Kepler's 2nd. Law

02122018, 09:53 AM
(This post was last modified: 02122018 10:43 AM by Ángel Martin.)
Post: #1




Kepler's 2nd. Law
From a recent conversation with a friend's on the Kepler's laws  his son's subject for a High school paper. The goal was to calculate the value of the swept area between two instants, as determined by the azimuth angles (a1, a2) of the segments linking the focus of the ellipse with the planet at those moments.
Initially I thought the formulas would involve the Elliptic functions, as elliptical sectors were involved  but it appears that's not the case when the coordinates are centered at the focal point, instead of at the center of the ellipse. I found that fact interesting, as it only involves trigonometric functions (not even hyperbolic). Here's the reference I followed to program it, a good article that describes an ingenious approach  avoiding painful integration steps. It may not be the simplest way to get this done, chime in if you know a better one. http://www.badoshanai.net/Platonic%20Dr...Sector.htm And here's the FOCAL listing for a plain HP41  no extensions whatsoever. The result is the area swept between the two positions defined by the angles a1 and a2; a2 > a1. The parameters a,b are the semiaxis of the ellipse, a>b. Code: 1 LBL "K2+" (*) the symbol "<)" is for the angle character. Example: calculate the area swept between a1 = pi/4 and a2 = 3.pi/4, if the ellipse parameters are a= 2, b= 3 Solution: A = 1.989554087 Once the area is obtained, and knowing the period of the orbiting (T), it's straightforward to determine the time taken by the planet to travel between the two positions, with the direct application of Kepler's 2nd. law: t = T . Area / pi.a.b Cheers, ÁM 

02132018, 01:07 AM
(This post was last modified: 02132018 02:41 AM by toml_12953.)
Post: #2




RE: Kepler's 2nd. Law
(02122018 09:53 AM)Ángel Martin Wrote: From a recent conversation with a friend's on the Kepler's laws  his son's subject for a High school paper. The goal was to calculate the value of the swept area between two instants, as determined by the azimuth angles (a1, a2) of the segments linking the focus of the ellipse with the planet at those moments. For the parameters a=2, b=3 and a1=.785398..., a2=2.35619... (Did I get the parameters right?) I get 5.89676233948396 which agrees with the calculator at http://keisan.casio.com/exec/system/1343722259 Code: DECLARE EXTERNAL FUNCTION F Tom L DM42 SN: 00025 (Beta) SN: 00221 (Production) 

02132018, 02:01 AM
Post: #3




RE: Kepler's 2nd. Law
Angel
The 143 page publication Introduction to Orbital Flight Planning (I) (NASACE165052) may be of interest as it includes a welldocumented {equations, diagrams, etc.}discussion with respect to orbital mechanics & Kepler as well as an HP67 based program for same. The discussion starts with elliptic orbits on page 51 then transitions to Kepler on page 54 with an HP67 program on page 57. BEST! SlideRule 

02132018, 08:06 AM
(This post was last modified: 02132018 08:07 AM by Ángel Martin.)
Post: #4




RE: Kepler's 2nd. Law
(02132018 01:07 AM)toml_12953 Wrote:(02122018 09:53 AM)Ángel Martin Wrote: Example: calculate the area swept between a1 = pi/4 and a2 = 3.pi/4, if the ellipse parameters are a= 2, b= 3 Mmm... something's fishy. I consistently get 1.989554087 using a1 = pi/4 (45 degrees) and a2 = 3.pi/4 (135 degrees). Besides, it's not possible to get the same as the web applet  which uses centered sectors, not focuscentered ones. In fact, using the angles above the applet shows an Area result of 3.5280156212854 Could you double check? 

02132018, 08:11 AM
Post: #5




RE: Kepler's 2nd. Law
(02132018 02:01 AM)SlideRule Wrote: The 143 page publication Introduction to Orbital Flight Planning (I) (NASACE165052) may be of interest as it includes a welldocumented {equations, diagrams, etc.}discussion with respect to orbital mechanics & Kepler as well as an HP67 based program for same. The discussion starts with elliptic orbits on page 51 then transitions to Kepler on page 54 with an HP67 program on page 57. Great reference, thanks for the link. I'll check it out ASAP. 

02132018, 07:32 PM
Post: #6




RE: Kepler's 2nd. Law
(02132018 02:01 AM)SlideRule Wrote: Angel Big thanks for the information! Saludos Saluti Cordialement Cumprimentos MfG BR + + + + + Luigi Vampa + Free42 BlackviewA7 '<3' I + + + 

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