Is the Orbit of a Satellite Affected by Changes in Speed?

[Abhishekdas]

Question on gravitation...

Homework Statement

A satellite is revolving around a planet in a circular orbit. Its speed is increased from v₀ to : (a) (1.5)^1/2 v₀
(b)2v₀

now in the book they have given that in case (a) the orbit becomes illiptical and in case (b) the satellite will escape.

Homework Equations

The Attempt at a Solution

The value of v₀ isn't given and nothing else is mentioned so how can one arrive to the above conclusions just like that..Is it just an error in the book? or sumthing else?

 

[gneill]

Determine a formula for the speed of a satellite in a circular orbit at some arbitrary radius. Next determine a formula for the escape speed of a satellite at the same radius. Compare. Conclude.

 

[Abhishekdas]

Hi gneill...
The speed of a satellite in a circular orbit is (GM/R)^1/2 whereM= mass of Earth and R= radius of earth. and escape velocity is root(2) times the above expression so that sorts out the second part...about 2 v0 but what about the first part...I don't get how and when a satellite moves elliptically...

Thank you for the reply ...

 

[gneill]

All you need to know is that the velocity has to be between that for circular motion and that for escape for the satellite to remain in a bound orbit (circular or elliptical). If Vo is the circular velocity, then escape velocity is √2 x Vo. Your job is to figure out where the given velocities lie...

 

[Abhishekdas]

Ya it is given in the book (without any explanation) that the velocity if lying between the escape velocity and velocity for circular motion is an ellipse...But why is it so? and is there any method to find the focus major/minor axes or anything of the ellipse?

 

[gneill]

Ya it is given in the book (without any explanation) that the velocity if lying between the escape velocity and velocity for circular motion is an ellipse...But why is it so? and is there any method to find the focus major/minor axes or anything of the ellipse?

Circles and ellipses are closed orbits; they repeat. Anything that escapes is an open orbit, and does not repeat; they pass by once and are gone for good. So if you want an elliptical orbit, it had best have a velocity that is less than escape velocity!

There are quite a few relationships between the various parameters of a given orbit, including the axes lengths, period, eccentricity, perihelion and aphelion distances, total mechanical energy, etc.

The basic equation is that of an ellipse. In polar form, where the origin is at the focus,

r = p/(1 + e*cos(ν))

where p is the semi-latus rectum, e is the eccentricity of the orbit, and ν is the angle from perihelion. p is related to the semimajor axis and the eccentricity by

p = a(1 - e²)

The perihelion distance is r_p = p/(1 + e) = a(1 - e)

The aphelion distance is r_a = p/(1 - e) = a(1 + e)

The total mechanical energy of an orbit is $$\xi$$ = v²/2 - μ/r

Orbit shape is determined by eccentricity or total mechanical energy

e < 1 --> $$\xi$$ < 0 --> bound orbit (elliptical or circular)
e = 1 --> $$\xi$$ = 0 --> unbound (parabolic); escape speed
e > 1 --> $$\xi$$ > 0 --> unbound (hyperbolic)

Beware, there are also degenerate orbits where the path is straight towards or away from the Sun (straight line). They may have any energy.

 

[Abhishekdas]

Hey...thanks...i think all these are new for me...