so that the rates of such events are limited by the requirement that they not overproduce the galactic 44Ca abundance. If this occurs, the rarity of these events in time could account for their absence in the COMPTEL data (see also the Discussion). The relevance of the X grains derives from the physical unlikelihood of condensation of X grains in such He caps. Their expansion is very fast, and the radiation environment intense, so that even the formation of molecules seems unlikely until the density has become too low for the growth of a 1-micrometer grain of SiC. Clayton et al. (1997b) note that even the possibility of grain condensation would seem to require 3-D explosive modeling in order that some of the He cap can remain at low velocity ("slow He''). This entire problem will require more study before plausibility of SiC X grains from SNIa can be admitted. We therefore conclude that the 44Ti-bearing X grains have arisen in SNII.

D.3 ISM Inhomogeneity of 44Ca from He-cap SNIa

We turn now to non-supernova Stardust whose 44Ca ratios reflect the initial compositions of their donor stars. The supernova X grains seem to support the assumptions that led to the conflict between the COMPTEL map (Fig. 1) and the expected hotspots (Fig. 2); namely, that X grains condensed within SNII, that the number of contributing SNII was large, and that most ejected alpha-rich freezeout matter. However, X grains do not demonstrate that the bulk ejecta of SNII contain sufficient mass of 44Ti  to account for the natural 44Ca abundance. To test for the presence of rare large 44Ti  producers (Woosley & Weaver 1994) responsible for making good the shortfall from SNII we turn to the mainstream SiC grains with the following original argument. Rare SNIa responsible for roughly 2/3 of galactic 44Ca plausibly result in Galactic inhomogeneities in the interstellar 44Ca/40Ca ratio that should be larger than those seen by astronomers for elemental-ratio variations attributed to inhomogeneous incorporation of SNII ejecta. It must be slower to homogenize the ISM if SNIa occurring every 3000 yr contribute 2/3 of 44Ca abundance than for homogenizing ISM from SNII occurring every 30 yr and making but 1/3 of the 44Ca abundance. ISM regions temporarily enriched in the rare SNIa ejecta may make stars having larger 44Ca/40Ca initial ratios. The mainstream SiC grains reflect the interstellar composition from which the carbon stars formed. That is, factor of two or more variations in 44Ca/40Ca ratio should be present in the initial compositions of stars.

It is this expectation that the mainstream grains may speak against. Hoppe et al. (2000) measured Ca isotopes in 28 mainstream SiC grains (see their Fig. 8) and found their 44Ca/40Ca ratios to be indistinguishable from solar despite large Si isotope variations in the same stars demonstrating that the grains come from different AGB C stars having distinct chemical evolution histories for their initial compositions (Timmes & Clayton 1996). Each mainstream grain was consistent with solar 44Ca/40Ca ratio and their average was enriched 3.8% in 44Ca, as expected for modest s-process enrichments of AGB atmospheres by the third dredge ups. Because of the small numbers of Ca atoms in mainstream grains, however, individual grain ratios were uncertain by at least 20%. Even so, 20% is a small inhomogeneity for rare events producing 2/3 of the bulk 44Ca. To improve this data base we call for high precision measurements of the 44Ca/40Ca ratio in a sample of mainstream grains with the new nanoSIMS ion microprobes. Such a survey could reveal more precisely any variations in initial 44Ca/40Ca ratios in intermediate-mass stars expected if the 44Ti synthesis has instead been the result of a few rare events of very large 44Ti yield. The present data speak against that. The new arguments that we have presented here can all be improved dramatically in the next few years. Although their message today is not statistically certain, their sense is to support the original conflict that we have displayed in Fig. 2. Because the 44Ca/40Ca ratio in mainstream grains from many AGB stars is very near to the solar ratio, we point out that it is no longer possible to entertain the idea that the sun may itself contain an anomalous 44Ca/40Ca ratio. One can not blame the inability of standard models of the chemical evolution of the galaxy to produce enough 44Ca by speculating that the sun is abnormally rich in its 44Ca content.

It should be noted that a thorough study of stardust isotopic inhomogeneity within the context of inhomogeneous galactic chemical evolution has been presented by Nittler (2005). He shows in particular that correlations between Si and Ti isotopic compositions found in stardust can not be accounted for by inhomogeneous GCE. He does not, however, discuss the variations in 44Ca/40Ca that are generated by his model owing to the He-cap SNIa. On the other hand, his calculations may not be ideal for this problem because his model admixes into a homogeneous ISM the same fraction for each supernova's ejecta (a =  $1.7\times 10^{-5}$) after Monte Carlo sampling to obtain N=70 supernovae. Nonetheless, that paper shows the potential power of stardust to delimit GCE inhomogeneities in general. This approach should be reconsidered carefully when a good data set for initial 44Ca/40Ca in stardust is available.

Mainstream SiC grains are probably not the best grains with which to measure initial stellar 44Ca/40Ca abundance ratios. The mainstream SiC grains contain very little Ca because Ca is much less favored chemically than is Ti within thermally condensing SiC grains. The hibonite Stardust grains described by Nittler et al. (2005), on the other hand, contain much more Ca because Ca is an essential ingredient of the hibonite crystal structure. Therefore, accurate measurements of 44Ca/40C in hibonite grains could reveal better information relevant to ISM inhomogeneity of the 44Ca/40Ca ratio.

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