5 Limits for planetary companions

We now continue with the quantitative determination of the sensitivity of the CES Long Camera survey for the discovery of planets orbiting the target stars. With this we ask the question: which planets would we have detected if they existed? Since they are not detected we can exclude their presence and hence set constraints. Starting from the null-hypothesis that the observed RV scatter is mainly caused by measurement uncertainties and/or additional intrinsic stellar effects, we establish the planet detection threshold for each star. This detection limit for each individual target is determined by numerical simulations of planetary orbits of varying amplitudes, periods and phase angles. These simulated planetary signals can either be recovered by a periodogram significantly or not and thus deliver the quantitative upper limits.

5.1 The method

Companion limits for different planet search samples were presented by Murdoch et al. (1993), Walker et al. (1995) and Cumming et al. (1999) for the Mount John Observatory, CFHT and Lick Observatory surveys, respectively. Nelson & Angel (1998) derived an analytical expression for detection limits and re-examined the CFHT data set. The sensitivity of RV surveys for outer planets with orbital periods exceeding the survey duration was studied in Eisner & Kulkarni (2001). They all used different approaches, most of them are based on the periodogram, to determine the detection capabilities of individual surveys. In Kürster et al. (1999b) we have already determined these limits for one target of the CES program, namely Prox Cen, using also a different method based on a Gaussian noise term.

The method we apply to the CES data was already described in Endl et al. (2001b) and in greater detail in Endl et al. (2001a). The latter paper also includes a comparison with a method based on Gaussian noise terms and discusses the differences and the advantage of the bootstrap approach. Our bootstrap-based method can be summarized as follows: for each star we perform numerical simulations of planetary orbits with K (the RV semi-amplitude), $\phi$ the orbital phase and P (the orbital period) as model parameters (orbital eccentricity e is set to 0). The maximum period represents the time span of CES observations of the target star (with a typical duration of $\approx$2000 days). For each set of K and P 8 different orbits are created with their phase angles shifted by $\pi/4$. These signals are added to the actual RV measurements (i.e. we use our own data set as noise term) and perform a period search in the range of 2-5000 days (using the Lomb-Scargle periodogram). We define a planet as detectable if the period is found by the periodogram and the statistical significance of its peak in the power spectrum is higher than $99\%$ (i.e. its false alarm probability (FAP) is lower than 0.01). If the FAP equals or exceeds 0.01 at only one of the trial phases, a planet is not classified as detectable at these P and K values. Again the FAP is estimated by using the bootstrap randomization method and for each simulated signal we perform 1000 bootstrap runs. By increasing the K parameter until the FAP is less than $1\%$ for all orbital phases at a given P value we obtain quantitative upper limits for planetary companions. By taking as the noise-term the obtained RV results for single stars (and the RV residuals for binary stars) we assure that the noise distribution is equal to the measurement errors of the CES survey for this star (obviously, the temporal sampling of the simulated signals are identical with the real monitoring of each individual star by the CES survey). The value of the parameter K can be transformed immediately into an $m\sin i$ value and the parameter Pinto an orbital separation a for the companion, thus we derive an $m\sin i - a$ companion limit function. The simulations are performed in the period range of P = 3 days and the total duration of observation for each individual target. This period range is sampled in such a manner that companion limits are computed every 0.25 AU (i.e. a P^2/3 spacing). One exception is the 1-year window (P = 365 days) where the companion limit is determined additionally in every case. The derived upper mass-limits are strictly correct only at the $P~\& K$-values where the simulations are performed (circles in Figs. .11-.17), the interpolated limits (dashed lines in the figures) are not mathematically stringent, since the frequency-space is too large to be sampled completely (i.e. two times for each independent frequency = Nyquist criterion).

The following stars are excluded from the limit determination due to insufficient observations: HR 753 (with only 6 RV measurements), HR 7373 (8 RV measurements), and $\zeta ^{1}$Ret (14 RV measurements). Another exclusion is $\iota$ Hor, where the planet was discovered, and the $\alpha$ Centauri system, in which case companion limits based on the CES data have already been presented in Kürster et al. (1999b) for Proxima and in Endl et al. (2001a) for $\alpha$ Cen A & B.

5.2 Stellar masses

In order to transform the obtained K value of detectable RV signals into an $m\sin i$ value for planetary companions using Kepler's third law, we have to assume a certain mass for the host star. The stellar mass values we take for the $m\sin i$ calculations are summarized in Table. 8. As an example for the dependence of the $m\sin i$ values on the stellar mass value, we take the case of $\beta$ Hyi ( $1.1~{M}_{\odot}$). For an orbital period of P = 44 days an error of $0.1~{M}_{\odot}$ leads to a small uncertainty in the $m\sin i$ of $\pm$ $0.03~{M}_{\rm Jup}$ and at a period of P = 1890 days the corresponding value is larger with $\pm$ $0.13~{M}_{\rm Jup}$.

 

 

Table 8: Assumed stellar masses for the companion limit determination. If not otherwise stated the mass estimates are taken from Gray (1988) for the according spectral type. Other references: 1. Porto de Mello, priv. comm., 2. Dravins et al. (1998), 3. Drake & Smith (1993), 4. based on the mass-luminosity relation from Henry et al. (1999).

Star	Mass [ ${M}_{\odot}$]	Star	Mass [ ${M}_{\odot}$]
$\zeta$ Tuc	1.061	HR 3259	0.9
$\beta$ Hyi	1.12	HR 3677	2.1
HR 209	1.1	HR 4523	1.04
$\nu$ Phe	1.2	HR 4979	1.04
HR 448	1.231	HR 5568	0.71
HR 506	1.17	HR 6416	0.89
$\tau$ Cet	0.89	HR 6998	1.0
$\kappa$ For	1.12	HR 7703	0.74
$\alpha$ For	1.2	$\phi ^{2}$ Pav	1.11
$\zeta ^{2}$ Ret	1.1	$\epsilon$ Ind	0.7
$\epsilon$ Eri	0.853	HR 8501	1.04
$\delta$ Eri	1.231	HR 8883	2.1
$\alpha$ Men	0.95	Barnard	0.164
HR 2400	1.2	GJ 433	0.42
HR 2667	1.04

5.3 Windows of non-detectability

There are two special cases which can render a simulated signal totally undetectable (using the criteria described above). These windows of non-detection are displayed in the figures as vertical lines which bracket a gap in the companion limits.

5.3.1 Insignificant signal

In the first case the data structure (total number of observation and sampling density) leads to the effect that for a certain phase angle the FAP of the input signal always exceeds the $1\%$ level regardless of the increase of the K parameter. This is the case for the gaps seen in the limits for instance for HR 209 at 2.25 AU (Fig. .11) and for HR 506 at the same separation (Fig. .12).

5.3.2 Aliasing

However, for most cases of non-detecability we find that due to aliasing the maximum power in the periodogram for certain phase angles is not located at the input period, again, regardless of the increase in K. The seasonal 1 year period is a typical window interval for every astronomical observing program and the problem of aliasing will occur if the signal is not properly sampled within this interval (in general sampling on irregular time basis can reduce the occurance of aliasing). This occurs for instance in the cases of HR 3259, $\beta$ Hyi, $\tau$ Cet, $\alpha$ Men and HR 4523, where the maximum power is found at the 1/2 P-value for input signals with P close to the seasonal 1 year period. The peaks in the power spectrum are significant (i.e. FAP > 0.01) but at the wrong P-values. In other cases the maximum power is found at a completely different P value for a certain orbital phase (e.g. at 2.9 days instead of the input value of 1425 days for HR 4523).

In a conservative approach we declare these cases as non-detections since the correct parameters of the planetary signal were not recovered (remember that one undetected signal out of the eight trial phases is sufficient to define a planet as undetectable although seven out of eight signals were successfully recovered). One would conclude though that a planet with the wrong orbital period is present and the correct period of its orbit remains unknown (though continued monitoring of this star would eventually lead to the correct orbital parameters).

5.4 Resulting companion limits

The derived upper mass-limits for planets orbiting the CES survey stars are displayed in Appendix B for each individual star. The horizontal dotted line in each plot shows (for better comparison) the $m\sin i=1.0~{M}_{\rm Jup}$ border. For most of the stars the CES limit line crosses this border at orbital separations less than 1 AU. This clearly demonstrates the need for a longer time baseline as well as a better RV measurement precision to detect "real'' Jupiters at 5.2 AU. The $\alpha$ Centauri system represents a special case in this limit-analysis: it allows the combination of observational constraints for planetary companions with dynamical limitations for stable orbits within the binary. In Endl et al. (2001a), Paper II of this series, we have combined the CES limits with the results from the dynamical stability study of Wiegert & Holman (1997) which led to strong constraints for the presence of giant planets orbiting $\alpha$ Cen A or B. Upper limits for giant planets around the third member of the $\alpha$ Centauri system, Prox Cen, based on the results of a different RV analysis of the CES data were presented in Kürster et al. (1999b). We therefore didn't include the plots for these 3 stars in this paper and refer the reader to the former articles.

In the cases of $\beta$ Hyi and $\tau$ Cet we have nights where a great number of spectra were taken in a short consecutive time. In order to distribute the sampling more evenly and to reduce the total number of RV measurements to save CPU time, we averaged the numerous RV measurements in those nights, to get a maximum number of 3 observations per night. This reduced the total number of measurements for $\beta$ Hyi to 94 (instead of 157) and for $\tau$ Cet to 62 (instead of 116).

The results for HR 448 are limited by the short monitoring time span of 438 d and the small number of observations (24 RV measurements). For this star only companion limits for orbital separations of a < 0.15 AU were found.

The candidate for a planetary companion to $\epsilon$ Eri (from Hatzes et al. 2000) is also indicated in Fig. .13 by an asterisk. It lies well inside the non-detectable region of the CES survey for this star. As described in Hatzes et al. (2000) the combination of several different RV data sets was necessary to find the signal of this companion. Moreover, the orbital period of 7 years is longer than the duration of the CES Long Camera survey.

For the binaries $\kappa$ For, HR 2400 and HR 3677 the simulated planetary signal is added to the orginial RV set, which still possess the huge variation due to the binary orbits. Thus, one additional step in the limit determination has to be performed. Before the periodogram analysis is started we subtract the binary motion (the preliminary Keplerian solution and trends from Sect. 3) by minimization of the $\chi ^{2}$-function. The same is done for the low-amplitude trends of $\alpha$. For, HR 6416 and HR 8501 caused by their stellar secondaries. Again the found trends are subtracted (by $\chi ^{2}$-minimization) from the synthetic RV sets and the periodogram analysis is performed on the RV residuals.

HR 5568 was only observed for 384 days and companion signals with periods >250 days ( $a \approx 0.7$ AU) were not detected at all orbital phases. The CES monitoring of HR 7703 spans over 1042 days and no gaps were found in the detectability of planetary companions within this time-frame (maximum separation $a \approx 1.75$ AU).

For HR 8323 we detected only 3 test signals at P=3,123 and 224 days, at all other P-values the test orbits were not recovered. This low detectability is due to the smaller number of observations (20 RV measurements) and the shorter time span of monitoring (1068 d). The results for HR 8323 are not displayed.

The numerous gaps in the CES detectability of companions of the giant HR 8883 (G4III) are a direct result of the large RV scatter ( $65.2~{\rm m~s}^{-1}$) and the irregular monitoring of this star (see Fig. .17).

In the case of GJ 433 (M2V) the small number of RV measurements (15) and the short duration of monitoring (337 days) prohibit determination of companion limits. No simulated planetary signal could be detected at all orbital phases at the trial periods of 3,10,40,100,150,175,250 and 333 days. Only at the trial period of 200 days a signal with a K amplitude of  $600~{\rm m~s}^{-1}$ was detected, corresponding to a companion of $m\sin i=9.8~{M}_{\rm Jup}$ with a semi-major axis of a=0.5 AU.

Barnard's star (M4V) constitutes another special case of the companion limit analysis. The limits determined from the CES RV data can be combined with the astrometric companion limits based on the HST Fine Guidance Sensor (Benedict et al. 1999). This combination is very effective since both methods are complementary to each other (the RV method is more sensitive to close-by companions while for astrometry the detectability of more distant companions is better). Figure .17 displays the companion limits derived from the CES data and combined with HST astrometric results. These combined limits demonstrate that - except for an aliasing-window near a=0.08 AU (corresponding to P= 20 to 27 d) - all planets with $m\sin i > 1.2~{M}_{\rm Jup}$ can be exluded.

5.5 Orbital eccentricity

One limitation of this method is the restriction to sinusoidal signals (e = 0), i.e. planetary system similar to our Solar System. The huge parameter space for eccentric orbits (including the two additional parameters of orbital eccentricity e and anomaly $\omega$) simply makes it impossible to sample all possible orbits at a given period P. The main motivation of examining the validity of the derived limits also for eccentric orbits is of course the fact that many extrasolar planets with longer periods known today have eccentric orbits.

5.5.1 The test case: HR 4979

In order to check the validity of the derived e = 0 limits also for eccentric orbits we select the special test case of HR 4979 and compare the limits for e = 0 orbits for certain sets of K and P values with e > 0 orbits. HR 4979 was chosen because the companion limits for this star are a smooth function without gaps (see Fig. .15). We want to emphasize that the following tests can only serve as an example and the results cannot be taken as generally valid for the complete survey. Furthermore, due to the above mentioned feasibility limitations, the sampling rate of the parameter space has to be kept low (e.g. the parameter $\omega$ is sampled only every  $90^{\circ }$). The test consists of the two following steps:

1.

Simulations are performed, where the orbital phase $\phi$ is kept fixed and the parameters P,e,and $\omega$ are varied, to obtain a rough estimate of the e > 0 limits validity;

2.

For the smallest and largest P-value, the orbital phase parameter $\phi$ is also varied and the detectability is determined for these cases.

For step 1 we create 110 simulated orbits with different e and $\omega$ values at the trial periods P = 46,130,365,1045, and 1452 days (with a fixed value of orbital phase angle $\phi$), and the K semi-amplitude of  22,20,35,26, and $30~{\rm m~s}^{-1}$ (these values represent the limits obtained for the e = 0 case). For each parameter-set of P, K and e we simulate 4 different orbits with $\omega$ shifted by  $90^{\circ }$. Again, we perform for each simulated signal 1000 bootstrap runs to obtain the FAP level. A signal is classified as non-detected if the FAP exceeds $1\%$ or the periodogram shows maximum power at a different period than the input value P. Figure 19 displays the detectability of these test signals as a function of orbital eccentricity. For smaller values of P these signals get undetectable at e = 0.5 and 0.6, while longer periodic signals are detectable until e = 0.8. The lower detectability at shorter periods might be explained by the fact that with increasing eccentricity the duration of the RV maxima (minima) of the orbits is getting smaller than typical observational time intervals. For longer periods these maxima or minima are better sampled than for short periods and hence better detectable for the periodogram.

In step 2 the orbital phase $\phi$ is introduced as additional parameter for two selected P-values (the minimum and maximum value of P: 46 & 1452 d). For each set of P,K,e and $\omega$ values the parameter space of $\phi$ is additionally sampled 5 times (equidistant). This means that for each pair of P and e parameter 20 synthetic planetary signals are generated (at 4 different $\omega$ and 5 different $\phi$ values). Figure 20 shows the result of this second test. The general form of the results from the first test is preserved.

These tests illustrate - at least in the case of HR 4979 - that the e=0 limits appear to be valid for longer periods and eccentricities of e<0.6, while for smaller P values the validity is constrained to low eccentricities.

Figure 19: Eccentricity-test part 1: detectability of e>0 orbits in the case of HR 4979 and fixed orbital phase. For each trial period and eccentricity 4 different orbits ($\omega$ shifted by $90^{\circ }$) are created with the K-value obtained from the sinusoidal simulations. The graph shows how many of the 4 test signals are detected. The detectability increases with orbital period. For the shortest period P = 46 days the detectability ends at e = 0.5 (no test signal is recovered) and for the longer periods at e = 0.8.

Figure 20: Eccentricity-test part 2: detectability of e>0 orbits for HR 4979 at P = 46 and 1452 d with the orbital phase $\phi$ varied. For each pair of P and e 20 synthetic orbits at the 4 different $\omega$-values, each with 5 different $\phi$-values, are created. The plot displays how many of the 20 signals are recovered successfully. The general form of Fig. 19 is preserved.

5.6 Average detection threshold

By computing the ratio F(P):

$$F(P) = \frac {K_{\rm detected}[{\rm m~s}^{-1}]} {RV~{\rm scatter}[{\rm m~s}^{-1}]}$$ (1)

with P the trial period, $K_{\rm detected}$ the semi-amplitude of the detected RV signal and the RV-scatter as the individual noise-term of the star and by taking the mean ratio $\bar{F}_{\rm Star}$ for all P-values where an upper mass limit was determined we obtain an average detection threshold for each star. For the 30 stars, where we determined companion limits, and at all P-values we found F(P) > 1., except in the case of $\beta$ Hyi where $F(3~{\rm d})=0.8$ (which also demonstrates the detectability of signals with smaller amplitude than the measurement precision by a sufficient number of datapoints). The main bulk of values of F(P) lies in the range from 1.2 to 4, with an average value of 2.75 ( $\bar{F}_{\rm CES}$) for all 30 stars and P-parameter values where companion limits were derived. The value of $\bar{F}_{\rm CES}$ corresponds to the average of the RV signals detected and does not represent the exact average detection threshold of the CES survey due to the presence of the non-detectability gaps, which cannot be taken into account in this statistic. Table 9 gives the average detection factor for each star ( $\bar{F}_{\rm Star}$) and the resulting mean $\bar{K}$-amplitude of detectable planetary signals (the presence of non-detectability gaps is also indicated).

The minimum value of $\bar{F}_{\rm Star}$ was found for HR 2667 ( $\bar{F}_{\rm Star}=1.54$) and the highest value for HR 448 ( $\bar{F}_{\rm Star}=16.4$). In terms of average detectable K-amplitude the minimum for the CES Long Camera survey is $19.9~{\rm m~s}^{-1}$ for HR 5568, which is not surprising since this is the star with the smallest RV-scatter ( ${\rm rms} = 7.7~{\rm m~s}^{-1}$), and the maximum at $369.4~{\rm ms}^{-1}$ for the highly variable star HR 8883.

 

 

Table 9: Average detection threshold $\bar{F}_{\rm Star}$ and resulting mean $\bar{K}$-amplitudes of detectable planetary signals for each star. Stars indicated with an asterix (*) possess windows of non-detectability.

Star	$\bar{F}_{\rm Star}$	$\bar{K}$	Star	$\bar{F}_{\rm Star}$	$\bar{K}$
		[ ${\rm m~s}^{-1}$]			[ ${\rm m~s}^{-1}$]
$\zeta$ Tuc*	3.3	71.3	HR 3259*	2.26	36.7
$\beta$ Hyi*	1.76	35.2	HR 3677	2.44	56.8
HR 209*	5.37	124	HR 4523*	3.57	53.6
$\nu$ Phe	2.16	38.6	HR 4979	1.66	23.2
HR 448*	16.4	280.3	$\alpha$ Cen A	1.88	22.2
HR 506*	3.79	90.5	$\alpha$ Cen B	2.14	26.9
$\tau$ Cet*	1.95	22.6	HR 5568*	2.58	19.9
$\kappa$ For	2.42	34.5	HR 6416	1.77	34.3
$\alpha$ For	1.85	94.2	HR 6998	2.67	52.4
$\zeta ^{2}$ Ret	2.04	44.4	HR 7703	2.8	37.3
$\epsilon$ Eri*	2.67	40.0	$\phi ^{2}$ Pav	2.25	79.5
$\delta$ Eri	2.12	32.9	$\epsilon$ Ind	3.35	45.2
$\alpha$ Men*	2.51	24.6	HR 8501*	1.73	40.4
HR 2400	1.69	42.0	HR 8883*	5.67	369.4
HR 2667	1.54	25.4	Barnard*	4.58	170.5

Figure 21 compares these CES-survey mean $\bar{K}$-amplitudes of detectable planetary signals with the K-amplitudes of known extrasolar planets as a function of B-V. For this purpose we selected only K-amplitudes less than $100~{\rm m~s}^{-1}$ (which excludes three CES stars: HR 448, HR 8883 and Barnard's star and 27 known extrasolar planets). Except for CES stars with B-V<0.6, most of the RV-signals of the known extrasolar planets are within the detection range of the CES Long Camera survey.

Figure 21: Average detectable K-amplitudes of the CES-survey (diamonds) as a function of B-V, compared to K-amplitudes of known extrasolar planets (cirlces). Only K-amplitudes less than $100~{\rm m~s}^{-1}$ are displayed (excluding HR 448, HR 8883 and Barnard's star). The list of amplitudes of the known extrasolar planets was compiled using the information on the websites of the Geneva observatory program ( http://obswww.unige.ch/$\sim$udry/planet/planet.html) and the California & Carnegie planet search ( http://exoplanets.org/) and the colors were derived from the SIMBAD database. Except for the blue part of the diagram (stars with B-V<0.6) most of the known extrasolar planets are located within the detection range of the CES Long Camera survey.

Figure 22: Comparison of detected simulated signals (circles) with an analytically derived detection threshold (solid line) based on the Lomb-Scargle periodogram and a False Alarm Probability (FAP) of 0.01 (using Eq. (15) in Cochran & Hatzes 1996). F denotes the S/N-ratio of the signal in the power spectrum and N the total number of measurements per star.

Figure 22 shows a comparison of our numerical simulations with an analytic detection threshold for a False Alarm Probability of 0.01 by using Eq. (15) from Cochran & Hatzes (1996). The theoratical curve (solid line in Fig. 22) is calculated on the basis of the Lomb-Scargle periodogram and gives the S/N-ratio ( $F=K/\sigma$) of signals detected with FAP $\le 0.01$ as a function of number of measurements N. Clearly, the plot shows that virtually no signals can be detected with N<20, consistent with our results, and that the curve constitutes a clear lower limit to our "real'' detectability (circles in Fig. 22). It is not surprising that the detected signals do not get closer to the theoratical limit, since they had to be recovered at all phase angles. In many cases the CES Long Camera survey successfully recovered signals at certain phase angles at lower F-values, which are not included in Fig. 22 due to the above mentioned criterion. Figure 22 demonstrates again that the sensitivity of RV planet search programs can also be improved by increasing N, i.e. by taking more data, beside raising the RV precision.

5.7 Mass range

The lowest companion mass which was detected during our simulations corresponds to an $m\sin i = 0.079~{M}_{\rm Jup} \approx 0.27~{M}_{\rm Saturn}$ or $25~{M}_{\rm Earth}$ planet in a P=3 d and a=0.036 AU orbit around HR 5568. Figure 23 shows the lower section of the limiting $m\sin i$-values of all simulated planetary signals which were determined as detectable by the CES survey. Planets with $m\sin i = 1~{M}_{\rm Jup}$ were found undetectable for the 30 CES survey stars at separations larger than 2 AU. However, at small orbital separations of $a \approx 0.04$ AU the CES survey would have detected "51 Peg''-type planets in all cases and for 22 survey stars even the presence of sub-Saturn-mass planets. The bias of RV planet searches towards short-period planets is obvious in Fig. 23. Also indicated in Fig. 23 is the location of the planet discovered around $\iota$ Hor. With $m\sin i = 2.26~{M}_{\rm Jup}$ and a=0.925 AU $\iota$ Hor b clearly lies above the main part of the limiting detectable $m\sin i$-values and was thus "easily'' detectable by the CES Long Camera survey.

Figure 23: Lower section of all detectable $m\sin i$-values (diamonds) of the CES Long Camera survey. The horizontal dashed lines show the mass values of Jupiter, Saturn and a hypothetical 50 Earth-mass planet. The location of $\iota$ Hor b ( $m\sin i = 2.26~{M}_{\rm Jup}$, a=0.925 AU) is also indicated. $\iota$ Hor b lies well above the main bulk of detectable $m\sin i$-values. At orbital separations larger than 2 AU no planets with $m\sin i < 1~{M}_{\rm Jup}$ are detectable. For 22 stars the detectable $m\sin i$-value at $a \approx 0.04$ AU reach down into the sub-Saturn mass regime ( $m\sin i < 0.3~{M}_{\rm Jup}$).