Calculus of prograde burns.

[ajburges]

Anyone know of a reference for integrating the system of a prograde burn WRT time? I would like the ability to more accurately predict long ejection burn trajectories, but I have a feeling that the problem needs higher level differential equations to solve.

I am somewhat rusty with my higher calculus and keep running into issues modeling the burn. My best was to model it as a sum of the acceleration vectors, but the thrust is dependant on the velocity direction and I can't recall a way to model that in the equations.

The reason I want this is to more accurately time high eccentricity ejections. Instantaneous impulse optimizations just don't work for 5 km/s burns and Pe kicks only saves 800 m/s from the final burn (sub-Munnar).

I know an Eve flyby would make the trip more manageable, but I wanted to preserve the explore contact. After taking 6 tries to place the burn correctly, I turned to math to solve the more general problem.

Edited August 30, 2015 by ajburges

[LordCorwin]

May be you're looking something like the JPL CSPICE library, although it is a huge library made with real kernels.

Also, I think there is the function RK78, Runge Kutta Aproximation of 7'th & 8th order, in C & Fortran. I know that many orbit problems were based on it, although I had problems to use it apropiately. After the first step it always gave me 0 values.

I know that it works with first order differential eqs, and that the orbit problem is a second order diff eq, but the idea is to split it in two one-order diff eqs (of 3 coordinates each one).

Hope this helps.

[GoSlash27]

ajburges,

I'd think you could fairly easily compute that with a simple spreadsheet without having to integrate.

The mass is going to decrease linearly over time and the force is constant. Therefore the acceleration will increase linearly over time.

The mean acceleration will be the same as the integrated acceleration, so finding time is straightforward.

t= DV/Am

Am= (A1+A2)/2

A1= T/M1

A2= T/M2

M2= M1/Rwd

Rwd= e^(DV/9.81Isp)

where

t= time in seconds

DV= delta vee in m/sec

Am= mean acceleration in m/sec^2

A1= initial acceleration in m/sec^2

T= thrust in kilonewtons

M1= initial mass in kg

A2= ending acceleration in m/sec^2

M2= ending mass in kg

e= Euler's number; approx. 2.718

Isp= specific impulse in seconds.

I just realized I totally misunderstood what you were after! I thought you wanted a precise burn time calculation. Sorry!

Best,

-Slashy

(please correct any errors if you spot them!)

Edited August 28, 2015 by GoSlash27

[ajburges]

Seems like my initial post had enough ambiguity for confusion.

Many are familiar with the function to model the first burn off a Hohmann transfer. My problem is that that equation models the thrust as an impulse. I would like to find or derive an equation (or set) to model a thrust over time.

Is there a good reference on how to model this. The most sensible approach I could conceive would be to model the acceleration as a sum of the forces and integrate WRT time, but the thrust acceleration direction is dependant on the velocity!

Is the only way to solve this problem numerical integration? Part of why I want this is so that I can solve for when to start a low TWR burn for a desired escape trajectory.

[Jimbimbibble]

Well, orbits are going to be eliptical or hyperbolic and there are equations to get orbit based on velocity and position. So, you can use a numerical method to calculate the result of a burn. Get current acceleration, velocity, and position, then compute your current trajectory. Increment velocity and position and recalculate orbit. With a small enough time step you should be able to get a decent answer. You're going to have to write code to do this, of course, but it shouldn't be too bad if you restrict it to 2D. Adding plane changes is possible with this method but it would be nastier.

[Nich]

It should be possible to obtain an approximation using laplas transforms and a lot of differential equations and a Phd in Mathmatics however Numerical integration will give you an answer that is just as accurate and is a lot easier to calculate. and yes it could be done in a spread sheet but I would use matlab

[GoSlash27]

Seems like my initial post had enough ambiguity for confusion.

Many are familiar with the function to model the first burn off a Hohmann transfer. My problem is that that equation models the thrust as an impulse. I would like to find or derive an equation (or set) to model a thrust over time.

Is there a good reference on how to model this. The most sensible approach I could conceive would be to model the acceleration as a sum of the forces and integrate WRT time, but the thrust acceleration direction is dependant on the velocity!

Is the only way to solve this problem numerical integration? Part of why I want this is so that I can solve for when to start a low TWR burn for a desired escape trajectory.

ajburges,

The burn time formula I posted would model the thrust over time. It would not predict the effect of cosine losses, which is what I think you're after.

Using the model I provided would tell you precisely when to start a low twr burn. It just won't tell you what happens as a result of starting it so early or which way to point.

Best,

-Slashy

Edited August 29, 2015 by GoSlash27

[Snark]

Without actually trying to solve the problem, it's worth noting that it's possible it may not actually be precisely solvable without numerical integration-- there are plenty of problems (even ones simpler than that) that fall into that category.

For example: What is the circumference of an ellipse? (Or just, what is the length of an elliptic arc?) For example, I have an ellipse of semiminor axis 1 meter and semimajor axis 2 meters. What's the circumference? Sounds easy, right?

Turns out that it's actually completely impossible to express in terms of elementary functions. Not just hard, but actually, provably not possible.

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

I'm not making any assertion that what you've asked for is impossible, just noting that in principle it might be.

[ajburges]

Thanks Snark. I have not completely digested that article, but it's enough to convince me that I won't be able to simply derive an equation for this.

Looks like I need to read up on sage math some more to learn the calls for numerical integration. If I get it to work I might post the code here.