Photometric Analysis of π Sco
by Bret Darby Lehmer
The π Sco spectroscopic binary system has recently been discovered to possess
photometric variations. The system (magv = 2.89) has been known for nearly a century
to be a binary with a period of 1.57 days. Parallax measurements from the Hipparcos
satellite imply the stars are 140.8 pc from earth. The stellar components of the π Sco,
spectral types B0.5V and B2V, are located in the upper left hand corner of the H−R
diagram. These stars are extremely hot (22,000 − 30,000 degrees Kelvin) and massive (5
− 10 solar masses) and thus relatively rare in nature. The intense radiation, mass loss,
and stellar wind collision effects combined with the tightness of the orbit make studying
the system difficult. By studying the variation in the system brightness however,
physical information about the stellar components can be deduced. Here the methods
and details of the photometric analysis of π Sco are discussed followed by an explanation
and final report of the determined physical characteristics.
The Hipparcos satellite has gathered photometric data of π Sco with good enough
phase coverage to make the analysis discussed above possible. Based on the inferences
above and radial velocity data obtained by Dr. Reed Riddle, initial parameters can be
formulated to create a model light curve. The technique implemented here is to take the
model based on the initial parameters and tweak the factors until the model fits the
Hipparcos data.
Two computer algorithms were used to fit the model light curve to the data. The
algorithms were Nightfall (R. Wichmann) and The Wilson − Devinney Code for
Computing Binary Observables (R. Wilson / E. Devinney). Nightfall is C based software
that uses a Simplex Algorithm to fit model light curves to photometric data sets. The
program has a graphical interface that creates plots and animations of orbits. The
graphical interface is set up to display parameter values and option buttons that make the
fitting process simple. The initial parameters (mass ratio, system inclination,
temperatures, and fill factors based on Roche Lobe filling) are entered into the main
interface screen. Other options can be included in the computation such as system mass,
separation, star spots, eccentricity, line profile measurement, limb darkening, reflection,
and model atmosphere. The parameters are then varied to optimize the fit of the light
curve and eventually yield a solution set of the orbit. The Wilson − Devinney Code for
Computing Binary Observables uses a different technique for fitting light curves. The
code consists of two programs. One program (LC) will take entries from a data file and
compute a light curve based on the entries. The second program (DC) takes the
parameter entries in combination with photometric data and varies the entries so a model
of the orbit can be formed.
When applied to π Sco the two algorithms provided an excellent means for
checking work. The starting point of the process began with Nightfall. By varying
parameters two at a time to optimize the fit of the model a nice steady convergence to a
"good" solution was reached. Once a pretty good idea of what the parameters were, it
was time turn to the Wilson − Devinney software. Instead of using the DC program to
vary the parameter values, a plotting program was written that would compare the model
light curve produced by LC with the actual data . The plots were compared by
computing a sum of errors and an RMS value of the errors. Then the parameters were
adjusted by hand in the input files to better the fit. The result of the painstaking tinkering
was a set of physical solutions for the system that came out to be consistent with both
programs (see figures at final pages).
An interesting result of the analysis is that the stars of π Sco do not eclipse (i = 42
degrees) despite the systems photometric variations. Instead the light intensity variations
are a product of the geometry of the stars themselves. The ellipsoidal shape causes the
effective temperature to be variable over the surface of each star. This in turn effects the
luminosity and since the stars are tidally locked the brightness variations are periodic.
Now a pretty solid picture of the π Sco system has been developed. The output
parameters from the analysis have been determined with good confidence. This,
however, is by no means the final word on π Sco. In the near future, radial velocity data
will be combined with the photometric data and a spectroscopic/photometric study will
be implemented to further narrow the physical parameters. Later on, more observations
of the system will be made for the purpose of analyzing the wind collision aspect.
Figures
On the pages to follow are printouts and plots created by both Nightfall and the
Wilson−Devinney code. Figures 1 and 2 are produced by the Wilson−Devinney and a
customized plotting program, while figures 3 and 4 come out of Nightfall.
Figure 1: Wilson−Devinney produced output of physical parameters.
mpage nref mref ifsmv1 ifsmv2 icor1 icor2 ld
1
JDPHS
2
1
2
0
0
J.D. zero
0
0
Period
1
dPdt
fract. sd.
noise
0.0000
0.0000D+00
1
8502.262560 0.1570080000D+01 0.000000D+00
JD start
JD stop
8500.000000
JD incr
8900.000000
0.010000
Ph start
Ph. stop
Ph incr
Ph norm
0.0000
1.0000
0.0100
0.2500
MODE IPB IFAT1 IFAT2 N1 N2 Arg. Per
2
Ph. shift
0
1
1
30 30
dPerdt
Th e
3.000000 0.00000D+00
Nspot1
Nspot 2
1
1
F2
Vgam
Incl
g1
.00000
14.8514
1.0000
1.0000
0.0000
42.570
1 .000
3.0116
wv lth
Alb 2
2.2433
1.000
1.000
L1
0.447100 5069.25820
L2
Pot 1
Pot 2
1756.30881 0.800
x2
0.800
y1
0.000
1.000
M2/M1
0.38564D+01 0.39506D+01
x1
V FAC
g2
F1
Alb 1
V UNIT(km/s)
1.00
s−m axis
T2
138472375.
0.00000
ecc
T1
seed
0.70500
y2
0.000
0.4788D+03
x1(bolo) x2(bolo)
0.799
el3
0.799
opsf
0.0000 0.0000D+00
y1(bolo)
0.000
m zero
2.810
y2(bolo)
0.000
factor
1.0000
STAR CO−LATITUDE LONGITUDE SPOT RADIUS TEMP. FACTOR
1
0.00000
0.00000
0.00000
0.00000
2
0.00000
0.00000
0.00000
0.00000
Star
M/Msun
(Mean Radius)/Rsun
M Bol
Log g (cgs)
1
10.49
4.80992
−5.79
4.09
2
7.40
3.78877
−4.00
4.15
Figure 2: Model light curve produced by Wilson−Devinneyplotted over photometric data.
Figure 3: Nightfall produced output of physical parameters.
Figure 4: Nightfall model light curve plotted over photometric data