Gravitational force on the earth and mars

[Oerg]

Homework Statement

A Hohman transfer is an orbit that takes a rocket from Earth to Mars
in the manner as shown by the figure below, with the minimum amount
of rocket fuel expended. The rocket fires its engine so that its velocity lies
tangential to the orbit of the Earth (A). It then shuts off its engine and
follows an orbit around to the other side of the Sun, where it intercepts the
orbit of Mars, at which point its velocity is tangential to the orbit of Mars
(B). Ignore the gravitational fields of the Earth and Mars, and assume that
both planets undergo circular orbits in the same plane around the Sun.

Homework Equations

The Attempt at a Solution

I have no idea how to attempt this question at all. I know a few variables, but somehow, I just can't piece together the puzzle. Through some astronomical data, I know

the gravitational force on the Earth and mars,
their velocities, period etc.
the gain in potential energy needed to reach Mars (which means the loss of kinetic energy of the satellite)

 

[Redbelly98]

I don't see any question here.

 

[Oerg]

oh I am so sorry

(a) How much time does the Hohman transfer take?
(b) Clearly, the Hohman transfer can only be carried out at certain times.
If this timing is not met, Mars will be at the wrong position in its orbit
when the rocket reaches the end of the Hohman transfer! How often can
a Hohman transfer be attempted?

 

[D H]

oh I am so sorry

(a) How much time does the Hohman transfer take?

Show some work. Hint: What are the characteristics of the transfer orbit? What are the relevant equations?

(b) Clearly, the Hohman transfer can only be carried out at certain times.
If this timing is not met, Mars will be at the wrong position in its orbit
when the rocket reaches the end of the Hohman transfer! How often can
a Hohman transfer be attempted?

Suppose we initiate a Hohmann transfer that will be successful. Where must Mars be at the end of the transfer, and thusly, where must Mars be at the start of the transfer? Think of it like an analog clock: How often are hour hand and minute hand separated by some specific angle? How does that pertain to the question at hand?

 

[Oerg]

erm for part a, i can't find any equations on transfer orbits, am I to use Kepler's third law? I am really confused, so no workings yet other than those mentioned in the first post.

 

[Oerg]

oh ok, so i use kepler's third law, and the semi major axis is the distance from the Earth to the sun + the distance from Mars to the sun divided by 2, right? I hope i am :S

 

[D H]

Correct.

 

[Oerg]

YES YES YES YES YES THANK YOU THANK YOU THANK YOU

Now I think question 2 should be easy, hopefully..

 

[Oerg]

Suppose we initiate a Hohmann transfer that will be successful. Where must Mars be at the end of the transfer, and thusly, where must Mars be at the start of the transfer? Think of it like an analog clock: How often are hour hand and minute hand separated by some specific angle? How does that pertain to the question at hand?

I don't understand this part, i did part 2 by taking the difference in angular velocities between Earth and mars, and then taking 2 pi over that to find the time needed to rearrange themselves at that angle. Am i right?

 

[D H]

No. Why 2 pi? It looks like you are just guessing.

 

[Oerg]

because I am thinking it this way, Earth is the faster planet in terms of angular velocity, so I am taking the relative angular velocity of Earth wrt Mars so 2pi divided by this relative angular velocity would give the time needed to make a loop back for the angle to launch.

 

[D H]

Forget the 2*pi business. How often do the is the angle subtended by the lines connect Mars with the Sun and and Earth with the Sun achieve some a specified value?

 

[Oerg]

everytime the faster planet, Earth makes it back a round to the intended angle?

 

[D H]

Correct. So how often is that? Hint: Would more or fewer opportunities occur if Mars orbited at 1.05 AU rather than 1.666 AU?

 

[Oerg]

I don't get it, isn't the mass of the sun the same?

 

[Redbelly98]

AU refers to the distance between a planet and the sun, not the sun's mass.

1 Astronomical Unit (AU) is the distance between the Earth and sun, by definition.