How to determine best method for inclination burn?

[FinalFan]

I recently desired to take a small manned craft into polar orbit, so on an unrelated mission I hauled up a pod stuck to 800 fuel and a tiny engine and hurled it into the void.  But going from equatorial LKO to polar LKO is not best done from LKO.  If I did a single maneuver the delta-V would have come close to 3200!  I know that with large inclination changes it can be cheaper to raise your apoapsis, do the inclination change, and then re-circularize, and that was certainly the case here, but is there any easy method to determine whether this is the case?  Or more importantly in the example, is there an easy way to approximate how much the apoapsis should be altered for the most efficient maneuver?  Or a ratio of altitude delta-V to inclination delta-V?  I just did a bunch of permutations of the two maneuvers until it seemed like the diminishing returns weren't really worth the effort anymore, but I was wondering if there was a "right" way to figure this out that doesn't require an actual rocket scientist. 

[Serenity]

Usually large inclination changes are done during the ascent of the rocket.

In real life there are some rare cases that reaching a high apoapsis first and then changing inclination is more efficient, in Kerbal though i doubt there is anything like that.

I am not very experienced with orbital mechanics but i know that if you want to change the inclination after you circularized 

then in no way you can be more fuel efficient than doing the inclination change during the ascent.

In the end if you are in orbit and want to do an inclination change its gonna cost you a lot no matter how high is your apoapsis.

It might be a bit better but the cost to get that apoapsis(and then lower it again) will negate most of the gain.

Maybe there are some more advanced orbital mechanics that you can use(with other planets etc) but thats what i gathered for Kerbal from various sources.

Edited April 30, 2018 by Boyster

[HebaruSan]

 

[Pand5461]

There is a formula to decide whether a bielliptic inclination change is better than a single-burn:

 

where r_a is the ratio of transfer orbit apoapsis (plus the body radius, of course) to the radius of initial circular orbit, i is the inclination change.

Consequence: for changes less than 2arcsin(1/3) = 38.94°, single burn is favourable. From 38.94° to 2arcsin(sqrt(2) - 1) = 48.94°, a bielliptic transfer with some finite apoapsis radius is optimal. For inclinations changes by more than 48.94°, a bi-elliptic change with any apoapsis radius will be better than a single-burn change.

Edited April 30, 2018 by Pand5461

[bewing]

Use the Mun to do your plane change.

 

[GrouchyDevotee]

... launch North in the first place?

Pand5461 has nicely laid out the math.