Finding the semi-major axis of a transfer orbit?

[kasra12321]

Homework Statement

Hi, I have a question on my homework that I'm absolutely stuck at. I'm not sure how to go about this. Can someone help me through the steps to solve it? there are more after this similar to it so learning how to do this one would help me do the other ones by my self. Thanks a lot!

Your trip to Mars is accomplished by using an elliptic transfer orbit going from Earth to Mars as shown in Fig. 1. This trajectory assumes that Earth at departure, the Sun, and Mars at arrival, are aligned. Also, we will assume that Earth's and Mars' orbits are circular, with radiuses R = 149,943,160 km and 228,189,693 km, respectively.

What is, in meters, the semi-major axis, a, of this transfer orbit? Hint: determine its radiuses at aphelion and perihelion.

http://as370.socialhwk.com/engr370/hw/midterm/264x340xfig1-hw3.gif.pagespeed.ic.sqq5I9yIl4.png

Homework Equations

I found these in a lesson online.
http://as370.socialhwk.com/engr370i/ch04/ch4_8/IMG00020.GIF
http://as370.socialhwk.com/engr370i/ch04/ch4_8/IMG00023.GIF
http://as370.socialhwk.com/engr370i/ch04/ch4_8/IMG00024.GIF
http://as370.socialhwk.com/engr370i/ch04/ch4_8/IMG00026.GIF

The Attempt at a Solution

I know R1 is 149,943,160 km and r2 is 228,189,693 km. However, I'm not sure what Gmp and therefore can't solve for V1 and V2. And even if I find out V1 and V2, how would I find out the semi-major axis? Or the aphelion or perihelion.

Thanks a lot for your help!

 

[SammyS]

Homework Statement

Hi, I have a question on my homework that I'm absolutely stuck at. I'm not sure how to go about this. Can someone help me through the steps to solve it? there are more after this similar to it so learning how to do this one would help me do the other ones by my self. Thanks a lot!

Your trip to Mars is accomplished by using an elliptic transfer orbit going from Earth to Mars as shown in Fig. 1. This trajectory assumes that Earth at departure, the Sun, and Mars at arrival, are aligned. Also, we will assume that Earth's and Mars' orbits are circular, with radiuses R = 149,943,160 km and 228,189,693 km, respectively.

What is, in meters, the semi-major axis, a, of this transfer orbit? Hint: determine its radiuses at aphelion and perihelion.

http://as370.socialhwk.com/engr370/hw/midterm/264x340xfig1-hw3.gif.pagespeed.ic.sqq5I9yIl4.png

Homework Equations

I found these in a lesson online.    Don't make this overly complicated.
[ IMG]http://as370.socialhwk.com/engr370i/ch04/ch4_8/IMG00020.GIF
[ IMG]http://as370.socialhwk.com/engr370i/ch04/ch4_8/IMG00023.GIF
[ IMG]http://as370.socialhwk.com/engr370i/ch04/ch4_8/IMG00024.GIF
[ IMG]http://as370.socialhwk.com/engr370i/ch04/ch4_8/IMG00026.GIF

The Attempt at a Solution

I know R1 is 149,943,160 km and r2 is 228,189,693 km. However, I'm not sure what Gmp and therefore can't solve for V1 and V2. And even if I find out V1 and V2, how would I find out the semi-major axis? Or the aphelion or perihelion.

Thanks a lot for your help!

The hint asks for the radii at aphelion and perihelion.


What are those two values?

 

[gneill]

The major axis is the long axis of the ellipse that describes the transfer orbit. Can you work out the length of the ellipse given the orbital radii of the starting and ending points of the transfer curve?

It looks as though Gmp is meant to be G*Mp, where G is Newton's gravitational constant and Mp the mass of the primary in the system (In your problem it's the Sun).

 

[kasra12321]

I think I need to find out the aphelion and perihelion but I am not sure how.

And gneill, I'm not exactly sure what you're asking but If you mean the radius that the ellipse makes, would it be the diameter of Earth's orbit + the radius of mar's?

 

[SteamKing]

The orbits are circular according to the problem statement, so aphelion and perihelion are not needed.

gneill was not asking what the radius of the ellipse is, but whether you can work out the length of the major axis of the transfer ellipse given the radii of the orbits of the Earth andMars?

 

[gneill]

I think I need to find out the aphelion and perihelion but I am not sure how.

And gneill, I'm not exactly sure what you're asking but If you mean the radius that the ellipse makes, would it be the diameter of Earth's orbit + the radius of mar's?

Using the Sun as a starting point, draw in line segments joining the Sun to the starting point and ending point of the transfer orbit. Consider these two line segments joined together. Do they lie along an axis of the transfer orbit ellipse? Which axis? What's its total length?

 

[kasra12321]

Using the Sun as a starting point, draw in line segments joining the Sun to the starting point and ending point of the transfer orbit. Consider these two line segments joined together. Do they lie along an axis of the transfer orbit ellipse? Which axis? What's its total length?

Like this?

if yes, I'm not sure what axis it lies on, but its total length should be:

earth's orbital radius plus mars'

149,943,160km + 228,189,693 km = 378,132,853 km.
But I'm not sure what that gives me.

 

[gneill]

Like this?

if yes, I'm not sure what axis it lies on, but its total length should be:

earth's orbital radius plus mars'

149,943,160km + 228,189,693 km = 378,132,853 km.
But I'm not sure what that gives me.

Yes, that's right. Is it the long axis or the short axis of the ellipse that is the transfer orbit? that is, is the the MAJOR axis or the MINOR axis of the ellipse?

 

[kasra12321]

The axis I have drawn is long, or the major axis? So I'm looking for the semi-major axis? and that would be half of it?

 

[gneill]

The axis I have drawn is long, or the major axis? So I'm looking for the semi-major axis? and that would be half of it?

Yup.

 

[kasra12321]

holy ****ing ****. I just entered it and it accepted it. You just walked me through the problem without me even realizing it... Thank you so much.

 

[gneill]

holy ****ing ****. I just entered it and it accepted it. You just walked me through the problem without me even realizing it... Thank you so much.

Glad I could help.

 

[kasra12321]

I have another quick question if you don't mind me asking. The next part of the question is asking for eccentricity of the orbit. Now I know I can figure this out if I figure out the minor axis. Is there an equation to find out the minor axis by knowing the major axis?
I'm looking online and I can't really find a way to find eccentricity with only the info I have.

 

[SammyS]

I have another quick question if you don't mind me asking. The next part of the question is asking for eccentricity of the orbit. Now I know I can figure this out if I figure out the minor axis. Is there an equation to find out the minor axis by knowing the major axis?
I'm looking online and I can't really find a way to find eccentricity with only the info I have.

The sun is at a focus.

That should help.

 

[gneill]

From geometry there are various measures of the ellipse which, in appropriate combination, are related to its eccentricity. These include the semi-major axis a, the minor axis b, and the distance between the foci c. Extrapolating to the jargon of orbits, the perihelion (periapsis) and aphelion (apoapsis) distances are also related. It would be worthwhile looking into the basic geometry of the ellipse, since it's very important in orbital mechanics.

You should be able to find expressions that include the values you're given or have worked out. In particular, I'd suggest looking for relationships that include the semi-major axis and the periihelion distance, the semi-major axis and the aphelion distance, or just the perihelion and aphelion distances. There's also a relationship involving the distance between the foci and the semi-major axis. All of these measures are accessible given the information at your disposal.

 

[kasra12321]

Thank you! I found the aphelion and perihelion and plugged it in an equation from there