Quick question regarding a derivation

[simonpro]

Right, I'm supposed to be teaching a class tomorrow, and one of the problems they have to do is:

Show that specific orbital energy at perigee is equal to specific orbital energy at apogee using the knowledge that the radius and velocity vectors are perpendicular at these points and that H = R x V is constant.

I can't for the life of me figure it out, even though it must be goddamn simple. I can do it easily through other methods, but not whilst making use of the two piece of information it gives.

Anyone see what I've missed? My diagnosis is that there's something obvious I haven't paid attention to; most likely a symptom of a large hangover.

 

[mjessick]

See chapter 1 of Bate Mueller and White. (Section 1.6)
The two pieces of info basically say: h = rp * vp = ra * va

BMW writes the energy equation in terms of h,rp then massages to find that energy is a function of semi major axis only (not radius), and thus is constant all around the orbit.

 

[simonpro]

Yeah, that's the way I did it. I'm pretty sure it's not the way I want them to do it though as the next question involved deriving that E=-mu/2a

Still, I can't figure it out so I'll just change the damned question
But thanks for the reply, much appreciated.

 

[Urwumpe]

Can't you also show this over the definition of the cross product?

As they are assumed perpendicular, you can essentially reduce the math a lot and write R x V much simpler in scalar form.

 

[simonpro]

Well, you can then say that the magnitude of r x v is equal to the product of the magnitudes of r and v, but that doesn't get me much further. Plug that into the equation and you just get a load of r and v magnitudes, but it doesn't simplify much.

 

[Urwumpe]

Well, do they already know what specific orbit energy is without the relation of semimajor axis to energy? In that case, the scalar solution should be pretty easy... r can be used for potential energy, v for kinetic...

 

[David]

E/m = (v^2)/2 - G*M/r.............[total (specific) mechanical energy=kinetic+potential]
= (r*v^2 - 2*G*M)/(2*r)..........[common denominator]
= (h*v - 2*G*M)/(2*h/v)..........[h=r*v can be substituted, as explained, above]
= (h*v^2 - 2*G*M*v)/(2*h)......[simplify by getting rid of compound fraction]
= (v^2)/2 - G*M*v/h................[further simplification (could have skipped stuff, but this is how I did it)]

does not depend upon r (so, it's the same at periapsis as at apoapsis, where the vector -> scalar substitution is valid and h is constant)

(Seems kind of weird, since it does depend upon v, but I suppose that the subtraction covers that)

 

[simonpro]

Heh, empirical evidence that hangovers ruin concentration. Gave this another go this morning and got it first time, as did about half of the students, for all I know the other half are still working on it

(edit) I'll post what I did (for anyone who is or is not interested) later on.


-----Posted Added-----


Here we are, I'd be interested if anyone has a shorter way. This is a little more laborious than I'd like: