Expert: Philip Stahl - 2/4/2005

Question
Hello Philip,
I was watching the 2nd DVD of Cosmos yesterday and Sagan was discussing Kepler's laws of planetary motion.  For he 2nd law, where all the orbits of planets are ellipses with the sun at one focus ("foci")...  I was wondering, what is the other foci then??  Based on the graphics on the dvd, the foci must be inside the orbit of Mercury.  But obviouisly there is no other astronomical body (and I would imagine it would have to be huge)inside that orbit.   If the sun is indeed the only gravitatinal force working here, then wouldn't with time all the elliptical orbits eventually become circular?  

Thanks!
Tim

Answer
Hello.

Carl Sagan's 'Cosmos' series is excellent, and I am glad you're able to see it again. It will probably remain an excellent resource for some time to come.

In terms of 'foci' for elliptical orbits, while the Sun occupies one of these, it is a common misconception to think an "alter" mass must occupy the other. In fact, not. Technically, in terms of analytic geometry, foci for ellipses (which can be constructed easily using the right mathematical equations) merely represent mathematical points.

Thus, for an elliptical planetary orbit, the "other focus" is simply a mathematical point. (Which can be located, again, if one knows how to apply the planetary data - e.g. the known eccentricity e, the semi-major axis a, etc. using the correct formula)

As to where a particular focus is located, this will depend on the particular planet being referenced. For example, in the case of the Earth, with eccentricity e = 0.016 (approx.) the other focus would definitely be interior to Mercury's orbit. However, Pluto's - with e = 0.248, would not.

The distance of "displacement" from the ellipse center is always given by D = e x a, where e is the eccentricity and a the semi-major axis (mean distance to the Sun, for example).

In the case of Earth, this means: D =  (0.016) x 1 A.U. =  0.016 A.U. (remember 1 A.U. = astronomical unit = 1.5 x 10^11 m, the mean Earth -sun distance), which is definitely inside Mercury's orbit. (Mercury's mean distance from the sun or semi-major axis = 0.39 A.U. )

Meanwhile, in the case of Pluto, at a mean distance of 39.5 astronomical units from the Sun:

D = e x  a =  (0.248)  x (39.5 A.U.) = 9. 79 A.U. which is definitely FAR outside Mercury's orbit!

Regarding the alteration of shape of orbits, bear in mind that the shape and dimensions of any particular orbit is really a representation of the *total energy* (kinetic and potential) of the system.

In a branch of astronomy called "celestial mechanics" we use a specific formula to describe this energy in terms of shape. It is known as "the energy constant" or C.

For all circular orbits (e = 0) we have:  C = - u/a

where a is the semi-major axis and u is a constant of the motion.

For all elliptical orbits:

C = - u/ 2a

Note that for a circular orbit, C cannot change - since both u and a are constants (a always equals r, the radius of the circular orbit)

However, note that for an elliptical orbit C will be defined by the particular values for a, the semi-major axis, say for a given planet.

Factoring into this is the angular momentum for the orbit, L - defined more specifically as:

L =  +/-  [a (e^2 - 1) ]

What all this means is that - in order for any elliptical orbit - with a defined L and a - to change to a CIRCULAR orbit, an external force is required such that it alters the angular momentum.

Simply allowing time to pass - even billions of years- will not permit or allow originally elliptic orbits to become circular. Rather, somehow, some agent must enter that alters L.

What that might be is anyone's guess. What we do know already, is that at the time of solar system formation, the Sun transferred most of its own spin angular momentum outwards to the planets - imparting to them their respective orbital velocities, energy parameters.

Thus, it is implausible any more of the "residual"  solar angular momentum could ever be transferred again.

The orbits we detect now - with their parameters- will likely remain so, certainly until the Sun begins its expansion to a red giant and incinerates all or most of the inner planets!