3. For a spaceship of mass m in a circular orbit around a star of mass M, a gravitational force on the spaceship by the star acts as a centripetal force keeping the spaceship in a uniform circular motion, Fcentripetal = Fgravity, v² GMm m r r2 (1) where vt is the tangential speed of the spaceship in orbit and r is the radial distance from the star to the spaceship. You can express vt in terms of the circumference 2πr divided by the orbital period T, Vt = 2πr T Then, by substituting the above expression for vt into Eq. (1), we obtain the following relationship between T and r: T2 = 4π2 GM 3 (2) Eq. (2) is known as the Kepler's 3rd law for planetary motion and is very useful in astronomy. It is applicable to any body orbiting another body. Let's try some examples. (a) The orbital period of Mars is 687 days. What is Mars' orbital radius in astronomical units (AU)? One AU is defined as the average distance between the Sun and the Earth and is about 1.50 × 108 km. [In Eq. (2), all quantities have to be in the metric units.] (b) Cygnus X-1 is an x-ray binary system consisting of an invisible black hole and a visible 25 Mo companion star orbiting around each other every 134 hours. The black hole and the companion are separated by 0.20 AU. Determine the mass of the black hole in solar mass. [Hint: If two bodies are comparable in mass, you need to replace M in Eq. (2) with the total mass of the system, Mtotal. After you have calculated the total mass of the system, find the mass of the black hole by itself.]

3. For a spaceship of mass m in a circular orbit around a star of mass M, a gravitational force on the spaceship by the star acts as a centripetal force keeping the spaceship in a uniform circular motion, Fcentripetal = Fgravity, v² GMm m r r2 (1) where vt is the tangential speed of the spaceship in orbit and r is the radial distance from the star to the spaceship. You can express vt in terms of the circumference 2πr divided by the orbital period T, Vt = 2πr T Then, by substituting the above expression for vt into Eq. (1), we obtain the following relationship between T and r: T2 = 4π2 GM 3 (2) Eq. (2) is known as the Kepler's 3rd law for planetary motion and is very useful in astronomy. It is applicable to any body orbiting another body. Let's try some examples. (a) The orbital period of Mars is 687 days. What is Mars' orbital radius in astronomical units (AU)? One AU is defined as the average distance between the Sun and the Earth and is about 1.50 × 108 km. [In Eq. (2), all quantities have to be in the metric units.] (b) Cygnus X-1 is an x-ray binary system consisting of an invisible black hole and a visible 25 Mo companion star orbiting around each other every 134 hours. The black hole and the companion are separated by 0.20 AU. Determine the mass of the black hole in solar mass. [Hint: If two bodies are comparable in mass, you need to replace M in Eq. (2) with the total mass of the system, Mtotal. After you have calculated the total mass of the system, find the mass of the black hole by itself.]