Gravitational potential energy traveling from earth to mars

[ago01]

Homework Statement

Imagine a trip from Earth to Mars in terms of gravitational potential energy.


Assuming you have a mass of 70 kg70, what is your change in gravitational potential energy in moving from the surface of the Earth to the surface of Mars in GJ? Don't forget the sun!

Relevant Equations

Universal gravitation

My attempt:

Let $M_e$ be the mass of the Earth and $M_m$ be the mass of the person. Let $D_{EM}$ be the distance from Earth to Mars and let $R_e$ be the radius of the earth.

Defining these constants (leaving off units for brevity):

Masses in Kilograms (G is not a mass but I'll leave it in this group)

$M_e = 5.97x10^{24}$

$G = 6.67x10^{-11}$

$M_m = 70$

Distances in meters...

$D_{EM} = 3.594x10^8$

$R_e = 6.38x10^6$

Then...

$\Delta{U} = U_{mars} - U_{earth}$

$= -\frac{GM_eM_m}{D_{EM}+R_e} - (-\frac{GM_eM_m}{R_e})$


$= -\frac{GM_eM_m}{D_{EM}+R_e} + \frac{GM_eM_m}{R_e}$

$= -\frac{(6.67x10^{-11})(5.97x10^{24})(70)}{3.594x10^{11} + 6.38x10^6} + \frac{(6.67x10^{-11})(5.97x10^{24})(70)}{6.38x10^6}$

$= 4.37x10^9 J$

$= 4.37 GJ$

But this is incorrect. I understand that when dealing with gravitational potential energy we "move" the mass from infinity to it's "destination" and take the difference to get the potential energy.

It's obvious here I didn't include the sun. I understand that it's supposed to be a hint but I cannot imagine a reason the sun would factor into the potential energy between two other planets. Maybe my professor meant something related to the calculation of the distances? I am unsure. Any help would be greatly appreciated.

 

[haruspex]

Your expression for $U_{mars}$ is quite wrong. You have neither the mass nor the radius of Mars in it.
You should check whether the change in distance from the Sun is significant.

 

[Janus]

It's obvious here I didn't include the sun. I understand that it's supposed to be a hint but I cannot imagine a reason the sun would factor into the potential energy between two other planets. Maybe my professor meant something related to the calculation of the distances? I am unsure. Any help would be greatly appreciated.

So you don't think that it is important that, in order to go from Earth to Mars, you have climb higher in the Sun's gravity well?

 

[ago01]

So you don't think that it is important that, in order to go from Earth to Mars, you have climb higher in the Sun's gravity well?

Well when you put it like that...

I suppose my (bad) logic was that this would've been "built in" to the earth-mars equation because the Earth's potential energy (from the equation) would reflect it's distance from the sun, so doing things relative to just the Earth made sense.

I'm going to go review the problem now.

 

[haruspex]

Well when you put it like that...

I suppose my (bad) logic was that this would've been "built in" to the earth-mars equation because the Earth's potential energy (from the equation) would reflect it's distance from the sun, so doing things relative to just the Earth made sense.

I'm going to go review the problem now.

Your mass has a GPE in respect of three larger masses: Earth, Mars and Sun. You need to find how much each is changed by the move.

 

[ago01]

Your mass has a GPE in respect of three larger masses: Earth, Mars and Sun. You need to find how much each is changed by the move.

So thinking about it w.r.t. gravity wells that makes sense. But why wouldn't I need to be concerned with say...venus? Wouldn't Mars also be encountering the gravity wells associated with other large masses?

 

[haruspex]

So thinking about it w.r.t. gravity wells that makes sense. But why wouldn't I need to be concerned with say...venus? Wouldn't Mars also be encountering the gravity wells associated with other large masses?

You don't need to worry about forces or GPEs between the planets and sun. They remain whatever they are, unaffected by your journey.
Likewise, the change in your GPE wrt other planets will be very small. It is much larger in respect of Mars and Earth because you get so close to them, and larger in respect of the Sun because that is so massive.

 

[Orodruin]

So thinking about it w.r.t. gravity wells that makes sense. But why wouldn't I need to be concerned with say...venus? Wouldn't Mars also be encountering the gravity wells associated with other large masses?

Technically, you need to consider the potential from all stars in the Universe to get the full potential. However, the change in those potentials is going to be absolutely negligible compared to the main contributor(s), where the main candidates are the Sun, Earth, and Mars. (In actuality, some of those are also negligible, but I will let you figure out which ...)