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In astrodynamics, any given pair of [[apsis|apsides]] can predict the [[semi-major axis]] and eccentricity of any orbit. Specifically, for periapsis q and apoapsis Q:

In astrodynamics, any given pair of [[apsis|apsides]] can predict the [[semi-major axis]] and eccentricity of any orbit. Specifically, for periapsis q and apoapsis Q:

a = (Q + q)/2

<math>a = (Q + q)/2</math>

''e'' = (Q - q)/(Q + q)

<math>e = (Q - q)/(Q + q)</math>

or

or

''e'' = 1 - 2/((Q/q) + 1)

<math>e = 1 - 2/((Q/q) + 1)</math>

By the same token, a and ''e'' can predict Q and q.

By the same token, a and ''e'' can predict Q and q.

Q/q = (1+e)/(1-e)

<math>Q/q = (1+e)/(1-e)</math>

and

and

Q + q = 2a

<math>Q + q = 2a</math>

Therefore

Therefore

Q - q(1+e)/(1-e) = 0

<math>Q - q(1+e)/(1-e) = 0</math>

and

and

Q + q = 2a.

<math>Q + q = 2a</math>

Subtracting the first equation from the second yields

Subtracting the first equation from the second yields

q(1 + (1+e)/(1-e)) = 2a

<math>q(1 + (1+e)/(1-e)) = 2a</math>

From the above:

From the above:

q = a(1-e)

<math>q = a(1-e)</math>

and

and

Q = a(1+e).

<math>Q = a(1+e)</math>

For ''e'' = 0, Q = q = a = r, the orbital radius, as one would expect.

For ''e'' = 0, Q = q = a = r, the orbital radius, as one would expect.

[[Category:Astronomy]]

[[Category:Astronomy]]