# TV Lib: temperature and radius variations from photometry

## Observations: 11, 13, 17, 18 March 2005

###### Michel Bonnardeau

31 March 2005

Revised and Updated: 22 April 2005, 8 May 2005.

### Abstract

* Two-color time-series were obtained for this pulsating RR Lyrae
star. The pulsation is too early when compared with the ephemeris. The
temperature and the radius variations are computed from different models.
*

### Introduction

TV Lib is a pulsating star with a period of 6.5 hours. It is classified
as a RR Lyrae RRab with an asymmetric light curve.

However, this star is considered as "deviant": when compared with others
RRab, its period is much smaller for its high metallicity (Bono et al
(1997)).

### Observations

The observations were carried out with a 203 mm SC telescope, Johnson
B and V filters in a filter wheel and a SBIG ST7E camera (KAF401E CCD).
The B and V filters are used alternatively. 95 images were obtained with
the V filter and 94 with the B filter. Each exposure is 200 second long
(except for two V exposures that are 60 s long).

The comparison star is Tycho 5581-00743 with Johnson magnitudes B=11.125
and V=10.303 (after conversion from the Tycho magnitudes, see Mamajek
et al (2002)). The check star is GSC 5581-00764, with the observed magnitudes
B=13.050 (standard deviation 0.081) and V=13.343 (0.015). A sample image
is here.

The following light curve allows to time the pulse:

*The error bars are the +/- one-sigma statistical uncertainties.*

The pulse maximum is at: 2,453,443.665 HJD.

All the light curves, including those for the check star, are here.

### Comparison with the ephemeris

The GCVS has a 91 yr old ephemeris for the pulse maxima:

T0 = 2,420,017.3015 (6 Sep 1913)

P0 = 0.269624031 day.

The pulse at 443.665 is then pulse number 123,974 and it should be at
443.671. __It is 9 minutes too early__.

### Computation of the temperature

The temperature (°K) of the star surface may be estimated from the B-V
magnitude owing to Reed (1998):

B-V = -3.684.log(T) + 14.551 (when T <9100°K)

I proceed the following way:

The measured magnitudes are folded in phase with the period P0. The
B and V magnitudes are not measured simultaneously, therefore the magnitudes
at any phase are computed by interpolation.

The measured magnitudes are corrected for the transformations coefficients:

B = b + Tb*Tbv*[(b - v) - (Bc - Vc)]

V = B - [(Bc - Vc) + Tbv*[(b - v) - (Bc - Vc)]]

where b, v are the measured magnitudes, Bc, Vc the magnitudes of the comparison
star. Tb and Tbv are the transformations coefficients. They were measured
on 7 Jan 2005 from the Landolt 95 field, of course with the same setup
as for the present observation:

Tbv = 0.967

Tb = -0.043

The temperature versus the phase is then:

*The red dot lines are the range of the temperatures corresponding to
the +/- 1-sigma statistical uncertainties on the magnitudes. The thick
red line is smoothed.*

### The distance

RR Lyrae stars are standard candles with all roughtly the same absolute
average V magnitude of 0.6. With an observed average V magnitude of 11.531,
the distance to TV Lib is then D=1.5kpc.

Layden (1997) has a more sophisticated distance estimation of D=1.42kpc.

### Computations of the radius

I use 3 different methods to compute the star radius as a function of the
phase:

The surface brightness method: Barnes and Evans (1976)
considered a large number of measured star diameters (from occultation,
interferometry, etc) and discovered an empirical relation between the
diameter, the V magnitude, the B-V color. The star radius may then be
computed, independantly of the temperature;
The bolometric method: I compare the absolute magnitude
of the star with the one of the Sun. Knowing the radius of the Sun and
the effective temperatures of the Sun and of TV Lib, I compute TV Lib
radius;
The photon counting method: I assume the star radiates like a black body
with the computed effective temperature. By counting the photons I detect at the
telescope I compute the radius.
### The radius from the surface brightness method

According to Barnes and Evans (1976) one has:

4.2207 -0.1*V -0.5*log(phi) = 3.964 - 0.333*(B-V)

where phi is the angular diameter in arc milliseconds. With the distance
D=1420pc, the radius can readily be computed:

*The dot lines are computed from the +/- 1-sigma statistical uncertainty
on the magnitude. The thick line is smoothed.*

### The radius from the bolometric method

The absolute V magnitude Mv (at 10pc) of TV Lib may be computed from the measured
V magnitude and the distance D=1420pc:

Mv = 5 + V - 5*log(D)

The Sun has an absolute V magnitude of Mov=4.89. The
luminosity L(t) as a function of phase t of TV Lib is then
(in Sun luminosity unit):

where To=5780°K is Sun effective temperature, and BC is the bolometric correction.
BC as a function of the effective temperature is given by Reed (1998) as:

BC(T) = C6*(log(T)-4)^4 + C7*(log(T)-4)^3 + C8*(log(T)-4)^2 + C9*(log(T)-4) + C10

with C6=-8.499 C7=13.421 C8=-8.131 C9=-3.901 C10=-0.438.

The luminosity being proportional to R^2 and T^4, the radius R(t) is
(in solar radii unit):

### The radius from the photon counting method

Assuming the star radiates like a black body, the radius may be computed
as a function of the distance to the star:

the number of photons Nv emitted by the star in the V band is, as a
function of phase t and per unit of time (Stefan-Boltzmann law):

where:
- R(t) is the star radius,
- sigma is Stefan constant,
- E is the energy of a photon in the V band: E=h.nu where h is the
Planck constant and nu the photon frequency. nu=c/lambda where c is
the velocity of light and lambda the wavelenght,
- alpha(t) is the proportion of the energy emitted in the V band.
alpha(t)=1 for the whole spectrum. It depends upon the phase t because
is it T(t) dependant;

alpha(t) is given by:

where:
- DELTAW(t) is the energy emitted by a black body in the frequency
band DELTAnu,
- k is Boltzmann constant,
- W(t) is the energy emitted by a black body in the whole spectrum;

the number of photons received at the telescope (per unit of area and
of time) is:

where:
- nV(t) is the number of ADU counted by the camera (actually,
the magnitudes are computed from this number),
- g is the number of electrons/ADU. According to SBIG, g=2.3 for a ST7E
camera,
- qV=0.52 is the quantum efficiency of the CCD in the V band,
- dzeta_t=0.8 is the telescope transmission,
- dzeta_a=0.405 is the atmospheric transparency (it is assumed to be 0.7 for
an air mass=1),
- d1=203mm and d2=81mm are the diameters of the telescope and of the
central obstruction,
- tau=200s is the duration of one exposure;

the star radius R(t) over the distance to the star D is given by:

where Nv0(t) is Nv(t) for R=1 solar radius.
The V filter is centered at lambda=0.545micron. I make the simplification
that its transmission curve is box-shaped with a width of 0.083micron
and a height of 1. The star radius in solar radii for D=1.42 kpc is then:

### Interstellar extinction

The interstellar extinction has not been taken into account. Its effect
is to increase V and B-V. Correcting for the interstellar extinction
would increase the computed temperature.

For the radius, correcting V would increase the radius, but the correction
B-V is to diminish it. Therefore the interstellar extinction would have
little effect on the radius computation. This was also noted by Barnes
and Evans (1976).

### Conclusions

One pulse is observed and found 9mn too early when compared with the
91-yr old ephemeris.
The temperature is found to increase when the star becomes brighter
(from 5800 to 7200°K).
The computations of the radius with the surface brightness and the bolometric
methods give a radius of 6 solar radii with an amplitude of 10%. The first
method is independant of the temperature determination. The radius goes
inwards just before the maximum of brightness and of temperature;
The computation of the radius with the photon counting method gives
the same behavior as the other two methods, but with a smaller radius
of 4 solar radii. This discrepancy may come from the star not being a
black body, the atmospheric transparency being overated (I have an oceanic
climate).
### References

Barnes T.G., Evans D.S. (1976) MNRAS __174__ 489

Bono G., Caputo F., Cassisi S., Incerpi R., Marconi M. (1997) ApJ __483__
811

Layden A.C. (1997) Astron. Journal __108__ 1016

Mamajek E., Meyer M., Liebert J. (2002) AJ 124 1670M (appendix C)

Reed C. (1998) J. RAS of Canada __92__ 36

### Technical notes

Telescope and camera configuration.

Computer and software configuration.

### Astronomical notes

Pulsating stars and
its links.

Similar determinations of the temperature, radius and luminosity are
HERE.