Reference no: EM133029154

Assignment -

Problem 1 - Tides

a. Tides on the Earth are dominated by the gravitational effect of the Moon, but the Sun also plays a role. Following our discussion in class, show that the relative tidal force on the Earth due to the Sun and the Moon depends only on M_Sun, M_moon, the distance from the Earth to the Moon, and the distance from the Earth to the Sun. Focus only on the relative forces along a line connecting the centres of the Earth, Moon, and Sun. Given the values of these parameters from Appendix C in your textbook, how much stronger is the tidal force due to the Moon than the Sun?

b. Calculate the tidal acceleration between your head and your toes due to the Moon. Compare this value with the tidal acceleration between your head and your toes due to the Earth, and the overall acceleration due to the Earth's gravity.

c. In the movie Interstellar, an astronaut crosses the event horizon of a 100 million solar mass black hole - and survives! Is this reasonable? Base your answer on calculating the tidal acceleration on an astronaut (head and toes!) at the black hole's event horizon (see C&O section 17.3, particularly Equation 17.27), and compare your result with your calculations from part b.

d. What would happen instead if the astronaut crossed the event horizon of a smaller black hole with a mass of only 10 solar masses? Show both your calculations and explain in words what the result implies.

Problem 2 - Mass loss in binary stars

Many stars are in binaries. Some stars have strong mass loss, at least for parts of their lives. Here, we will investigate how mass loss can affect the orbital parameters of binaries, and look at the eventual fate of the Earth-Sun system.

a. Show that if star 1 with mass m₁ is losing mass at a rate m^·₁, the orbital separation a will evolve as a^·/a = m^·₁/M, where M = m₁ + m₂ is the total mass of the system.

It will be easier if you start with substituting Ω = 2π/P into Kepler's third law, where P is the orbital period and Ω is the angular frequency. Write Kepler's third law in terms of Ω, a, G, and M.

Next, show that you can rewrite the orbital angular momentum L as:

L = ((m₂/m₁)/(1 + m₂/m₁)²)a²MΩ (1)

where is the reduced mass of the binary (start with C&O Eqn. 2.30 for L in a binary system, and assume e = 0).

Then show that

3(a^·/a)+2(Ω^·/Ω) = M^·/M (2)

From (1), find a similar relation for L^·/L. You can ignore the effects of mass loss on m₁ in the (m₂/m₁)/(1 + m₂/m₁)² term; this would be a second order effect that we won't consider here.

The initial change in L, however, is driven by the change in mass only, so write another expression for L^·. Then the system responds to this change.

With the above relationships, you should be able to solve for a^·/a in terms of m^·₁ and M only.

b. How much do you expect the Earth-Sun system to expand over the 10 Gyr main-sequence lifetime of the Sun, given that the Sun is losing mass in the form of light? And how much given the mass-loss rate of 3 x 10^-14 M·yr^-1 associated with the solar wind?

c. As a giant, the Sun will lose mass much faster, ending its life as a 0.6 M· white dwarf. At what distance will this leave the Earth in its orbit?

Problem 3 - Jupiter's luminosity

The Sun is in thermal equilibrium because its thermal timescale (the time to radiate away its gravitational potential energy, or the Kelvin-Helmholtz timescale) is short compared to its age. However, smaller objects need not be in thermal equilibrium, and their radiation can be powered entirely by gravity.

a. Jupiter radiates more energy than it receives from the Sun by 8.7 x 10^-10 L·. Given Jupiter's radius and mass from C&O, compute its thermal timescale. Could gravitational contraction power this luminosity for Jupiter's entire lifetime of 4.5 Gyr?

b. Use conservation of energy to estimate the rate at which Jupiter's radius is shrinking to power this radiation. Assume Jupiter is a sphere of constant density. Give your answer in both cm year^-1, and km Gyr^-1.