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9 Nov 2019




These are parametric equations for a Keplerian orbit:

tan(φ/2) = √{(1+e)/(1−e)} tan(ψ/2)

r = a ( 1 − e cosψ) and t = ( T/ 2π) (ψ− e sinψ)

Here (r,φ) are plane polar coordinates, t is time, and ψ is aindependent variable parameter. Also, T = period of revolution, a =semimajor axis, and e = eccentricity.

(A) From the parametric equations, express r as a function of φ.That is, eliminate the variable ψ, and hence obtain an equation forthe orbit in space. (Your answer cannot depend on ψ.)

(B) Determine the Cartesian coordinates (x,y) and time t for thepoint P shown on the figure.

(C) Determine the Cartesian coordinates (x,y) and time t for thepoint R shown on the figure.

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Reid Wolff
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17 Jun 2019

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Related textbook solutions

Physics: Principles with Applications
7th Edition, 2013
Giancoli
ISBN: 9780321625922

Related questions

These are parametric equations for a Keplerianorbit:

tan(φ/2) = √{(1+e)/(1−e)} tan(ψ/2)

r = a ( 1 − e cosψ) and t = (T/ 2π) (ψ− e sinψ)

Here (r,φ) are plane polar coordinates, t istime, and ψ is a independent variable parameter. Also, T =period of revolution, a = semimajor axis, and e =eccentricity.

(A) From the parametric equations, express r as a functionof φ. That is, eliminate the variable ψ, and hence obtain anequation for the orbit in space. (Your answer cannot depend onψ.)

(B) Determine the Cartesian coordinates (x,y) andtime t for the point P shown on thefigure.

(C) Determine the Cartesian coordinates (x,y) andtime t for the point R shown on thefigure.

Figure 1 illustrates a Keplerian orbit, with Cartesiancoordinates (x,y) and polar coordinates (r,φ).

Parametric equations for a general Keplerian orbit are r(ψ) = a(1 − e cosψ)

t(ψ) = *(T)/2π+ ( ψ− e sinψ)

tan(φ/2) = *(1+e)/(1−e)+1/2 tan(ψ/2)

Here ψ is an independent variable. The solution is periodic in ψwith period ∆ψ = 2π. Also, a, e, T are constants that determine theparameters of the orbit.

Figure 2 shows the relation between ψ and the spatialcoordinates. In the figure, the coordinates (ξ,η) are defined by ξ= x + ae and η = y a /b,

withb=a*1−e2 ]1/2.

(A) What kind of curve is the orbit?

(C) Determine x(ψ) and y(ψ).

(D) Determine ξ(ψ) and η(ψ).

(E) Express r as a function φ .

(F) The angular momentum is L = m r2dφ/dt. Determine L in termsof {a,e,T} Hence verify that L is a constant of the motion.

(G) The energy isE = [1/2] m ( dr/dt )2+ [1/2] m r2(dφ/dt)2− GMm/r.Determine E in terms of {a,e,T}.

(H) E must be a constant of the motion. Hence determine T and Efrom the equation that you obtained in (F).

(I) Write T in terms of the spatial orbit parameters and theforce constant GM.

(J) Now express L and E in terms of a, e, GM.

(K) In Figure 1, determine the coordinates of the points P, A,and R

(L) In Figure 1, determine the time t at the point R.



A particle of mass m moves under the influence of a central force.The potential energy function is
V(r) = {
K (a2−r2) /2 if r ≤ a
0 if r > a

as illustrated in the figure. The initial position is (x,y) = (a/2,0) and the initial velocity is (vx,vy) = (0, v0).
[Data: m = 1.0 kg; K =17.1 N/m; a = 1.0 m; r0 = 0.5 a; v0 = 2.0m/s.]

(A) Determine the time ta when the mass crosses the boundary r =a.

(B) Determine angular position φa where the particle crosses theboundary r = a.

***HINT GIVEN***

Use the constants of the motion to analyze the orbit.

Write L = mr^2 dφ/dt = the initial value.
What is the initial value?


Write E = (1/2)m (dr/dt)^2 + L^2/(2mr2) +V(r )
= the initial value. What is the
initial value?

Part A. In the energy equation,
change the variable from r to u,
where r =sqrt(u). The result can be
solved by u(t) = c1+c2 cosh(αt).
Determine c1, c2, and α.

Part B.
φ = Integral 0 to t [L/(mr^2)] dt

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