TV Lib: temperature and radius variations from photometry
Observations: 11, 13, 17, 18 March 2005
31 March 2005
Revised and Updated: 22 April 2005, 8 May 2005.
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.
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
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
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.
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
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
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):
alpha(t) is given by:
- 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;
the number of photons received at the telescope (per unit of area and
of time) is:
- DELTAW(t) is the energy emitted by a black body in the frequency
- k is Boltzmann constant,
- W(t) is the energy emitted by a black body in the whole spectrum;
the star radius R(t) over the distance to the star D is given by:
- 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
- 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
- tau=200s is the duration of one exposure;
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:
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).
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
Barnes T.G., Evans D.S. (1976) MNRAS 174 489
Bono G., Caputo F., Cassisi S., Incerpi R., Marconi M. (1997) ApJ 483
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
Telescope and camera configuration.
Computer and software configuration.
Pulsating stars and
Similar determinations of the temperature, radius and luminosity are