Astrophysics PS7

Author

Will St. John + Caedan Miller + Kiki Murphy

Problem 1

Consider a spectroscopic binary system with a period of 2.17 days. The orbits have zero eccentricity. The maximum Doppler shifts of the fainter and brighter stars are 112.6 km/s and 71.4 km/s, respectively.

  1. How can an astronomer deduce the eccentricity of the orbits in a spectroscopic binary system? What observations, measurements, modeling, etc. are needed?
  2. Assuming a statistical value for , determine the total mass of the spectroscopic binary system (in solar masses). Explain your choice of statistical value.
  3. What is the mass of the fainter star?
  4. What is the mass of the brighter star?

Answer:

import astropy.units as u
import astropy.constants as c
import numpy as np

P = 2.17 / 365 * u.yr
V1 = 112.6 * u.km / u.s
V2 = 71.4 * u.km / u.s
sin3i = 2/3

total_mass = P / (2 * np.pi * c.G) * (V1 + V2) ** 3 / sin3i
print(f"total mass: {total_mass.to(u.M_sun):.2f}")

ratio21 = V2 / V1
m2 = total_mass / (ratio21 + 1)
m1 = ratio21 * m2
print(f"mass of fainter star: {m1.to(u.M_sun):.2f}")
print(f"mass of brighter star: {m2.to(u.M_sun):.2f}")
total mass: 2.10 solMass
mass of fainter star: 0.82 solMass
mass of brighter star: 1.29 solMass

Problem 2

Assume that two stars, “1” and “2”, are in circular obits about a common center of mass and are seperated by a distance “a”. Assume an orbital inclination of i and stellar radii of “R1” and “R2” for the two stars.

  1. Find an expression for i that just barely produces an eclipse. Hint: consider the Figure 7.8 from the textbook

The Kepler space telescope revolutionzed our understanding of extrasolar planets. Hundreds of scholarly manuscripts have been published about these results. One intriguing results is the Kepler-1661 system presented by Socia et al. (2020). Use the information in the paper and apply the equation you deduce for i to calculate:

  1. the minimum inclination angle for an eclipse to occur at the minimum separation of the primary and secondary stars,
  2. and the minimum inclination angle for an eclipse to occur at the maximum separation of the primary and secondary stars.
  3. Do your constraints on i agree with the results presented by Socia et al. (2020)?

Answer:

\[i > \cos^{-1}\left(\frac{R_1 + R_2}{a}\right)\]

R1 = 0.76 * u.R_sun
R2 = 0.276 * u.R_sun
P = 28.2 / 365 * u.yr
m1 = 0.84 * u.M_sun
m2 = 0.262 * u.M_sun
e = 0.112
a = (P ** 2 * c.G * (m1 + m2) / (4 * np.pi ** 2)) ** (1/3)
i = np.arccos((R1 + R2) / a)
i.to(u.deg)

\(88.526554\mathrm{{}^{\circ}}\)

Problem 3

  1. Consider an eclipsing, spectroscopic binary system. The orbital period is 5.51 years and the eccentricity is zero. The maximum radial velocity of Star A is 4.7 km/s and the maximum radial velocity of Star B is 17.4 km/s. The time period between first contact (or beginning of eclipse) and minimum light is 0.48 days. The length of the primary minimum is 0.84 days. The apparent bolometeric magnitudes of maximum, primary minimum, and secondary minimum are 6.41, 9.12, and 6.66, respectively. Find the following:
  1. The ratio of stellar masses
  2. The sum of stellar masses (in solar masses)
  3. The individual masses of Star A and Star B
  4. The radii of Star A and Star B
  5. The ratio of the effective temperature of the two stars.

Answer:

e = 0
P = 5.51 * u.yr
v_A = 4.7 * u.km / u.s
v_B = 17.4 * u.km / u.s
v = v_A + v_B
dt_fc_to_ml = 0.48 / 365 * u.yr
primary_minimum_length = 0.84 / 365 * u.yr
B_0 = 6.41
B_p = 9.12
B_s = 6.66

mAmB_ratio = v_B / v_A

total_mass = P / (2 * np.pi * c.G) * (v_A + v_B) ** 3 / sin3i
print(f"total mass: {total_mass.to(u.M_sun):.2f}")


m_B = total_mass / (mAmB_ratio + 1)
m_A = mAmB_ratio * m_B
print(f"Mass of Star A: {m_A.to(u.M_sun):.2f}")
print(f"Mass of Star B: {m_B.to(u.M_sun):.2f}")

R_s = v * dt_fc_to_ml / 2
R_l = v * primary_minimum_length / 2 + R_s

print(f"Radius of Star A: {R_l.to(u.R_sun):.2f}")
print(f"Radius of Star B: {R_s.to(u.R_sun):.2f}")


temp_ratio = ((B_0 - B_p) / (B_0 - B_s)) ** (1/4)
print(f"Temperature ratio (B to A): {temp_ratio:.2f}")
total mass: 3.38 solMass
Mass of Star A: 2.66 solMass
Mass of Star B: 0.72 solMass
Radius of Star A: 1.81 solRad
Radius of Star B: 0.66 solRad
Temperature ratio (B to A): 1.81