Elliptical Orbit Homework: Calculate r1/r0

[Cryphonus]

Homework Statement

The equation of the elliptical orbit of Earth around the sun in
polar coordinates is given by
r =ep/1 − e cosa
where p is some positive constant and e = 1/60. Let r0 and r1
denote the nearest and the furthest distance of the Earth from
the sun. Calculate r1/r0

Homework Equations

the one that is provided with the question

The Attempt at a Solution

I simply tried to give the max and min values for Cosa, which is 90 and 0 degrees.But I am not really sure if its right,Glad if you can help me here...

Thanks a lot

Cryphonus

 

[D H]

I simply tried to give the max and min values for Cosa, which is 90 and 0 degrees.But I am not really sure if its right,Glad if you can help me here...

That's not right. Try drawing a picture with the Sun at one of the foci of the ellipse. For what angles does the distance between the Earth and Sun reach minimum and maximum?

 

[Cryphonus]

0 - 180 degrees?

 

[D H]

Don't guess!

Do you know calculus? If you do you should easily be able to determine these critical angles. Even without calculus, a bit of critical thinking is all that is needed. The value of $\cos a$ ranges between -1 and +1. Given that, what are the minimum and maximum values for the denominator in your equation, $r=ep/(1-e\cos a)$? Finally, are the extrema in the denominator related to the extrema of the radial distance?

BTW, that equation does not look quite right. The orbit equation in standard form is $r=p/(1+e\cos\theta)$.

 

[Cryphonus]

I didnt guessed it :) .It just i took the max and min values as 0 and 1 which is ofcourse not true, so silly of me (: . I don't know about the equation it is given in the question.. but if you have any idea about what the question says "where p is some constant" i would be happy to hear.I never heard such a constant called "p" about this subject...

 

[D H]

One way to express the radial distance as a function of angle for an elliptical orbit is

$$r=\frac {a(1-e^2)}{1+e\cos \theta}$$

where a is the semi-major axis, e is the eccentricity of the orbit, and θ is the "true anomaly", the angle between the line from the focus to the closest approach ("perifocus") and the line from the focus to the current position.

An alternative parameter to the semi-major axis a for characterizing the size of an ellipse is the semi-latus rectum, $p=a(1-e^2)$. The semi-latus rectum is also given by

$$p=\frac{h^2}{GM}$$

where h is the specific orbital angular momentum, G is the universal gravitational constant, and M is the mass of the central object (e.g., the Sun).

Note that there is no factor of e in either form of the orbit equation.

 

[Cryphonus]

Ok thanks a lot i will ask around in the collegea about e