Download Homework 3

AST111 PROBLEM SET 3
Useful equations:
A Hohmann transfer orbit between two circular orbits with radius R (for the inner
orbit) and radius R0 (for the outer orbit) about a mass M involves two velocity kicks
(high thrust, short burn)
!
r
r
GM
2R0
∆v =
−1
R
R + R0
!
r
r
GM
2R
1−
∆v 0 =
R0
R + R0
Synchronous orbit radius about a planet with mass M and spin period Pspin
2
1
2
3
rSync = Pspin
(GM ) 3 (2π)− 3
Hill radius of a planet with mass Mp about a star with mass M∗ at orbital semimajor axis a
1
Mp 3
rH = a
3M∗
Tidal force on the surface of a body of mass m with radius R caused by a body of
mass M at a distance d
GM mR
Ftide ∼
d3
Ratio of radiation to gravitational forces for a particle of density ρ and radius a
in orbit about a star of luminosity L∗ , mass M∗
Frad
3L∗ Q
β≡
=
Fgrav
16GM∗ cρπa
where c is the speed of light and Q is the fraction of star light absorbed by the
particle.
Poynting-Robertson timescale for radial orbital decay for a particle in orbit
v −1
tP R = Porb β
c
1
2
AST111 PROBLEM SET 3
where v is the orbital circular velocity, c is the speed of light and Porb is the orbital
period about the star.
Drag Force on a particle of cross sectional area A caused by a fluid of density ρ
and velocity v is
1
Fdrag = CD ρv 2 A
2
Here CD is a unitless drag coefficient.
Homework problems
1. On Hohmann transfer orbits
Consider a satellite of mass m with initial circular orbit of radius R about a
planet of mass M . We make a Hohmann transfer to a final circular orbit of radius
R0 , using two velocity kicks ∆v, ∆v 0 (two high thrurt short burns).
(a) What is the semi-major axis of the transfer orbit?
(b) How long do you have to wait in the transfer orbit to go between pericenter
and apocenter?
(c) Show that the increase in orbital energy (for the first burn)
GM m 1
1
−
2
R a
where a is the semi-major axis of the transfer orbit.
(d) Show that the sun of orbital energy increases (for the two burns) is
GM m 1
1
−
2
R R0
Notice that the total expended energy is independent of the transfer semimajor axis.
2. On tidal disruption
AST111 PROBLEM SET 3
3
(a) Show that a comet of mass mc and radius rc grazing the surface of a planet
with mass Mp and radius Rp can be tidally disrupted if its density is lower
than the mean density of the planet.
Hint: equate tidal force by Mp on mc to gravitational binding force of mc .
(b) Compute the mean density of Jupiter. What is the mean density of a comet
that can be tidally disrupted by Jupiter?
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Workshop problems
1. Tidal disruption by a black hole
The event horizon of a black hole of mass M is
rs =
2GM
.
c2
This radius is also known as the Schwartzschild radius.
(a) What density object can be tidally disrupted by the black hole outside the
event horizon (where we can see it being disrupted!)?
(b) The black hole at the Galactic center has a mass of 2 million solar masses.
What density object can be disrupted outside its Schwartzschild radius?
(c) How does the minimum density of an object that cannot be disrupted outside
a Schwartzschild radius depend on black hole mass? Write your relation like
this
α
M
ρmin = X
2 × 106 M
where you need to find X in g cm−3 and exponent α.
2. Small Particles blown out of star systems
(a) Show that a dust particle released at perihelion from a body on an eccentric
Keplerian orbit will escape from the Solar system if β > 1−e
. Here where
2
β is the ratio of radiation pressure to the Sun’s gravity force, and e is the
eccentricity of the body from which the particle escapes.
4
AST111 PROBLEM SET 3
Hint: consider the energy per unit mass of a dust particle under the forces
of gravity and radiation pressure
E
v2
GM
=
− (1 − β)
m
2
r
(b) How does β scale with stellar mass, luminosity and particle radius?
(c) How does the Poynting-Robertson drag lifetime for radial decay depend on
stellar mass, luminosity, orbital radius and particle radius?
3. On the stationary synchronous orbit for Mars
For Earth the geosynchronous orbit is at a radius of 42,164 km. The day for
Mars is about 40 minutes longer than the day for Earth but Mars’s mass is about
1/10th of that of the Earth.
(a) Approximately what is the radius of the synchronous orbit around Mars?
(b) Should the sidereal day or the solar day be used to calculate the exact location
of the geosynchronous orbit?
4. Drag Force from a Stellar wind
Low mass stars have low luminosity but strong winds. For these we can consider
a β that is the ratio of drag force (from the wind) to the gravity force.
Consider a spherical wind with mass flux Ṁw = 4πr2 ρw uw where ρw is the wind
density at radius r and uw is the wind velocity. We assume that Ṁ is constant
and that uw is also constant with radius r from the star but ρw (r) depends on r.
Show that the drag force on a particle of radius a is proportional to 1/r2 and
that the associated ratio of wind drag force to gravity force (independent of radius)
3Ṁw uw
16πGM∗ ρa
where ρ is the density of the particle.
βw =