help with understanding non-hohmann interplanetary transfers

[Uncertaintea]

I want to get to Jool in about 40 days. I am well aware that it will cost a lot of delt-v, what i am wondering is how do i find out exactly how much delta-v is needed for that leg of the journey?

In an additional but related question: how do i determine an orbit specifically, by using the time and position vectors between two points. I know it can be done thanks to several tutorials i found out there but, for some reason, i just cannot wrap my head around what they are saying and am looking for another description.

thanks

sources i have used:

http://www.braeunig.us/space/

http://www.ltas-vis.ulg.ac.be/cmsms/index.php?page=ad

[Mr Shifty]

I just wrote a tutorial for this using alexmun's web app calcualtor:

http://forum.kerbalspaceprogram.com/showthread.php/33547-Tutorial-Non-Hohmann-Interplanetary-Transfers

Let me know if you can't figure it out using that.

[Uncertaintea]

I saw your post a day or so ago and thought it was well put together and would do everything i want, but i wouldn't learn anything from it except how to punch in numbers. I want to do it myself not have it done for me. My goal is to learn how to do the necessary calculations myself, maybe build my own calculator someday.

[falofonos]

wikipedia has a good page on porkchop plots

http://en.wikipedia.org/wiki/Porkchop_plot

Following links from there through http://en.wikipedia.org/wiki/Lambert%27s_problem gives you the maths.

Calculating a specific launch time and arrival time will require you to know the location and velocity of the origin and destination but it should be doable.

[Mr Shifty]

The Lambert's Problem wikipedia page describes the problem pretty clearly. Gravity exerts an attractive force between two objects proportional to the product of their masses:

F_{earth-on-ship} = (-r^hat)*G*m_earth*m_ship/r² = m_ship*a_ship

(r^hat is the unit vector pointing from the earth to the ship. It's there just to indicate the direction of the vector. Notice that it's negative because the force on the ship is toward the earth.)

a_ship = (-r^hat)*G*m_earth/r²

Since the acceleration is both the second derivative of the radius vector, and inversely proportional to the square of its magnitude, this is a differential equation:

r'' = (-r^hat)*G*m_earth/r²

All 2-body orbits satisfy this equation. What Lambert (and Gauss) were trying to find was the specific orbit r(t) that would satisfy certain boundary conditions:

Given an initial position vector r_i at time t_i and a final position vector r_f at time t_f, such that r(t_i)=r_i and r(t_f)=r_f, find the solution to the differential equation. There is no analytic solution, which means that you can only solve it by rearranging terms to make it easier, guessing at a solution, checking to see how close you were, modifying your guess, then iterating until your solution converges. Alexmun's app does that process for every point on a 300x300 point grid (90,000 times) for each of the graphs his app creates. It might be worthwhile to work through the math by hand once, but the app is not there to erase the math; it's to alleviate tedium.

Edited June 4, 2013 by Mr Shifty

[Uncertaintea]

excellent! that is exactly what i was looking for. I didn't know that there was no analytic solution which explains why i have had such a hard time with it. thank you