[VA] SRC #012f - Then and Now: Angle

[Valentin Albillo]

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Hi, J-F Garnier, pavel nemec cz, Gjermund Skailandand and vaklaff,

J-F Garnier Wrote:Thanks Valentin, for this last episode. It was particularly interesting, and I'm surprised too by the low participation. Especially, graphic calculators able to display the trajectories are missing.

You're welcome, J-F, but I'm not surprised in the least, it's just as expected. Some seven months ago I posted a simple Probability problem which required no special knowledge and I got ~ 90 replies, but now I post a little "physics" problem which requires some very basic knowledge about simple concepts such as force, acceleration, velocity, etc., and I get about 5 replies or less. Oh, well, that's life in this New Forum.

As for vintage graphic calculators missing, consider that all of them are RPL models so no surprise either. ’Nuff said.

J-F Garnier Wrote:See how close starting angles such as 40.8 and 43.3 produce completely different trajectories.

Yes, and to further demonstrate it, in that same figure you include the trajectory for Angle = 80.03º = 1.3968 rad, namely:



but you don't include the one for the very close Angle = 80.93º = 1.4125 rad (less than 1º apart,) which is much shorter as it doesn't backtrack, though (a) still far from the minimum time, (b) outside of my scanning range, and (c) even if it were included it would nevertheless be rejected by my heuristics.

pavel nemec cz Wrote:First of all thank you Valentin for all your excellent contributions via interesting and fun masterpieces professionally plotted, written and explained here with so special care for all details aiming to our education. [...] Be sure it does not discourage me to silently follow those threads [...]

Thanks for your kind words, pavel, I'm glad you like my contributions and even somewhat profit from them. As for "silently", as long as you care to post your appreciation here for me to know, your alleged "silence" speaks volumes !

Gjermund Skailand Wrote:Thanks for the challenges. [...] Is it possible to calculate longer paths, say circling the star cluster once twice, or would numerical inaccuracies quickly make any such results rubbish?

You're welcome, Gjermund, thanks for your interest. As for your question, yes, it's quite possible (if difficult,) see Direct gravitational N-body simulations in N-body simulation.

To minimize numerical inaccuracies while computing the path for extended time periods, it's usually mandatory to use multiprecision with a high-order integration method. For instance, the 33-34 digits I posted were obtained using not RK4 but an RK14 method (which actually produced as many as 50 correct digits that I then rounded down to 33-34 digits for posting here.)

vaklaff Wrote:+1 to what Pavel wrote, from another silent observer. Thanks to Valentin and all the contributors!

Thanks for your appreciation, vaklaff, but as long as you've posted here to voice it, you're not silent anymore. Much appreciated.

Regards to all of you.
V.