---
abstract: 'We have estimated the ages of a sample of A–type Vega–like stars by using Strömgren *uvby$ \beta  $* photometric data and theoretical evolutionary tracks. We find that 13 percent of these A stars have been reported as Vega–like stars in the literature and that the ages of this subset run the gamut from very young (50 Myr) to old (1 Gyr), with no obvious age difference compared to those of field A stars. We clearly show that the fractional IR luminosity decreases with the ages of Vega–like stars.'
author:
- 'Inseok Song, J.-P. Caillault,'
- 'David Barrado y Navascués,'
- 'John R. Stauffer'
title: 'Ages of A–type Vega–like stars from *uvby$ \beta  $* Photometry'
---

Introduction
============

There are several unusual sub–groups among the A–type stars, such as the metallic–line stars (Am), the peculiar A stars (Ap), $ \lambda  $ Bootis type stars, and shell stars [@AbtMorrell95]. Another class of stars with many members amongst the A dwarfs is that of the Vega–like stars. Vega–like stars show excess IR emission attributable to an optically thin dust disk around them. These disks are believed to have very little or no gas [@LBA99]. It is very important to know the ages and, hence, the evolutionary stages of these stars, since they are believed to be signposts of exo–planetary systems or of on–going planet formation. However, determining the ages of individual A–type stars is a very difficult task. Some indirect age dating methods for A–type stars include the use of late–type companions if any exist (HR 4796A and Fomalhaut; see @Stauffer95 [@Barrado97; @myPhD]) or using stellar kinematic groups (Fomalhaut, Vega and $ \beta  $ Pictoris; see @David98 and @BSSC). The use of Strömgren *uvby$ \beta  $* photometry [@ATF97], however, provides a more direct and general determination of the ages of A–type stars.

The photometric *uvby*$ \beta  $ system as defined by @Stromgren63 and @CM66 allows for reasonably accurate determination of stellar parameters like effective temperature $ T_{eff} $, surface gravity $ g $, and metallicity for B, A, and F stars [@crawford79; @NSW93 and references therein]. The $ T_{eff} $ and $ g $ values can then be used to estimate directly the ages of stars when they are coupled with theoretical evolutionary tracks (though for individual stars these estimates have relatively large error bars).

In this letter, we describe our application of this technique to a volume limited sample of 200 A stars.

Method
======

$ T_{eff}\protect $ and $ \log g\protect $ determination
--------------------------------------------------------

Extensive catalogues of *uvby*$ \beta  $ data have been published by @HM80, @Olsen83, and @OP84. We have used these catalogues and WEBDA[^1] databases to find *uvby$ \beta  $* photometry data for our sample of A–type stars.

Numerous calibration methods of effective temperature and surface gravity using *uvby$ \beta  $* photometry have been published . @MD85, in particular, demonstrate that their calibration yields $ T_{eff} $ and $ \log g $ to a 1 $ \sigma  $ accuracy of $ 260 $ K and $ 0.10 $ dex, respectively. However, as pointed out by @NSW93, $ \log g $ from @MD85’s calibration depends on the $ T_{eff} $ value while the most desirable calibration method should not. Therefore, we used the @MD85 grids with Napiwotzki et al.’s gravity modification to eliminate the $ \log g $ dependence on $ T_{eff} $ for early–type stars. The subsequent temperature calibration is in agreement with the integrated–flux temperatures $ \left( T_{eff}=(\pi F/\sigma )^{1/4}\right)  $ from @Code, @Beeckmans, and @Malagnini at the 1% level and the accuracy of $ \log g $ ranges from $ \approx 0.10 $ dex for early A stars to $ \approx 0.25 $ dex for hot B stars [@NSW93].

A rapidly rotating star has a surface gravity smaller at the equator than at the poles and both the local effective temperature and surface brightness are therefore lower at the equator than at the poles. Thus, in comparing a rotating star with a non–rotating star of the same mass, the former is always cooler. But the apparent luminosity change of a rotating star depends on the inclination angle ($ i $) such that a pole–on $ \left( i=0^{\circ }\right)  $ star is brighter and an edge–on $ \left( i=90^{\circ }\right)  $ star is dimmer than a non–rotating star [@Kraft]. In all cases, the combination of the luminosity and temperature changes result in an older inferred age compared to the non–rotating case. This effect is prominent in spectral types B and A in which most stars are rapidly rotating ($ v\sin i\geq 100km/sec $). Recently, @FB98 simulated the effect of stellar rotation on the Strömgren *uvby$ \beta  $* photometric indices. They concluded that the effect of stellar rotation is to enhance the stellar main sequence age by an average of $ 40\% $. Therefore, we included the stellar rotation correction suggested by @FB98. However, their rotation correction schemes are available only for stars with spectral type between approximately B7–A4. We extended the range of rotation correction such that for stars earlier than B7, we used the correction scheme for B7 stars and for stars later than A4, we used the correction scheme for A4 stars. Therefore, stars earlier or later than the Figueras & Blasi’s (1998) range will have more uncertain ages.

Large uncertainties in estimated ages are mainly due to the large error in $ \log g $. However, using a rotation correction scheme based on the projected stellar rotational velocities $ \left( v\sin i\right)  $ rather than a scheme based on the true stellar rotational velocities $ \left( v\right)  $ may have resulted in uncertainties also. The stellar rotation decreases the effective temperature depending on the inclination angle (small change of $ T_{eff} $ for $ i\approx 0^{\circ } $ but large change of $ T_{eff} $ for $ i\approx 90^{\circ } $) but the current rotation correction scheme cannot distinguish between the case of large $ v $ with small $ i $ and the case of small $ v $ with large $ i $. Thus, rotation correction using $ v\sin i $ instead of $ v $ may cause uncertainty in stellar ages.

Ages of Open Clusters
---------------------

The theoretical evolutionary grids of @Schaller92 were used to estimate ages of stars from $ T_{eff} $ and $ \log g $. To verify that our age dating method is working, we applied the method to a few open clusters with ages determined by other methods – $ \alpha  $ Perseus (80 Myr), Pleiades (125 Myr), NGC 6475 (220 Myr), M34 (225 Myr), and Hyades (660 Myr). The ages for the first two clusters are based on recent application of the lithium depletion boundary method (LDBM) (@PerAge and @Stauffer99 for $ \alpha  $ Perseus; @Stauffer98 for Pleiades). The ages for the other clusters are from upper main sequence isochrone fitting (UMSIF), and are taken from @JonesProsser or @Lynga. The age scales based on the two different methods (LDBM and UMSIF) are not yet consistent with each other, and both have possible systematic errors. The current best UMS isochrone ages for $ \alpha  $ Perseus and the Pleiades are in the range 50–80 Myr and 80–150 Myr.

In Figure \[OpenCluster\], one can see that the isochrones of these open clusters are fairly well reproduced. However, there are some deviations from the expected values. Stars that are younger than or close to 100 Myr, like stars in $ \alpha  $ Perseus, tend to locate below the theoretical 100 Myr isochrone. So we assigned an age of 50 Myr for the stars below the 100 Myr isochrone. At intermediate ages, the open cluster data provide a mixed message – the M34 Strömgren age appears to be younger than the UMSIF age, whereas the NGC 6475 Strömgren age seems older than the UMSIF age. This could be indicative of the inhomogeneous nature of the ages (some from LDBM, some from relatively old UMS models, some from newer models) to which we are comparing the Strömgren ages.

If we could use $ v $ data instead of $ v\sin i $ and if one could make a rotation correction scheme by using $ v $ values, then the new correction scheme would tighten more stars for a given cluster to the locus of the cluster compared to the uncorrected case. However, the $ v\sin i $ rotation correction scheme used in this study shifts the loci of clusters and only moderately reduces the standard deviations of ages (see, e.g., the case for the Pleiades in Figure \[roteffect\]).

Field A stars and Vega–like stars
=================================

We have identified 200 A dwarfs within 50 pc with known $ v\sin i $ values and measured *uvby$ \beta  $* photometric indices. The distance limit of 50 pc was chosen so that the photospheres of most A–type stars within the given volume should be detected in the 12 $ \mu  $m IRAS band and that the volume should contain enough A–type stars to draw a statistically significant result. Since rotation greatly affects the estimated stellar ages, we only included stars with known $ v\sin i $ values (from SIMBAD) throughout this study. $ T_{eff} $ and $ \log g $ values were calculated and corrected to account for the rotation effects as described in the previous section. Among these A–stars, 26 have been identified as possible Vega–like stars by cross–indexing the current list with Song’s [-@myPhD] master list of “proposed” Vega–like stars. Estimated ages, along with other data — spectral type, fractional IR luminosity $ f $, *uvby$ \beta  $* photometric **data*,* and $ v\sin i $ — are summarized in Table \[AVegas\]. The frequency of Vega–like stars in our sample is 13% in good agreement with the results from other volume limited surveys: $ 14\pm 5\% $ from @Plets99’s [-@Plets99] survey of the incidence of the Vega phenomenon among main sequence and post main sequence stars and about $ 15 $% or more from the review article on the Vega phenomenon by @LBA99.

More than 95% of our sample stars are listed in the IRAS Point Source Catalog and/or Faint Source Catalog and were detected at least at 12 $ \mu  $m, and about 75% of them were detected at 12 and 25 $ \mu  $m. Based on the 12 and 25 $ \mu  $m IRAS fluxes, we checked whether there could be more IR excess stars besides the 26 already reported in the literature. Photospheric IR fluxes at the IRAS bands were calculated by using

$$\label{eqn}
F_{\nu }=6.347\times 10^{4}\frac{\pi ^{2}R^{2}}{\lambda ^{3}}\frac{1}{\exp \left( \frac{14388}{\lambda T}\right) -1}\, [Jy]$$

where $ \pi  $ is parallax in arcseconds, $ R $ is stellar radius in solar radii, $ \lambda  $ is wavelength in $ \mu  $m, and $ T $ is stellar effective temperature in Kelvins [@myPhD]. In Equation \[eqn\], $ R $ and $ T $ values were calculated from the $ M_{v} $ versus $ R $ or $ T $ relations [@AQ] where $ M_{v} $ values were determined from apparent visual magnitude (from SIMBAD) and *Hipparcos* distance data. Uncertainties of IR fluxes ($ \Delta F_{\nu } $) were calculated from

$$\label{Ferror}
\Delta F_{\nu }=F_{\nu }\left( \pi _{\circ },R_{\circ },T_{\circ }\right) \left[ \frac{2\Delta \pi }{\pi }+\frac{2\Delta R}{R}\right] \, [Jy]$$

where flux uncertainty due to $ \Delta \mathrm{T} $ is negligible (less than 0.02% for a given 1% error in T at 10,000K). Average flux uncertainties due to $ \pi  $ and $ R $ uncertainties are 3% and 4%, respectively. If we define the significance of IR excess ($ r_{\nu } $) as excess IR flux normalized by the uncertainty, then it can be calculated by $ r_{\nu }=(F_{IRAS}-F_{\nu })/\Delta F $ where $ \Delta F $ is the total flux uncertainty due to $ \Delta F_{\nu } $ and $ \Delta F_{IRAS} $ ($ F_{IRAS} $ and $ \Delta F_{IRAS} $ stand for flux value and flux uncertainty value from the IRAS catalog, respectively). $ \Delta F_{\nu } $ and $ \Delta F_{IRAS} $ were added in quadrature to calculate the total flux uncertainty ($ \Delta F $). We define the *bona–fide* Vega–like stars to be those that show significant IR excesses, $ r_{\nu }\geq 3.0 $, at three or more IRAS bands, with the most prominent excess at 60 $ \mu  $m. We have found that 51 additional stars show significant IR excesses ($ r_{\nu }\geq 3.0 $) at both 12 and 25 $ \mu  $m. However, only 14 of them turned out to be legitimate Vega–like star candidates. The other 37 stars are either luminosity class $ III $ stars (whose IR excesses would not arise because of a circumstellar dust disk) or stars whose excess radiation can easily be explained with a nearby companion star within the IRAS beam. The new Vega–like candidates are summarized in Table \[new\] with their $ r_{\nu } $ values at 12 and 25 $ \mu  $m. Determining the $ f $ values for the Vega–like candidates with only 12 and 25 $ \mu  $m IR flux measurements is difficult, because, for most of the cases, stellar photospheric flux dominates compared to any excess at these wavelengths; thus a slight error in the photospheric flux calculation results in a large error in $ f $ values. For this reason, we have not taken these stars into account in our consideration of $ f $ versus age relation (see below).

The photospheric flux calculated from the @Plets99’s [-@Plets99] empirical relation between the visual magnitude and the IRAS 12 $ \mu  $m magnitude is always higher than the photospheric flux values calculated by using equation \[eqn\]; thus the significance of the IR excess for most of the new Vega–like stars falls below the 3 $ \sigma  $ threshhold when @Plets99’s method used. Therefore, these 14 new candidates have to be treated with care. We considered two different sets of Vega–like stars: (1) using *al*l proposed Vega–like stars (case A, N=26) and (2) using only the *bona–fide* stars (case B, N=20). The second column of Table \[AVegas\] indicates the case(s) to which the star belongs. Our conclusion, discussed below, does not depend on the choice of case.

We assume that all of the A stars in the sample are post–ZAMS stars. We make that assumption because of simple timescale arguments (the ratio of $ <10 $ Myr old stars to the number of 100–300 Myr old stars should be of order $ <10/200 $ or 5%), and because we expect pre–ZAMS A stars to be located generally in star forming regions, which would make them easy to identify. There is no obvious age difference between field A–type stars and A–type Vega–like stars within 50pc with both groups running the gamut from very young (50 Myr) to old (1 Gyr). This result (and those in @Silverstone and @Song00) contrasts with @Habing99’s (1999) claim of the Vega phenomenon ending sharply at around 400 Myr.

We have checked whether a correlation exists between ages and dust properties by comparing our estimated ages of A–type Vega–like stars and their fractional IR luminosities, $ f\equiv (L_{IR}/L_{*}) $, found in @myPhD. Unfortunately, a plot of $ f $ versus age is not very informative mainly because of the large uncertainties of the estimated ages for individual stars. Therefore, we divided the Vega–like stars into two groups, one for the stars younger than 200 Myr and the other for stars older than 200 My, and calculated each group’s average $ f $–value (Table \[fages\]). Clearly, the younger A–type Vega–like stars have higher $ f $ values compared to those of the older ones (case–independent). However, we cannot more accurately quantify this relation because the uncertainties in $ T_{eff} $ and $ \log g $ are large.

Summary and Discussion
======================

In an attempt to determine the ages of A–type Vega–like stars, we have used a technique involving *uvby$ \beta  $* photometry and theoretical $ \log T_{eff}-\log g $ evolutionary tracks. In addition, we have applied corrections for the effects of rapid rotation. As a test of this procedure, we have estimated the ages of a few open clusters and find that our values are in good agreement with their standard ages. We then applied this age dating method to the 200 A–type stars within 50 pc with known $ v\sin i $ values. Thirteen percent of these A stars have been reported as Vega–like stars in the literature and their ages run the gamut of very young (50 Myr) to old (1 Gyr) with no obvious age difference compared to the field A–stars. The younger Vega–like stars have higher $ f $ values compared to those of the older ones.

Vega–like stars are closely related to the $ \lambda  $ Boo stars. These are metal–deficient A–stars with IR excesses. Vega itself is discussed as a possible member of the $ \lambda  $ Boo class [@HR]. An age determination of $ \lambda  $ Boo stars was presented by @IB based on the assumption that $ \lambda  $ Boo stars are main sequence stars. However, @HR argue that $ \lambda  $ Boo stars are probably pre–main sequence stars. If $ \lambda  $ Boo stars are indeed closely related to the Vega–like stars, then, based on our determination of the main sequence nature of Vega–like stars, it is likely that most of the $ \lambda  $ Boo stars are main sequence stars.

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  ------------ ---------- ------------ ------------------------------ ------------ ----------------- --------------- ---------------- ------------ -------------------- ---------------- ------------- ------------- --------------
     [HD]{}     [Case]{}    [Sp.]{}     [$ f\equiv L_{IR}/L_{*} $]{}                [$ v\sin i $]{}                                                                                                                  
                            [type]{}       [$ \times 10^{3} $]{}                     [$ m_{1} $]{}    [$ c_{1} $]{}   [$ \beta  $]{}   [(km/s)]{}   [$ \log T_{e} $]{}   [$ \log g $]{}    [lower]{}     [best]{}      [upper]{}
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  ------------ ---------- ------------ ------------------------------ ------------ ----------------- --------------- ---------------- ------------ -------------------- ---------------- ------------- ------------- --------------

  : A–stars with IR excesses\[AVegas\]

  -------- ------------------- --------- -------------- -------------- --------
     HD           other           Sp.                       Remark     
   number         name           type     12 $ \mu  $m   25 $ \mu  $m  
    2262     $ \kappa  $ Phe      A7V         3.8            3.3       
    6961         33 Cas           A7V         3.7            3.6       
   18978         11 Eri           A4V         3.9            3.5       
   20320         13 Eri           A5m         4.1            3.1          SB
   78209         15 UMa           A1m         5.0            3.7       
   87696         21 LMi           A7V         3.4            3.3       
   103287    $ \gamma  $ UMa      A0V         4.8            4.2          SB
   112185   $ \epsilon  $ UMa     A0p         7.5            6.4          SB
   123998     $ \eta  $ Aps       A2m         5.0            4.0       
   137898        10 Ser          A8IV         4.0        4.0$ ^{*} $   
   141003    $ \beta  $ Ser      A2IV         5.0            4.0        Double
   192696        33 Cyg         A3IV–Vn       5.7            4.7          SB
   203280    $ \alpha  $ Cep     A7IV         10.2           5.5       
   214846    $ \beta  $ Oct     A9IV–V        7.1            5.8       
  -------- ------------------- --------- -------------- -------------- --------

  : New A–type Vega–like candidates\[new\]

$ ^{*} $100 $ \mu  $m excess, 25 $ \mu  $m excess $ r_{\nu }=1.9 $

  --- ------- ---------- --------------------- ------
              Number         Average $ f $     
              of stars    ($ \times 10^{3} $)  
   A  $ < $   200 Myr             12            1.79
      $ > $   200 Myr             14            0.18
   B  $ < $   200 Myr              7            0.71
      $ > $   200 Myr             13            0.17
  --- ------- ---------- --------------------- ------

  : Ages and $ f\protect $ values of Vega–like stars\[fages\]

[^1]: Web version of BDA (Open clusters database, @BDA) http://obswww.unige.ch/webda
