How to calculate how fast the craft will be going at a given point in orbit?

[guitarxe]

So I'm launching my ship into orbit, to an apsis of, say, 80km and want to circularize at that altitude.

Using either of these two equations I know how fast I need to be going to have a circular orbit:

v = SQRT ( GM ( ( 2 / r ) - ( 1 / a ) ) )

or

v = R * SQRT ( g / ( R + h) )

Either one gives us a speed of 2,278.9 m/s for a circular orbit of 80km. That's how fast I want to be going to have a circular orbit. So knowing how fast I should be going, and how fast I am going, I can calculate the difference to get the dV required to do a circularization burn.

The problem, then, is knowing how fast will I be going? I can't really just take a look at my speedometer, because I need to know this ahead of time. I am launching from the surface, aiming for an apoapsis of 80km, and need to know how fast I will be going when my ship reaches the apoapsis. How can this be calculated?

Edited September 25, 2014 by guitarxe
changing to answered

[mhoram]

The speed of an object at periapsis and apoapsis is quite eays to calculate.

Have a look at chapter 11 Orbital Mechanics of the PDF here:

https://github.com/mhoram-kerbin/ksp-physics-documentation/releases

I did not yet try to calculate the speed on other points on the orbit.

[Rhomphaia]

If you are using an informational mod such as engineer, or raise your periapsis above the suface, then you can work out the semimajor axis of your current trajectory and put that into the first equation to find out how fast you will be going at apoapsis

if not you can rearrange the first equation to solve for a at a given point before you reach apoapsis, say just after you leave the atmosphere, then put that value of a back in and solve for r=680km

[Laie]

What are you trying to do? Are you writing your own autopilot?

[guitarxe]

Not quite. I just like to be precise. In this case, I want to figure out a way of doing an orbital insertion that gives me a nice circular orbit at 80km. As precise as is practical, anyway.

So I calculated then, that for an 80km orbit, I will need to burn 1173.7dV to circularize it. I also calculated that it will take 107seconds of burn time for my ship that has one LV-909 engine and a mass of 5.294 tons.

That all looked very reasonable to me.

So I waited until I was 53seconds to Apoapsis, pointed the ship into prograde and did a 107second burn. At the end of the burn I was at 300km+ apo and ~85km peri.

I realize I probably shouldn't have been burning prograde the entire time, but at a vector that is somewhere in between of where my prograde is at the start and where it will be at the end. I shudder to think how much more I will need to learn and understand to calculate something like that.

[UmbralRaptor]

If you know your speed at any given altitude (well, radial distance), you can find it at any other via conservation of energy: E = v²/2 - µ/r

If you know at least some orbital parameters, you can mess around with the Vis-viva equation

[AlexinTokyo]

Your speed at apoapsis before circularisation will depend on the altitude of the periapsis, which will depend (a lot) on your ascent profile.

As extreme examples, imagine the difference between the circularisation burns for a 'point-straight-up' launch with an apoapsis of 80 km and a 'point-to-the-horizon-as-soon-as-possible' ascent. In the former you'll have a huge burn required, in the latter it could be as low as double digits.

In general, in stock KSP, a reasonable ascent profile should cost about 4,500 m/s, divided between the boost/climb phase and the circularisation phase. In general, if you use less on climb (a steeper profile) you'll need more to circularise, and if you use more on climb (a shallower profile) less.

The result of this is that you can only know your circularisation ÃŽâ€v requirement before launch if you know, what your flight profile will be, specifically what your periapsis will be when you reach MECO with your apoapsis at 80 km.

[Laie]

So I calculated then, that for an 80km orbit, I will need to burn 1173.7dV to circularize it. I also calculated that it will take 107seconds of burn time for my ship that has one LV-909 engine and a mass of 5.294 tons.

So I waited until I was 53seconds to Apoapsis, pointed the ship into prograde and did a 107second burn. At the end of the burn I was at 300km+ apo and ~85km peri.

For purposes of circularization, a reasonable direction would be to point just at the horizon. Also, over 1000m/s for circularization is /a lot/. You must have come up pretty steep, which also explains why pointing prograde was so horribly wrong in your case.

Forgive me that I ask, but are you familiar with the concept of gravity turns, and have you heard about maneuver nodes?

[monophonic]

Not quite. I just like to be precise. In this case, I want to figure out a way of doing an orbital insertion that gives me a nice circular orbit at 80km. As precise as is practical, anyway.

So I calculated then, that for an 80km orbit, I will need to burn 1173.7dV to circularize it. I also calculated that it will take 107seconds of burn time for my ship that has one LV-909 engine and a mass of 5.294 tons.

That all looked very reasonable to me.

So I waited until I was 53seconds to Apoapsis, pointed the ship into prograde and did a 107second burn. At the end of the burn I was at 300km+ apo and ~85km peri.

I realize I probably shouldn't have been burning prograde the entire time, but at a vector that is somewhere in between of where my prograde is at the start and where it will be at the end. I shudder to think how much more I will need to learn and understand to calculate something like that.

Others have mentioned many things you may or may not have considered, but I'll add one: When calculating your burn time for circularization, did you consider that as your LV-909 burns fuel, your ship keeps getting lighter and lighter. It certainly did not have a mass of 5.294 tons after the burn.

[mhoram]

Others have mentioned many things you may or may not have considered, but I'll add one: When calculating your burn time for circularization, did you consider that as your LV-909 burns fuel, your ship keeps getting lighter and lighter. It certainly did not have a mass of 5.294 tons after the burn.

I am certain that he did. The mass of the used fuel can be calculated using the Tsiolkovsky rocket equation.

And by using the fuelconsumption (kg/s) one can calculate the burntime.