Re: Area of ellipse between major axis and ray through focus, given angle
- To: mathgroup at smc.vnet.net
- Subject: [mg71673] Re: Area of ellipse between major axis and ray through focus, given angle
- From: "Narasimham" <mathma18 at hotmail.com>
- Date: Sun, 26 Nov 2006 03:48:59 -0500 (EST)
- References: <ek96ij$fd5$1@smc.vnet.net>
Kelly Jones wrote:
> Given:
>
> 1) an ellipse with eccentricity "ec", one focus on the origin, and
> the major axis along the x-axis
>
> 2) a ray through the origin at angle theta to the x-axis
>
> Question:
>
> What Mathematica function gives the relation/inverse relation between
> the angle theta and the area of the ellipse between the x-axis and the ray?
dA/dt is constant as per Kepler's second Law.Theta to time t relation
is through Jacobi Elliptic integrals(Newton's differential equation of
planetary motion,not in terms of elementary functions).
(* Combine two equations 1) r^2 d theta = 2 dA ; 2) p/r = (1 +
ec*Cos[theta[A]]). Simplify output of Dsolve with Boundary Condns theta
= 0, A = 0 *)
p = 1 ; ec = .7071;
EQ = {theta'[A] == 2 ((1 + ec*Cos[theta[A]] )/p) ^2, theta[0] == 0} ;
NDSolve [EQ, theta, {A, 0, 10}]
TH[u_] = theta[u] /. First[%]
Plot[TH[A], {A, 0, 10}]
Clear[p, ec];
eq = {theta'[A] == 2 ((1 + ec*Cos[theta[A]] )/p) ^2} ;
DSolve [eq, theta, A];