To find the average speed, relative to the primary, for an object in an elliptical orbit, one divides the circumference of the ellipse by the period of the orbit.

v = C/P

The period of an elliptical orbit is found from

P = 2π sqrt{ a³ / (GM) }

The circumference of an ellipse is found, approximately, from

Q = [1 − sqrt(1−e²)] / [1 + sqrt(1−e²)]

q₁ = Q² / 4
q₂ = Q⁴ / 64
q₃ = Q⁶ / 256
q₄ = 25 Q⁸ / 16384
q₅ = 49 Q¹⁰ / 65536
q₆ = 441 Q¹² / 1048576
q₇ = 1089 Q¹⁴ / 4194304
q₈ = 184041 Q¹⁶ / 1073741824
q₉ = 511225 Q¹⁸ / 4294967296
q₁₀ = 5909761 Q²⁰ / 68719476736

C = πa [1 + sqrt(1−e²)] { 1 + q₁ + q₂ + q₃ + q₄ + q₅ + q₆ + q₇ + q₈ + q₉ + q₁₀ }

where...
a = the semimajor axis of the ellipse
e = the eccentricity of the ellipse

You can also approximate the average speed of an object in an elliptical orbit as its speed when the distance from the primary is equal to the semimajor axis of the ellipse.

v ≈ sqrt(GM/a)

However, this approximation will always give a speed that is a bit too high. A satellite will have its average orbital speed at a point a little farther from the primary than the length of its semimajor axis.

EXAMPLE.

Let's find the average speed of Mercury, relative to the sun.

GM = 1.32712440018e20 m³ sec⁻²
a = 5.79091e10 meters
e = 0.20563

v ≈ sqrt(GM/a) = 47872.1 m/s

Q = 0.010800499863027321053
q₁ = 0.0000291626993228133
q₂ = 0.0000000002126157579
q₃ = 0.0000000000000062004
q₄ = 0.0000000000000000003
Higher terms are essentially zero.
C = π (5.79091e10 meters) [1.9786298090187116306] {1.0000291629119447720}
C = 3.5997629e11 meters
P = 7600536.9 seconds
v = 47362.0 m/s

So, contrary to the opinion of some, the average speed around an elliptical orbit is not quite the same as the speed relative to the primary when the separation is equal to the semimajor axis.

In general, the speed of an object in an elliptical orbit, relative to its primary, is found from the Vis Viva equation:

v = sqrt{ GM ( 2/r − 1/a ) }

The distance of the object from the primary when its speed is the average for its orbit is found from

r = 2aP²GM / { aC² + P²GM }

For Mercury, this distance is 5.852948e10 meters, or 1.010713 times its semimajor axis.

Jerry Abbott