homework2 - Astronomy 362 Problem Set#1 Due Friday January...

Astronomy 362 Problem Set #1
Due Friday, January 24 at 12 noon
In the homework problems below, “C&O” refers to your textbook. I have selected the
problems with the expectation that they will take up to 1 hour of hard thinking/calculating
each. If you are taking longer than this, feel free to come by my office to hash out where
you are stuck.
Note on collaborating: You may work together on this problem set, but all work
presented here must be your own. You must clearly acknowledge any people you
collaborated with.
ASSUMED READING: Please finish C&O Chapters 2 and 3 before Friday, January 24.
1. [C&O 2.4 (tweaked)] Derive E =
1
2
μ v 2 −
GM μ
r
(Eq. 2.25), the expression for the
total energy of a two-body system in terms of reduced mass from the expression
for total energy of a two-body system in general cartesian coordinates
E =
1
2
m
1
!
v
1
2
+
1
2
m
2
!
v
2
2
−
Gm
1
m
2
!
r
2
−
!
r
1
.
HINT: Write down expressions for
!
r
1
and
!
r
2
in terms of
!
r and the masses m
1
and
m
2
first. Use those to find expressions for velocities
!
v
1
=
d
!
r
1
dt
and
!
v
2
=
d
!
r
2
dt
and go
from there. You may assume constant masses.
2. [C&O 2.7 (tweaked)] Calculate the escape speed from the Solar System, starting
from Earth's orbit. This is the minimum speed spacecraft must reach to escape
directly from Earth to interstellar space (e.g. - New Horizons ). You can assume
that the Sun constitutes all of the mass of the Solar System. NOTE: For full
credit, clearly explain your reasoning.