A curious problem with Hohmann transfers

[jedidia]

EDIT: After some more thinking it through, it turns out that this question makes no sense whatsoever (see my next post). I'm still leaving it here for posterity. In the end, it's just another case of "you don't get how orbits work!".


So, I wanted to write a simple limiter for my game, when generating an object with the requirement to be reachable in a certain time.
The math for this looks extremely simple from the outset, so I wasn't expecting any issues, until I got completely stomped by something I can't seem to wrap my brain around. Let's take you through it:

Given:

• R1, A current orbital radius
• t, a maximum time the transfer may take

Wanted:

• The highest possible orbit radius reachable within the maximum transfer time
• The lowest possible inner orbit radius reachable in the maximum transfer time

Since in a hohmann transfer, transfer time is always half the orbital period of the transfer orbit, this gives us the period of the transfer orbit:
Pt = 2*t

The semi-major axis of any orbit can be calculated from its period, so this gives us:
at = cubeRoot( Mü * (Pt / (2 * Pi)^2 )

Calculating the semi-major axis of a Hohmann transfer is very simple, being the average of its inner and outer radius:
at = (R1 + R2) / 2

Since we have the semi-major axis and R1, and are looking for R2, this is very easily inverted to:
R2 = 2at - R1

This gives us the maximum outer orbit, how nice. But: How on earth do I get the minimum inner orbit?
The equation turns out to be exactly the same, so it gets me the same result. I can't wrap my head around what exactly I need to do to get the inner radius. I'm just hitting a wall, ending up with the same equation and the same result, and I can't figure out why. Can anybody tell me how I get the inner radius?

What I'm realising is that for constant R1, a cannot possibly be the same for R2>R1 and R2<R1. But a is simply the result from the maximum time the transfer may take, which is the same for inwards and outwards transfer. Somewhere here there's a fundamental flaw in my thinking...

 

[jedidia]

Ok, I figured out where I was being stupid!

• Since a Hohmann transfer always takes half an orbit, the minimum transfer time to transfer to any higher orbit is always greater than current P/2
• For the same reason, the maximum amount of time it takes to reach any inner orbit is always smaller than current P/2
• In other words, What I was asking makes no sense whatsoever. If you have enough time to reach any higher orbit at all, you have enough time to reach any inner orbit you like.
• The question I should have been asking, and that my math did indeed give the answer to, is "what is the highest orbit I can reach in the given time"

 

[Urwumpe]

Also, you would have noticed the same if you made a drawing of the "two" transfer orbits.

 

[jedidia]

Also, you would have noticed the same if you made a drawing of the "two" transfer orbits.

I did a drawing, but no, that was not enough for me to notice it. I'm really, really bad at visualising numerical relations. I saw it on my drawing once I came to the conclusion logically, but alas not before...

 

[Urwumpe]

I did a drawing, but no, that was not enough for me to notice it. I'm really, really bad at visualising numerical relations. I saw it on my drawing once I came to the conclusion logically, but alas not before...

Well, better late than never. I also needed a moment to realize that you can only make a difference if you search a maximum transfer time with less than half of your current orbit period. In that case, your only solution is a lower orbit, if you assume a hohmann transfer.