View Full Version : Parallelogram Law Summations

Goeton

2015-Mar-15, 12:49 PM

Hi all. Is there a way to automate Parallelogram Law summations, the way I have shown in the drawing below. [see the link below]. The sun is at the origin. I start at 1, and the arms of the parallelogram law, are v and g. I compute the resultant [shown as bold black arrow], I then use this at 2, where the arms of the second summation are, r and g, and so.

http://3.bp.blogspot.com/-tFDkItTrHWY/Ug-bh0x_NRI/AAAAAAAAE6U/PKx81419uBA/s1600/Kepler%27s+Second+Law_4.bmp

Cougar

2015-Mar-15, 01:39 PM

Hmm. That appears to work for any particular time step (defined by v). So the acceleration parameter must be hidden in there someplace. Of course, the g vector is constantly changing, because the distance to the Sun is always changing, so you're dealing with g1, g2, etc.

Programming a Newtonian orbit is not too hard (I've even done it), but typically uses a different method. Given an initial position, velocity, and direction, you figure the acceleration and just update the position, velocity, and direction for each step. It's iterative.

Goeton

2015-Mar-15, 05:32 PM

Cougar, yes you right. But this summation method does not predict closed loops [as can be seen in the drawing too] and I wanted to determine the rate of precession per revolution in relation to distance [inverse square]. If it's not too much trouble could you perhaps write a program that can compute these summations on html canvas? Or does this need a special software?

Goeton

2015-Mar-15, 05:34 PM

Here's a demonstration of the precession. Please go to 1:55 and you will see the marble tracing a flower orbit [aka precession].

http://www.youtube.com/watch?v=MTY1Kje0yLg

grapes

2015-Mar-16, 02:29 PM

If you're just modeling two simple masses, there won't be any newtonian precession, except from artifacts of the modeling process.

Goeton

2015-Mar-17, 03:19 AM

grapes, but that's not what the vectors predict. Do you believe there are errors in my summation method? If so, please tell me what they are.

grapes

2015-Mar-17, 09:57 AM

Since the equations for two body motion can be solved analytically exactly, of course I believe there's an error somewhere.

It's possible, by discretizing the process you've introduced an artifact into the calculations. I dunno, I don't see any details. There could be a thousand other ways to get a wrong answer.

mkline55

2015-Mar-17, 01:34 PM

Hi all. Is there a way to automate Parallelogram Law summations, the way I have shown in the drawing below. [see the link below]. The sun is at the origin. I start at 1, and the arms of the parallelogram law, are v and g. I compute the resultant [shown as bold black arrow], I then use this at 2, where the arms of the second summation are, r and g, and so.

Use a spreadsheet. I use Excel. Put your initial values in row 1 (or row 2 if you like titles). In the next row use formulas to calculate the new values. Repeat that row as many times as you need. Then, if you want to graph some result column(s) just select those columns and create a chart.

Cougar

2015-Mar-18, 09:28 PM

....but that's not what the vectors predict. Do you believe there are errors in my summation method? If so, please tell me what they are.

Well, from your first graphic, error is introduced mostly from the time step, which is not accounted for. The combination of the v and g vectors cause the orbiting body to move along in its orbit, right? But the velocity is constantly changing - every second. No every nanosecond. No, every infinitesimal amount of time. In any case, the value of v has to be updated at very short intervals. Summing parallelograms formed by the v and g vectors does not incorporate this time consideration.

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