The orbital period of the binary system containing A0620-00 is \(0.32\) day, and Doppler shift measurements reveal that the radial velocity of the X-ray source peaks at \(457 \mathrm{~km} / \mathrm{s}\) (about 1 million miles per hour). (a) Assuming that the orbit of the X-ray source is a circle, find the radius of its orbit in kilometers. (This is actually an estimate of the semimajor axis of the orbit.) (b) By using Newton's form of Kepler's third law, prove that the mass of the X-ray source must be at least \(3.1\) times the mass of the Sun. (Hint: Assume that the mass of the \(\mathrm{K} . \mathrm{V}\) visible star- about \(0.5 \mathrm{M}_{\odot}\) from the mass-luminosity relationship-is negligible compared to that of the invisible companion.)

Step by step solution

01

Convert radial velocity and period to SI units

The given radial velocity is 457 km/s, which is equal to \(457 \times 10^{3} m/s\). The given orbital period is 0.32 day, which is equal to \(0.32 \times 24 \times 60 \times 60 s = 27648 s\).

02

Calculate the circumference of the orbit

For a circular orbit, the circumference is equal to the velocity times the period, \(C = v \times T\). Therefore, the circumference of the orbit is \(457 \times 10^{3} m/s \times 27648 s = 1.262 \times 10^{10} m\).

03

Calculate the radius of the orbit

The radius can be determined from the circumference by dividing by \(2\pi\). \(r = \frac{C}{2\pi} = \frac{1.262 \times 10^{10} m}{2\pi} = 2.01 \times 10^{9} m = 2.01 \times 10^{6} km\).

04

Use Kepler's Third Law

The simplified form of Kepler's third law is \(P^2 = a^3/M\), where \(P\) is the period in years, \(a\) is the semi-major axis (orbital radius) in AU (astronomical units, the average distance from Earth to the Sun), and \(M\) is the mass of the more massive body in solar masses. After substituting the orbital radius in AU, the period in years and rearranging, the mass becomes \(M = a^3/P^2\).

05

Convert radius to astronomical units and period to years

1 AU = 1.496 x 10^{8} km, and 1 year = 3.154 x 10^{7} s. The orbital radius in AU is \(2.01 \times 10^{6} km / 1.496 x 10^{8} km/AU = 0.013 AU\), and the period in years is \(27648 s / 3.154 x 10^{7} s/year = 0.000876 years\).

06

Calculate the mass

Substitute the values of the orbital radius and period into the rearranged form of Kepler's third law to find the mass of the X-ray source. \(M = (0.013)^3 / (0.000876)^2 = 3.1\) solar masses. By assuming that the mass of the visible star is negligible compared to that of the invisible X-ray source, this result suggests that the mass of the latter must be at least 3.1 times the mass of the Sun.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!