Launching to an inclined orbit

[RizzoTheRat]

If I want to launch to an inclined orbit, how do I calculate the launch angle?

Say I want to get to a 45 degree orbit, orbital velocity is 2270m/s, so that would be 2270xSin(45)=1612m/s North and 2270xCos(45)=1612m/s east.  As I'm already doing 175m/s East on the launch pad, is it just a case of subtracting that from the easterly component and launching at aTan(1612/1437)=48.2 degrees, or it more complex than that?

I can't quite get my head around how launching to higher altitude, and therefor a lower orbital velocity, would impact it, or how I time the launch to end the burn as I enter the plane (I think that's 175 x Time to Orbit rather than being related to other speeds isn't it?).  The real world doesn't help either as nobody launches from the equator, making it even more complicated to work out, but I believe a launch from Cape Canaveral .

Currently I just launch at the angle of the plane I want and do a small correction burn in orbit, and that correction burn is pretty small when I'm actually heading for a 6 degree Minmus intercept, but I'd like to know how to work it out properly.

[VoidSquid]

Unless you know exactly your ascend trajectory plus and more important the exact time until you reach the desired orbit, I don't think it's possible to make any serious calculations.

You maybe want to talk with the developers of MechJeb as MJ does provide the option to make a timed launch into a specific target plane, naturally they know how they do the calculations.

Edited January 22, 2020 by VoidSquid

[RizzoTheRat]

Sorry I should have said I'm using kOS so I can launch at very precise angles and times, and can make a pretty good estimate on how long it takes a vessel to get to orbit (by doing a test launch of the same vehicle).

 

I've since found this, which suggests my angle is about right, but I still can't get my head around the launch altitude bit

https://www.orbiterwiki.org/wiki/Launch_Azimuth 

[Zhetaan]

2 hours ago, RizzoTheRat said:

If I want to launch to an inclined orbit, how do I calculate the launch angle?

You're looking for the launch azimuth equation.  The Orbiter wiki is a good place to find that information, so I won't quote the page here.

21 minutes ago, RizzoTheRat said:

I've since found this, which suggests my angle is about right, but I still can't get my head around the launch altitude bit

I think you mean launch latitude, since there is no mention of launch altitude on that page.  The basic idea is that launching from any latitude other than zero imposes a minimum inclination in your orbit equal to the latitude.  This is because all orbital planes must pass through the centre of mass of the primary (Kerbin in this case), so if we assume that the centre of mass is also the centre of the planet (which is usually the case), then the orbital plane, in order to pass through the centre of mass, must be tilted at an angle equal to or greater than the amount that the launch site is 'tilted' with respect to the equator and the centre of the planet.

It is necessary to account for this for two reasons:  first, high-latitude launch sites can only launch vessels to high-inclination orbits (for example, launching from the north pole needs must result in a 90-degree polar orbit), and second, the rotational velocity of the surface varies with your distance to the axis of rotation.  In other words, your eastward velocity on account of the planet's rotation decreases as you approach the poles.  This should be fairly obvious once you think about it:  a point ten metres from the pole takes one day to trace a circle around it.  A point on top of an equatorial mountain (specifically, as far from the axis of rotation as possible) must trace a circle thousands of times larger, but still must do so in one day.

When calculating your launch azimuth, you need to account for the amount of inclination that you get as a consequence of your launch site's latitude and for the amount of velocity you receive from planetary rotation at that latitude.

When you restrict yourself to equatorial launches, these considerations simplify immensely.

[RizzoTheRat]

Yeah, not a problem launching from the KSC but ideally I'd like to get this working properly when launching from other bodies too.

I did mean Altitude though, which isn't on that page.  If I launch to a higher altitude I require a lower orbital velocity, which will result in a launch angle even further North I think, which is starting to make sense to me now.

I think I kind of understand it, next job will be to implement it in to my launch script...kOS is a never ending job

 

So supplementary question...doglegs.

I've heard some mention of launches to the ISS doing them, presumably this is a mid flight course correction?  Is this due to launch direction restrictions (ie only launching out over the sea) or is there more to it?  Presumably if I want to launch from 50 degrees North in to a 40 degree plane, I'd launch at 50 and then do a course correction in to the 40 degree plane when my path crosses it, is this what a dogleg is?

[KerballingSmasher]

1 hour ago, RizzoTheRat said:

Is this due to launch direction restrictions (ie only launching out over the sea) or is there more to it?  Presumably if I want to launch from 50 degrees North in to a 40 degree plane, I'd launch at 50 and then do a course correction in to the 40 degree plane when my path crosses it, is this what a dogleg is?

That's what I've always done in this kind of a scenario with needing an inclined orbit. There is one exception to this, though, and that's launching into a polar or a nearly-polar orbit. Instead of doing my gravity turn to the east, I do it to the north (or south). I think you could probably do the same with any degree, but since I usually only launch within about 10° of either polar, for scanning satellites around Kerbin, or equatorial, for easier transfers to the Mun and Minimus, I have yet to expirement with that.

[Starman4308]

I worked out the relevant spherical geometry issues a while back, working under the assumption that launch was instantaneous and occurred at a fixed altitude. Since neither is true, I'd have your kOS script switch to a PID loop once you're within tolerance of your desired azimuth.

I... don't remember the equations I came up with, but I do have the spreadsheet! I know I posted it somewhere, so I'll edit this post once I find where I did the math.

Note that the spreadsheet is configured for this strange planet called "Earth".

https://www.dropbox.com/s/ljtxh571ll9qy05/Azimuths.xlsx?dl=0

EDIT: Found my original post: 

 

 

Edited January 23, 2020 by Starman4308

[RizzoTheRat]

Interesting stuff, cheers. 

I think I have a handle on the launch angle now.  The biggest problem remaining is the timing, I can launch on the right bearing, but if the timing isn't right I'll end up at the right inclination but a different longitude of ascending node.

If I aim to launch a fraction late I should be able to dogleg on the ascent as my path passes through the target plane.  In theory I can calculate my closure speed and acceleration to the target plane so I shouldn't even need a PID to do it.  This is likely to need some thought and several sheets of paper

Just need to get my launch window calculator working properly first... 

[Zhetaan]

1 hour ago, RizzoTheRat said:

I think I have a handle on the launch angle now.  The biggest problem remaining is the timing, I can launch on the right bearing, but if the timing isn't right I'll end up at the right inclination but a different longitude of ascending node.

I was going to reply to your earlier question about doglegs but you seem to have figured out the concept on your own.  To be clear, however, a dogleg is a manoeuvre that uses some of the propellant margin to adjust the longitude of ascending node.  This is useful because it allows one to have an actual launch 'window' as opposed to a launch 'arrow-slit', to coin a term.  In other words, it allows one to launch a few minutes to either side of the perfectly ideal alignment of the target orbital plane with the launch site and give some flexibility to the launch time thereby.  One can also use it to change inclination:  adjustments to longitude of ascending node and inclination are similar in their execution.  Efficiency is a separate question.

[Flibble]

This is my kOS code for launch azimuth, given an arbitrary inclination (that is greater than or equal to the launch site latitude):

// Pe and AP in km.

local targetPe is Pe * 1000 + Ship:Body:Radius.
local a is (Pe + Ap) * 500 + Ship:Body:Radius.
local sinInertialAz is max(-1, min(cos(targetInclination)/cos(Ship:Latitude),1)).
local vOrbit is sqrt(2 * Ship:Body:Mu / targetPe - Ship:Body:Mu / a).
local vEqRot is 2 * Constant:pi * Ship:Body:Radius / Ship:Body:RotationPeriod.
set launchAzimuth to mod(arctan2(vOrbit * sinInertialAz - vEqRot * cos(Ship:Latitude), vOrbit * sqrt(1 - sinInertialAz^2)) + 360, 360).

Seems to work fine - I've used it to launch into a variety of orbits in RSS.

[RizzoTheRat]

Cheers.  I think my understanding now matches your LaunchAzimuth equation so I'll give something a similar a go.

It's all on hold at the moment though as I got fed up with the mess my kOS scripts were in and have started a complete rewrite so expect to spend a week or more filling in full stops I've missed out before I get to implementing new code

[ElWanderer]

If you're still trying to work out timing, I'm pretty happy with my solution. Here's a link to my documentation where I go through my launch azimuth/ETA calculations:

https://github.com/ElWanderer/kOS_scripts/blob/v1.1_draft_changes/documentation/lib_launch_geo_readme.md

[RizzoTheRat]

Nice writeup, cheers.  I'll have a read through in depth later.