3 Deriving V-I from Hipparcos Hp

The central idea of this study is to derive new sets of V-I color indices for red stars bypassing all various methods used in the original derivation of V-I (ESA 1997). We abandon the calibration methods based upon the ground-based B-V or Tycho $B_{\rm T}-V_{\rm T}$ for two reasons. First, the B-V color index, at least for carbon stars, is a poor representative of effective temperature due to the severe blanketing effect by molecular bands (Alksne et al. 1991) in the BV bandpasses. Second, many Hipparcos red stars have such a large B-V color index that their measurements are uncertain or, in the case of Tycho magnitudes, missing due to extremely low fluxes in the $B_{\rm T}$bandpass. In this sense the potential of Tycho $B_{\rm T}V_{\rm T}$ photometry for red stars is limited. However, there is a color index, $Hp-V_{\rm T}$, which to our knowledge, has been used neither in the Hipparcos reductions nor the following studies.

Figure 2: Normalized response curves for the Hipparcos Hp (solid line), Tycho $V_{\rm T}$ (short-dashed line), and Cousins I(long-dashed line) bandpasses. The corresponding curves are taken from Bessell (1990, 2000).

The normalized Hp and $V_{\rm T}$response curves provided by Bessell (2000) indicate only a 21 nm difference in the mean wavelength (see Fig. 2). This wavelength is calculated assuming a flat spectral energy distribution (SED) which is definitely not the case for late-type stars. If we account for the observed spectral energy distribution, e.g., from Gunn & Stryker (1983), then for an M7III spectral-type star (HIP 64569) the difference in the effective wavelengths of the two filters reaches 150 nm. The SEDs for the two carbon stars HIP 99 and 95777 yield an 84 and 94 nm difference in the effective wavelength, respectively. It is the extended red response of the S20 photocathode of Hipparcos main detector - Image Dissector Tube, which makes the $Hp-V_{\rm T}$ index fairly sensitive in the K-M spectral range (see ESA 1997, vol. 1, Fig. 1.3.4). We employ this property to calibrate V-I for late-type stars using $Hp-V_{\rm T}$.

3.1 Tycho photometry

First trials using the published Tycho $V_{\rm T}$ photometry indicated two problems. First, a large fraction of red stars lack Tycho photometry. Second, the $V_{\rm T}$ photometry shows a progressively increasing bias at faint magnitudes ( $V_{\rm T}>9$). This effect is illustrated by Fig. 3 where $Hp-V_{\rm T}$ values are abnormally small at Hp>8, equivalent to the "brightening'' of $V_{\rm T}$ at these Hp magnitudes.

Figure 3: Bias in the $Hp-V_{\rm T}$ at Hp>8 originating from original Tycho $V_{\rm T}$ magnitudes for a bright Mira  T  Cep  = HIP  104451 (top panel). If the $V_{\rm T2}$ epoch photometry is used, the bias disappears (bottom panel). A straight line is fitted to the data in the bottom panel and then just overplotted in the top panel.

It is suspected that the de-censoring technique (Halbwachs et al. 1997) has failed to completely correct the faint-magnitude bias. Therefore, it was decided to make use of the Identified Counts Data Base, ICDB (Fabricius & Makarov 2000b) - a by-product of the Tycho-2 data re-processing (Høg et al. 2000).

All transits of about 2.5 million stars included in the Tycho-2 Catalogue are represented in the ICDB by sequences of 13 time-ordered photon counts, separately for the inclined and vertical slits, and the $B_{\rm T}$ and $V_{\rm T}$ bandpasses. Combined with some instrument calibration files, this data base is sufficient to reproduce a complete astrometric solution for any Tycho-2 star, including its possible binarity status, photometric variability, etc. In this paper, we exploit the possibility to extract epoch photometry for selected stars by estimating the signal at the pre-computed, mission-averaged astrometric position.

The working version of Tycho-2 epoch photometry was derived some time ago for a search of a particular kind of variable stars, although it has not been implemented in the construction of the Tycho-2 Catalogue. It should be noted that, even though based on the same observational data, the Tycho-2 epoch photometry used here differs significantly from the published Tycho epoch photometry (ESA 1997). Nevertheless, the global calibrations of our current epoch photometry are consistent with the Tycho mission-average calibrations. On the star-by-star level, the Tycho-2 processing (both astrometric and photometric) is based on a single so-called Maximum Cross-Correlation estimator, while the original Tycho epoch photometry is the result of a series of successive linear and non-linear filterings (Halbwachs et al. 1997; ESA 1997, vol. 4). The main difference in the reduction procedure is that for a given star in Tycho-2, the determination of astrometric parameters was done over all collected transits at once; whereas in Tycho, a complete cycle of astrometric and photometric reductions was performed for each transit.

The latter method proved to be unreliable at a low signal-to-noise ratio, as the noise may mimic a signal from the star and produce a spurious astrometric detection and a subsequent false photometric estimate at the derived location. Such false detections tend to be abnormally bright, which then produce a bias in the faint magnitudes and hence necessitate the de-censoring analysis (Halbwachs et al. 1997) as the lesser of two evils.

The Tycho-2 epoch photometry is largely free of this de-censoring bias, since all photometric estimations are made at the correct location of a star image (within the astrometric precision), and all observations are retained. Still, Tycho-2 epoch photometry can only find restricted applications due to a possibly high background and contamination from other stars which could be present in the $40\hbox{$^\prime$ }$-long slits of the star mapper.

We will denote the re-processed Tycho photometry as $V_{\rm T2}$ to distinguish it from the original Tycho $V_{\rm T}$ epoch photometry.

  
3.2 Relationship Hp-V $_{\mathsfsl {T2}}$ vs. Hp

Due to the differences in spectral features, we kept the processing of carbon and oxygen- and zirconium-rich (M, S) stars separately. There are 321 carbon stars and 4464 stars of M and S spectral type, which have a pair of Hp and $V_{\rm T2}$ values. These stars were selected according to the listed spectral type in the Hipparcos Catalogue (field H76) but not fainter than Hp=11. In the case of a missing spectral type, we included the stars having Hipparcos V-I>1.5. Finally, the stars of K spectral-type were also considered if their V-I>2. Note that for the Hipparcos photometry we used the so-called $Hp_{\rm dc}$ magnitude estimate derived from the unmodulated part of a signal intensity (ESA 1997), since the mean photometric parameters have been obtained from $Hp_{\rm dc}$. In addition, the ground-based photoelectric photometry is always integrated over some aperture (usually with ${\ \hbox{$\scriptstyle/\mkern-13mu\mathchar''20D $ }}=15{-}30\hbox{$^{\prime\prime}$ }$) centered onto the target and hence, the flux from any object within this aperture is going to be included. However, in Tycho-2 photometry, if the star was found to be a binary (minimum separation $\sim$ $0\hbox{$.\!\!^{\prime\prime}$ }4$), only the brightest component has been retained and subsequently used for this study. Because of that, the color index $Hp-V_{\rm T2}$ of resolved binaries could be biased to some degree and thus, should be considered with caution.

For each star, the color index $Hp-V_{\rm T2}$ was visually examined as a function of Hpignoring the listed status flags. A pair of $Hp,V_{\rm T2}$ photometry was deleted if it deviated from the mean trend by more than $3\sigma$.

Figure 4: The distribution of mean standard errors for red stars from Hp photometry (solid dots) and Tycho-2 $V_{\rm T2}$ data (crosses) as a function of median Hp magnitude. The large scatter in the distribution of $V_{\rm T2}$ errors is due to the variability - observations of Miras generate the largest scatter. The lower envelope of the same error distribution reflects the contribution by photon noise.

As seen in Fig. 4 the precision of $Hp-V_{\rm T2}$ is driven by the precision of the $V_{\rm T2}$ photometry. A rapidly deteriorating error budget at Hp>9actually poses a problem of reliability of calculated slopes in the $Hp-V_{\rm T2}$ vs. Hp plot. We opted for an interactive and iterative linear fit to find a slope, i.e., gradient $\nabla_{HpV_{\rm T}}=\Delta(Hp-V_{\rm T2})/\Delta{Hp}$ and an intercept. It was decided to keep all datapoints unless any were clearly deviant or there was a peculiar trend usually due to very faint or corrupted $V_{\rm T2}$ epoch photometry. It should be noted that we were not able to find a perceptible difference in the color of variable stars observed at the same magnitude on the ascending or descending part of a lightcurve. In the case of a constant star or large uncertainties in the $V_{\rm T2}$ photometry, only the mean $Hp-V_{\rm T2}$ has been calculated. We note that Hp can be predicted for any $V_{\rm T2}$ via

$$Hp={b_0+V_{\rm T2} \over 1-b_1},$$ (2)

where b₀ is the intercept and b₁ is the slope from a linear fit. This simple relationship is crucial in bridging the ground-based VI photometry and Hipparcos Hp photometry (see Sect. 3.3).

Figure 5: Calculated gradients $\Delta (Hp-V_{\rm T2})/\Delta Hp$ as a function of observed Hp amplitude within the 5-to-95 percentile range, Hp95-Hp5, for 136 carbon stars (top panel) and 906 M and S stars (bottom panel). For M and S stars, the gradient is correlated with the amplitude of the brightness variation as indicated by a linear fit.

The calculated color gradients $\nabla_{HpV_{\rm T}}$ vs. the observed amplitude in Hp within the 5-to-95 percentile range, Hp₉₅-Hp₅, are shown in Fig. 5, separately for 136 carbon and 906 M and S stars. For both groups of stars, the color gradient ranges between -0.1 and -0.45. For carbon stars, the mean gradient is $\langle\nabla_{HpV_{\rm T}}\rangle=-0.24$, whereas it is -0.26 for the M and S stars. This indicates that on average the gradient $\nabla_{HpV_{\rm T}}$ is only marginally sensitive to the C/O ratio in the atmospheres of red stars. On the other hand, for M and S stars, the gradient is definitely correlated with the amplitude of a brightness variation in Hp - the color gradient increases at the rate -0.025 per mag of amplitude. Similarly, the gradient is correlated with the median V-I for M and S stars: this merely reflects another correlation between the amplitude of brightness variation and median V-I.

  
3.3 V - I calibration curves

We have not been able to find any ground-based $V-I_{\rm C}$ data for the red stars concurrent with the Hipparcos lifetime. To relate the ground-based V-I observations to Hipparcos/Tycho photometry we postulate that a star's luminosity-color relation (encapsulated by parameters b₀ and b₁ in Eq. (2)) is constant over several decades and adopt the $V_{\rm T2}$ magnitude as a proxy to tie ground-based observations into the Hipparcos $HpV_{\rm T2}$ system. In practice, it involves two important steps. First, the ground based V magnitude should be transformed into the system of Tycho $V_{\rm T}$. This is not trivial for red stars, therefore we provide step-by-step instructions explaining how to do that for carbon and M, S stars. Second, the derived $V_{\rm T2}$ magnitude now allows us to find the corresponding Hp value using Eq. (2) and thus, the color $Hp-V_{\rm T2}$. Only then, it is possible to relate a ground-based measurement of V-I to the corresponding $Hp-V_{\rm T2}$ value and be reasonably certain that both measurements are on the same phase of a light curve in the case of variable stars. As demonstrated by Kerschbaum et al. (2001), there is no phase shift between the variability in the V and $I_{\rm C}$ bandpasses for asymptotic giant branch stars, a dozen of which can also be found in Table 3. A small and consistent rms scatter of the residuals in the linear fits given in Table 3 for additional M stars and a few carbon stars, is another reassuring sign of the lack of a phase shift - a crucial assumption in the calibration procedure.

3.3.1 Carbon stars

Many carbon stars are too faint in the $B_{\rm T}$ bandpass, hence their $B_{\rm T}-V_{\rm T}$color index is either unreliable or is not available at all. Therefore, we first derived a relationship between the ground-based $(V-I)_{\rm C}$ and $(B-V)_{\rm J}$ using the Walker (1979) data:

$$(B-V)_{\rm J}=1.59-0.942(V-I)_{\rm C}+0.5561(V-I)_{\rm C}^{2}.$$ (3)

Then, the $B_{\rm T}-V_{\rm T}$ can be easily estimated using Eq. (1.3.31) in ESA (1997), vol. 1:

$$(B_{\rm T}-V_{\rm T})=1.37(B-V)_{\rm J}-0.26.$$ (4)

Finally, knowing the ground-based V-magnitude and employing Eq. (1.3.34) in ESA (1997), vol. 1, we derive

$$V_{\rm T2}=V_{\rm J}-0.007+0.024(B_{\rm T}-V_{\rm T})+0.023(B_{\rm T}-V_{\rm T})^{2},$$ (5)

which in combination with Eq. (2) yields the corresponding $Hp-V_{\rm T2}$.

3.3.2 M and S stars

Owing to some, albeit weak, dependence of TiO absorption upon the surface gravity, the stars of spectral type M can be divided into giants and dwarfs (main sequence stars). All stars in our sample with Hipparcos parallaxes smaller than 10 mas are considered to be giants. For M giants, $V_{\rm T2}$ follows directly from Eq. (1.3.36) (see ESA 1997, vol. 1):

$$V_{\rm T2}=V_{J}+0.20+0.03(V-I-2.15)+0.011(V-I-2.15)^{2}.$$ (6)

To calculate a similar relationship for M dwarfs, we used the data from Koen et al. (2002):

$$V_{\rm T2}=V_{J}+0.20+0.042(V-I-2.15).$$ (7)

As expected, Eqs. (6) and (7) are very similar so that, considering the uncertainties involved, our V-I photometry is not sensitive to the surface gravity. Equation (6) or (7) in combination with Eq. (2) then yields $Hp-V_{\rm T2}$.

3.3.3 Calibration curves

From the sources listed in Table 4, we have chosen 274 measurements of V-I for carbon stars and 252 for M and S stars. Quite often there is more than one V-I measurement for a given star. In the case of multi-epoch ground-based V-I data, we first obtained a linear fit to V-I as a function of V (e.g., Table 3). The coefficients of that fit were used to estimate the V-I index of variable stars at maximum brightness. The corresponding $Hp-V_{\rm T2}$ color index at maximum brightness has the advantage of being relatively insensitive to the uncertainties affecting the $Hp-V_{\rm T2}$ vs. Hp relation at its faint end (see Figs. 3 and 4). This is especially important at the blue end of the relationship between V-I and $Hp-V_{\rm T2}$ (corresponding to the maximum brightness in the case of variable stars) requires more care due to its steepness.

The calibration curves for oxygen (actually M and S) stars and carbon stars are presented in Fig. 6.

Figure 6: Color-color transformation for M and S stars (left panel) and carbon stars (right panel). The red end of this transformation ( $Hp-V_{\rm T2}<-1.5$) for carbon stars is uncertain due to the lack of intrinsically very red Hipparcos calibrating carbon stars.

Since many calibrating stars are fainter than Hp=8, the scatter is mainly along the $Hp-V_{\rm T2}$ axis (see also Fig. 4). The relationship between $V-I_{\rm C}$ and $Hp-V_{\rm T2}$ cannot be represented by a single polynomial, hence we provide segments of calibration curves along with a color interval of their validity (Table 6). Within this interval, a Hipparcos $(V-I)_{\rm H}$ is

$$(V-I)_{\rm H}=\sum_{k=0}^4 c_k (Hp-V_{\rm T2})^{k}.$$ (8)

 

 

Table 6: Polynomial transformation from $Hp-V_{\rm T2}$ to $V-I_{\rm C}$.

Spectral Type	Color Range	c0	c1	c2	c3	c4
M, S	$-0.20>Hp-V_{\rm T2}\geq-0.80$	1.296	-6.362	-5.128	-1.8096	0.0
M, S	$-0.80>Hp-V_{\rm T2}\geq-2.50$	2.686	-1.673	0.0	0.0	0.0
C	$-0.20>Hp-V_{\rm T2}\geq-1.77$	1.297	-4.757	-4.587	-2.4904	-0.5343
C	$-1.77>Hp-V_{\rm T2}\geq-2.00$	3.913	0.0	0.0	0.0	0.0

To calculate an epoch $(V-I)_{\rm H}$, one should use the epoch Hp photometry and obtain $Hp-V_{\rm T2}=b_0+b_1\times{Hp}$ (see Eq. (2)). Then, a polynomial transformation given by Eq. (8) and Table 6 leads directly to the desired $(V-I)_{\rm H}$ color index. However, there are numerous cases when it was not possible to determine a slope b₁ in the $Hp-V_{\rm T2}$ vs. Hp plot, although the amplitude of Hp variations indicated a likely change in $Hp-V_{\rm T2}$ as well. Therefore, for all such stars with a light amplitude having the range between maximum and minimum luminosities, $\Delta Hp>0.15$ (see entries H50-H49, ESA 1997, vol. 1), we adopted the mean slope, i.e., the mean gradient given in Sect. 3.2. A difficulty then is to find a point in the $Hp-V_{\rm T2}$ vs. Hp plot, to which the mean slope can be applied in order to estimate an intercept b₀. The median of the 3-5 brightest values of Hp and the corresponding median $Hp-V_{\rm T2}$ color were adopted for such a "reference'' point.

An important issue is to verify the system of our $(V-I)_{\rm H}$ photometry for red stars. The differences between the new median $(V-I)_{\rm H}$ and the best available Hipparcos V-I photometry (entry H40) are plotted in Fig. 7.

Figure 7: Hipparcos median V-I (ESA 1997, entry H40) vs. newly derived median $(V-I)_{\rm H}$ in this study. The reasons for some very large discrepancies are discussed in Sect. 4.1.

On average the two systems are consistent. The very red carbon stars are an exception because their $(V-I)_{\rm H}$ color indices reach saturation, whereas the Hipparcos V-I index is not restrained. Then, there are numerous cases where the newly derived $(V-I)_{\rm H}$ values differ considerably from those in the Hipparcos Catalogue - in extreme cases up to 3-4 mag. A closer look at these cases indicates various reasons for such discrepancies. It could be duplicity, an incorrect target, severe extrapolation in color, etc. Noteworthy is the fact that the $I_{\rm C}$ bandpass given in ESA (1997) is $\sim$30 nm wider on the red side than the one published by Bessell (1979). Uncertainty in the location of the red-side cutoff of the $I_{\rm C}$-bandpass owing to different detectors is known to be a major source of a small color-dependent bias (<0.1 mag) in the ground-based photometry of red stars.

3.3.4 Verification of the new ${\mathsfsl V}$ - ${\mathsfsl I}$ color

From the variety of available sources, we have chosen the two largest sets of ground-based Cousins V-I data to test our $(V-I)_{\rm H}$ color indices; that is Koen et al. (2002) for M stars and Walker (1979) for carbon stars. We also selected the data of Lahulla (1987), which is an independent source of V-I, albeit in the system of Johnson VI which was not used in the calibration.

Figure 8: Differences between our instantaneous $(V-I)_{\rm H}$ color index and those of Koen et al. (2002); Lahulla (1987); Walker (1979). The upper two panels represent M stars, whereas the bottom panel contains carbon stars. The accuracy of our calibrated V-I colors is clearly insufficient in the case of faint M dwarfs, which represent a large fraction of the Koen et al. (2002) sample.

The differences, $(V-I)_{\rm H}-(V-I)_{\rm C}$, are plotted as a function of ground-based V (Fig. 8). For the Walker (1979) and Lahulla (1987) datasets, the mean offset $\langle(V-I)_{\rm H}-(V-I)_{\rm C}\rangle$ is not more than +0.01 mag; the scatter of individual differences is 0.12 mag. The Koen et al. (2002) data are instrumental to test the reliability of $(V-I)_{\rm H}$ for early-type M stars, both dwarfs and giants. We note that at $V-I\approx2$ the calibration curve is very steep (left panel, Fig. 6). At this V-I, a variation in $Hp-V_{\rm T2}$ by only 0.01 mag corresponds to a 0.05 mag change in V-I. For relatively bright Hipparcos stars (V<9), the mean offset $\langle(V-I)_{\rm H}-(V-I)_{\rm C}\rangle$ is +0.04 but it increases to +0.20 for fainter stars (9<V<11). The scatter also rises from 0.13 to 0.40 in these two intervals. A noticeable bias in the mean $(V-I)_{\rm H}$ towards faint magnitudes might be an indication of some residual systematic error either in the Hipparcos Hp epoch photometry or in Tycho-2 $V_{\rm T2}$ magnitudes. As expected, rapidly increasing errors in $V_{\rm T2}$ as a function of magnitude (Fig. 4) clearly set a limitation on the accuracy of $(V-I)_{\rm H}$.