Re: Astronomical Unit (AU)

[David Bowman <David_Bowman@GEORGETOWNCOLLEGE.EDU>]

Regarding Don Polvani's question:

> I have seen one astronomical unit (AU) defined as:
>
> 1)      the mean distance between the earth and the sun
> 2)      the semi-major axis of the earth's orbit around the sun
>
> Would someone please tell me the correct definition of one astronomical
> unit?

Both of these two definitions are very close to each other in numerical
value, but (as alluded to by Leigh) neither is actually rigorously
correct because of the effect of the presence of the Moon on the Earth's
orbit.  The AU is (as far as I know) really defined as the length of
the semimajor axis of the center of mass of the Earth-Moon system in its
orbit around the Sun.  According to the JPL/NASA web site
http://ssd.jpl.nasa.gov/astro_constants.html the numerical value of the
AU is 149597870691 +/- 3 m.

Even if we only consider the Earth-Moon system's center of mass there is
still a problem with definition 1) because it is ill-defined without
further specification of the averaging process.  Let a be the length of
the semi-major axis of the appropriate ellipse, and let e be the orbit's
eccentricity.  Then the following expressions are *all* average distances
which differ from each other by the method of averaging used to take the
average:

r_1 = a

r_2 = a*(1 - e^2)^(1/2)

r_3 = a*(1 - e^2)^(1/4)

r_4 = a*(1 + (e^2)/2)

r_5 = a*(1 - (e + (1 - e^2)*arctanh(e))/(2*E(e)))

Average r_1 is merely the arithmetic average of the perihelion and the
aphelion distances r_1 = (r_p + r_a)/2 where we average only over the
orbit's extremal points.

Average r_2 is the *angular* average of the orbit's distance from the
Sun.

Average r_3 is an area average of the orbit's radius.  IOW, it is the
radius of the circle whose area is the same as the area enclosed by the
orbit.

Average r_4 is the *time* average of the orbital distance.

Average r_5 is the average distance of the orbit from the Sun where the
average is weighted by the *arc length* of the orbital path.  In this
last case the arctanh(...) function is the inverse hyperbolic tangent
function and the E(...) is the complete elliptic integral of the 2nd
kind.  This is the only one of these averages that differs from a by a
term of order e.  All the other averages listed only differ from a by
terms of at least order e^2.

In the limit of e --> 0 all these averages (and a, too) just boil down
to the orbital radius for a circular orbit.

David Bowman
David_Bowman@georgetowncollege.edu