What is "Mean Radius" for Orbits

Hi,

I've been wondering about the definition of "Mean Radius" for orbital
parameters. On Wikipedia I got these orbital data for Uranus Satellites:

Titania
Semi-major Axis: 435,910 km
Mean Radius : 436,300 km
Eccentricity : 0.0011

Oberon
Semi-major Axis: 583,520 km
Mean Radius : 583,519 km
Eccentricity : 0.0014

If the "Mean Radius" is defined as the radius of a circle whose area
equals to the area of a given ellipse, then the relationship between the
semi-major axis and the mean radius is given by:

a = R / ((1 - e^2)^(1/4))

where a = semi-major axis
R = mean radius
e = eccentricity

However, using the above equation yields to different results, especially
for Titania.

Where do I get wrong, the definition of the derivation?

TIA

Hi,

Perhaps what is meant by mean radius is the following:

Let R(t) be the distance from, e.g. Titania, to the barycenter of the
orbit (for this case, the barycenter is very, very close to Uranus'
center.), and let P be the period of Titania's orbit.

Then the mean value of R(t) along one orbit is just 1/P * integral(R(t)
dt) from zero to P where the location of the ephemeris is arbitrary.

Hope that helps.

Jason Harris

Ricky Romaya wrote:
Hi,

I've been wondering about the definition of "Mean Radius" for orbital
parameters. On Wikipedia I got these orbital data for Uranus Satellites:

Titania
Semi-major Axis: 435,910 km
Mean Radius : 436,300 km
Eccentricity : 0.0011

Oberon
Semi-major Axis: 583,520 km
Mean Radius : 583,519 km
Eccentricity : 0.0014

If the "Mean Radius" is defined as the radius of a circle whose area
equals to the area of a given ellipse, then the relationship between the
semi-major axis and the mean radius is given by:

a = R / ((1 - e^2)^(1/4))

where a = semi-major axis
R = mean radius
e = eccentricity

However, using the above equation yields to different results, especially
for Titania.

Where do I get wrong, the definition of the derivation?

TIA

In article ,
Ricky Romaya writes:
On Wikipedia I got these orbital data for Uranus Satellites:

Titania
Semi-major Axis: 435,910 km
Mean Radius : 436,300 km
Eccentricity : 0.0011

Oberon
Semi-major Axis: 583,520 km
Mean Radius : 583,519 km
Eccentricity : 0.0014

There's something very odd here. I don't know of an official
definition of "mean radius," but if it means something sensible, it
ought to be equal to semi-major axis for a circular orbit. In the
above data, the more elliptical orbit has mean radius nearly equal to
semi-major axis. That doesn't make sense. I also don't see why mean
radius should be greater than semi-major axis in one case and less in
the other.

_Allen's Astrophysical Quantities_ gives the eccentricities as 0.0022
and 0.0008, respectively, and agrees with the above semi-major axes.
Does that work out better? Perhaps mean radius should just not be
taken seriously.

--
Steve Willner Phone 617-495-7123
Cambridge, MA 02138 USA
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