−4 −3 −2 −1 0 1 2 3 4x 10 −3 Epoch O−C (days)
Figure 4.2: BP Vel (O-C) residuals of minimun light showing the ten epochs in the four filters.
P = Pest +b where b is the slope in the line. The new period was found to be P = 0d.265034
±0d.000048. From a linear least-squares fit of the ten times of observed minimum
in our data, the ephemeris was found to be
HJD0 = 2454147.934346(±0.00002) + 0.265034(±0.00005)
The reduced data was phased with the new period and then modeled with PHOEBE.
4.3
Modeling with PHOEBE
The PHOEBE configuration option of an over-contact system of the W UMa-type was attempted for the first fit. The results were not satisfactory however, the difference in the minimum depths was not sufficiently matched and the temperature of the primary diverged rather dramatically.
An over-contact system without thermal contact was found to be the best configuration option for BP Vel. This option, while complying with the contact configuration condition, that the potentials are equal Ω1 = Ω2, allowed the temperatures of the primary and
the secondary to differ. The different temperatures drove the model to match BP Vel’s observed minimum inequality. The primary minimum is around 0.15 mag deeper than the secondary eclipse in all filters.
24 CHAPTER 4. BP VELORUM
method would be employed for further light curve analysis. From Figure 4.1, a range of mass ratio values to test was decided upon. The values of q that we used, ranged from 1.2≤q ≤2.4. 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 1.46 1.48 1.5 1.52 1.54 1.56 1.58x 10 −5 mass ratio (q) S
Figure 4.3: Goodness of fit parameter,[S(q) =P
ωr2 =P
ω(O−C)2]vs. the mass ratio
q=m2/m1, determined using models produced by PHOEBE for BP Vel. The circled value
is theq used as a start point in the unspotted and spotted models.
The WD code was implemented on fixed values of q within this range, leaving the relative monochromatic luminosities (L1(B, V, R, I)), the primary and secondary tem-
peratures (T1, T2), the potential (Ω1 = Ω2), the inclination (i) and the limb-darkening
coefficients (X1(BV RI) and X2(BV RI)) free to converge. The gravity-darkening coeffi-
cient and the bolometric albedo were fixed at g1 = g2 = 0.32 and A = 0.5. The values
for q and A follow the convective envelope model for contact binaries taken from Lucy (1968a) and Ruci´nski (1969). The spectral type of BP Vel is KI so the initial value chosen for the temperatures of both components was 5000K (Lapasset et al., 1996). The limb- darkening coefficients were calculated using a square root limb-darkening law after each set of iterations.
The fit of the model was judged on the sum of the squared weighted residuals (S = Σ(wr2) = Σ(O
−C)2). The values of S were plotted against q in Figure 4.3. The q that
gives the best model isq = 2.1 compared with q= 1.9 found by Lapasset et al. (1996).
4.3.1
Unspotted Model
4.3. MODELING WITH PHOEBE 25 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Total Flux Phase (a) Blue filter
0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Total Flux Phase (b) Visual filter 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Total Flux Phase (c) Red filter 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Total Flux Phase (d) Infrared filter Figure 4.4: The unspotted WD model inBV RI filters.
The unspotted model was generated leaving the parameters described in the preced- ing section and the mass-ratio free. The goodness of fit for the unspotted model was
S = 1.41×10−05
. The observed light curves show unevenness at their maxima. This is more pronounced in the visual and red filters which rules out systematic errors from the instruments. If it were a case of a systematic error, one would expect it to be continued through the other filters and that there would be just as much scatter at minimum light. Lapasset et al. (1996) discovered a model with a star spot added to the secondary com- ponent produced the best fit. The placing of the star spot on the secondary component at the equator was repeated for this analysis.
4.3.2
Spotted Model
As shown in Figure 4.4 it is obvious there is some effect toying with the light curve near phase −0.25 in the Visual and Red filters. A star spot was placed on the secondary component at a colatitude of 90◦
. The other spot parameters (longitude, angular radius and temperature), were allowed to vary with the WD code through PHOEBE, until con- vergence. The system’s other parameters which had previously converged, were adjusted
26 CHAPTER 4. BP VELORUM 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Total Flux Phase (a) Blue filter
0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Total Flux Phase (b) Visual filter 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Total Flux Phase (c) Red filter 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Total Flux Phase (d) Infrared filter Figure 4.5: The spotted WD model in BV RI filters.
(a) phase = -0.25 (b) phase = 0
(c) phase = 0.25 (d) phase = 0.5 Figure 4.6: 3-D plots of the surface of BP Velorum (with spot).
4.4. SOLUTION 27