Gravity turn physics

[henryrasia]

Hey guys, I thought I'd post this question here since it's a very specific spacey physics thing.

How do gravity turns work?

I know what they are (btw, not KSP's turning to orbit, I mean the actual natural pitchover caused by gravity alone), but it's really puzzling me how the trajectory works out. I've tried solving it parametrically, but even the horizontal part get to a dead end: the velocity vector depends on the acceleration vector, which depends on the velocity vector's pitch, etc etc etc. It's a loop that reminds me of how computer code handles this situation, with new value = previous value + change in value. But it can't be this complicated. NASA used gravity turns way before computers could handle long simulations, so there must be an analytical solution. Any tips?

[Yasmy]

But it is this complicated. There are no analytic solutions in an atmosphere. You have to chug through integration/approximation methods for a set of coupled differential equations, either by hand or by machine. Computers (people with tables, adding machines, pencils and paper) were doing simulations like these long before governments had electronic computers.

[Bill Phil]

There were some decently powerful computers in the 60s. Yeah they're terrible by today's standards, but they were good enough.

I'm pretty sure it uses gravity and aerodynamics. Could be wrong, though.

[Superfluous J]

(btw, not KSP's turning to orbit, I mean the actual natural pitchover caused by gravity alone)

While most people (and most rockets) manually guide the pitchover, it is possible to do a (at least more) real gravity turn in KSP now that the atmosphere is kinder to fast moving, sleeker craft. I have a lot of "press space for space" rockets where all I have to do is stage - and never touch the steering - until I'm so close to orbit it's not worth worrying about. If that ain't a gravity turn, I don't want to do one

[GeneCash]

Wikipedia has the interesting note that a reentry is a reverse gravity turn. I never thought of it that way. And about the math:

This results in a coupled system of equations which can be integrated to obtain the trajectory. However, for all but the simplest case of constant n over the entire flight, the equations cannot be solved analytically and must be integrated numerically

Edit: OOPS, these equations also ignore drag, which is kind of an important gotcha.

https://en.wikipedia.org/wiki/Gravity_turn

- - - Updated - - -

OK, I found the discussion of "ascent flight mechanics equations of motion" in Griffin/French "Space Vehicle Design"

Turns out this in Google Books:

https://books.google.com/books?id=31GqndM3fk8C&pg=PA214&source=gbs_toc_r&cad=4#v=onepage&q&f=false

9 coupled equations with 20 variables. Ouch, mah brane.

[YNM]

You will need something numerical. One thing I can imagine is by drag-induced torque, or gravity induced torque (but I guess this one is a tad useless). But either thing needs some numerical analysis - that is, you can't be sure what design will take itself 200 km up or 70 km up just by looking at the equation - you must insert numbers.

And, yeah, I tell you, past scientists are hard workers, counting numbers carefully for each stage of their result ! Like, how does past astronomers draw the map for a solar eclipse while no computer (and printer) is around ?

[Kerbart]

There were some decently powerful computers in the 60s. Yeah they're terrible by today's standards, but they were good enough.

I'm pretty sure it uses gravity and aerodynamics. Could be wrong, though.

Let's not forget that if you work really hard you need a lot less computing power than we have today. A 1980s micro loaded with visicalc could already go a long way, and those computer had less power on board than the average wireless mouse these days (yes that's an unfounded claim I pulled out of where the sun don't shine but you get the drift).

[K^2]

Gravity turn is pretty complicated even before you take aerodynamics into consideration. With aerodynamics, even numerical solutions are very difficult to achieve.

[lajoswinkler]

Launchpads are places even more meteorologically probed than airports. I'm pretty sure today there are radiosondes being sent like every 15 minutes or so and the data is being crunched by supercomputers.