What is the Period of a Near Rectilinear Orbit with Zero Minor Axis?

[LCSphysicist]

Homework Statement

I don't know how to calc the period of a parable orbit.

Relevant Equations

third kepler's law: T = 2pia^3/2*(m/k)^0.5

I made and understood the letters a b c, but i don't understand the letter d.
I thought it would be the double of the answer C [

], but seems that's not right, since the book says the answer is two times the answer of B [ two times this[

]

 

[phyzguy]

The case described is not a parabolic orbit. The comet will return to it's starting position and velocity, and the comet's path repeats itself and is periodic. It is actually an extremely elongated ellipse. A parabolic orbit does not repeat.

 

[LCSphysicist]

The case described is not a parabolic orbit. The comet will return to it's starting position and velocity, and the comet's path repeats itself and is periodic. It is actually an extremely elongated ellipse. A parabolic orbit does not repeat.

It has been confused to me, since that area's law say basically dA/dt = L/2m, and being L = 0, what type of motion is this? I can't see.

 

[phyzguy]

It has been confused to me, since that area's law say basically dA/dt = L/2m, and being L = 0, what type of motion is this? I can't see.

The area of the orbit is zero. The comet justs moves back and forth along a line, kind of like a weight on a spring. Think of an elliptical orbit where the width of the orbit (the small direction) gets compressed to zero.

 

[LCSphysicist]

The area of the orbit is zero. The comet justs moves back and forth along a line, kind of like a weight on a spring. Think of an elliptical orbit where the width of the orbit (the small direction) gets compressed to zero.

OMG now i see, it really makes sense,but, unfortunately, i don't get yet the book's answer
We could apply the third kepler's law with great accuracy:

T = 2pi*(r/2)^3/2*(1/G*Ms)^0.5

T = pi*(r^3/(2*G*Ms))^0.5

DO you see any mistake?

 

[Filip Larsen]

Its a degenerate elliptical orbit that some authors label as rectilinear orbits.

 

[LCSphysicist]

Its a degenerate elliptical orbit that some authors label as rectilinear orbits.

Yes Now i see.
DO you agree with my result?

 

[Filip Larsen]

I assume you are still trying to understand why the answers in your textbook disagrees with your own result and, if so, we will probably better be able to spot where you go astray if you can specify what parameters are given in Problem 8.20 and what equation (4.58) is all about.

As already mention in this thread, the (near) rectilinear orbit is really "just" an elliptical orbit where the minor axis is (near) zero, or similarly, where the apoapsis (r_max) is finite (as for an elliptical orbit) and periapsis (r_min) is zero. By "just" I mean that the most of the equations for an elliptical orbit, e.g. equations speed or period, can be used directly by applying the right value for semi-major axis (which luckily relates fairly simple to the apo- and periapsis distance).

(Edit for wording).