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## Homework Statement

(a) Consider a planet orbiting a star of mass M?, on a circular orbit with circular

velocity vp. What is the planet's orbital radius, in terms of vp and M?

(b) Now suppose that the star (carrying the planet along with it) enters a nearly radial

orbit around a black hole of mass M, with total energy E = 0. What is the star's

velocity, v?, relative to the black hole, as a function of M and the star's distance R

from the black hole?

(c) At what black hole distance R is the planet tidally stripped from the star by the

black hole? That is, solve for the radius R where the star's Hill radius equals the

planet's orbital radius. Using your answer from part (b), what is the star's velocity v?

at this radius? Express this velocity in terms of M?, M, and vp. (2 points)

(d) Assume that the planet was moving at velocity vp relative to the star, in the same

direction as v?, at the time that it was tidally stripped. Assuming that the star and

planet are at roughly the same distance from the black hole, what is the planet's total

orbital energy? Show that the planet is now unbound from the black hole, and can

therefore escape to large distances.

(e) Using the total energy you calculated in part (d), what is the asymptotic velocity

of the planet at large distances from the black hole, v1? Again, express your answer

in terms of M?, M, and vp. Assuming that the planet's original orbital velocity was

vp 300 km/s, and that the black hole was roughly a million times more massive than

the star, M = 106M?, what is the maximum ejection velocity v1 that the planet can

achieve through this mechanism?

## Homework Equations

F=ma, binary self attraction: [tex] a_b=\frac {Gm_b}{r^2_b}[/tex]

Tidal force pulling on binary: [tex] a_t=\frac {2GM}{R^3}r_b[/tex]

## The Attempt at a Solution

The circular orbit is simple enough for part A:

[tex] \frac{v^2m_p}{r_p} = \frac {GM_\star m_p}{r^2_p}[/tex]

[tex] r_p=\frac {GM_\star}{v^2_p}[/tex]

And, if total energy is zero then kinetic and potential must be equal for part B:

[tex] \frac{1}{2}M_\star v^2_\star = \frac {GM_\bullet M_\star}{R}[/tex]

[tex] v_\star=\sqrt{\frac {2GM_\bullet}{R}}[/tex]

Now, more interesting is the radius at which the black hole pulls the planet and star apart for part C:

[tex] \frac {Gm_\star}{r^2_p}\leq\frac {2GM_\bullet}{R^3}r_p[/tex]

[tex]R \geq r_p(\frac{2M_\bullet}{M_\star})^{\frac{1}{3}}[/tex]

Recall that [tex]r_p=\frac{GM_\star}{v^2_p}[/tex]

Part D has me a little flustered. If I understand it conceptually it is saying that now that the binary system has been disrupted by the black hole the planet should have enough energy to escape the black hole system. So I believe that:

[tex] \frac{v^2_p m_p}{r_{p\bullet}} > \frac{GM_\bullet m_p}{R^2_{p\bullet}}[/tex]

where the velocity of the planet is now the velocity of the star plus the orbital velocity of the planet around the star.

I plugged everything in but it is one ugly looking equation. Once R and the two velocities are plugged in it is pretty unwieldy. I just want someone elses opinion on my methodology. Or perhaps I made a mistake in my algebra.

Thanks for any help.