Tangetial acceleration in elliptical orbit

[bkraabel]

Homework Statement

Is there any position in an elliptical orbit where the tangential component of the acceleration is greater than the component perpendicular to the tangential component? If so, what conditions on the orbit must there be for such a position to exist?

Homework Equations

For the tangential component of acceleration to be greater than the perpendicular component, the angle between R and v has to be greater than 45 degrees.
The semimajor axis is 2a.
The distance from the Sun (focus) to the satellite is R
The mass of the Sun is M and the mass of the satellite is m
The half-semimajor axis is related to the (constant) mechanical energy E=K+U of the orbit as
a=-\frac{GmM}{2E}
The eccentricity of the ellipse is related to the (constant) angular momentum L as
e^2=1+\frac{2EL^2}{G^2m^3M^2}
Combining the two previous expressions gives
a^2(e^2-1)=\frac{L^2}{2Em}

The Attempt at a Solution

From the geometry of an ellipse and the law of cosines, the best I can do is
R(R+\frac{GMm}{2E})<\frac{L^2}{Em}

But I can't seem to find a criterion that involves only constants of the motion. Any suggestions would be appreciated.

 

[tiny-tim]

hi bkraabel!

you're making this too complicated

once you've decided that …

For the tangential component of acceleration to be greater than the perpendicular component, the angle between R and v has to be greater than 45 degrees.

… this is no longer a physics problem, it's just a geometry problem

so forget energy, forget angular momentum:

what is the maximum angle that the line from the focus makes with the curve, for eccentricity e? ​