Selected Questions - Answers From All Experts Astronomy Forum (Abolish Kepler's 2nd Law?)

Question:  If we relocated the Sun in one focus of the orbit - say from the diagram in the link below

http://starchild.gsfc.nasa.gov/docs/StarChild/solar_system_level2/activity/song.html

we will discover that the straight line between the Earth and the Sun sweeps the interval of time between March 21 and Jun 21 is much bigger than that swept between September 21 and December 21, and the distance crossed by the Earth from March 21 to Jun 21 is also bigger than that crossed from September 21 to December 21.  This is  contrary to Kepler's second law so I say it must be  abolished.   Try it by yourself.. it is inevitable!  Do you concur?
regards,     Dr. Mohammed Barzaq

Answer:

The foci-adjusted mean daily motion for the Earth, from a celestial mechanics table, is 0.9856874 deg/day. When this is multiplied by the correct time interval for each given “quarter” orbit the same area will be obtained. There is no "violation" of the 2nd law.

With sufficient accuracy each area mapped out in accord with the 2nd law is given by:

½  r ²  (D  Θ)

Where Θ is the angular difference between t2 and t1 in the orbit..

The rate of areal description (mapping)  is then usually divided by (D t) so:

A =   ½  r ²  (D  Θ)    / (D t)

But, since this rate is constant (by the 2nd law) we may write:

h  =   r ²   d ( Θ) / dt

where h, a constant is 2x the rate of mapping of area by the radius vector.  We may also express it in terms of three constants c1, c2 and c3 defining the orientation of the orbital place, e.g.

h =  [c1 ² +  c2 ²   +  c3 ²]  ^½)   

 Thus, n, the mean daily motion is the mean value of  d ( Θ) / dt

 For all points in the orbit.

Re: your specific examples of time intervals, i.e. March 21 to Sept. 21, and Sept. 21 to Dec. 21, it is evident to me that your construal of “unequal” areas is based on the misconception that the dates of solstices and equinoxes are absolutely  fixed when they are not.

 For example, the date for the winter solstice which you fix as Dec. 21 can actually be any of the dates: Dec. 20, 21, 22 or 23. That is as much as a 3 day spread which will make a significant difference in your computations.

The same applies to your dates from March 21 (vernal equinox)  – which can also vary, and June 21 (summer solstice) , ditto. My point is that when the proper specific dates are used (for the given year – they change year to year) you will find the same areal ‘map out’ for each quarterly interval of the orbit – even with the differing distances factored in.

Thus, it follows that when the greater distance (radius vector) is entered for one interval, the comparison interval (in days, and hence degrees per day) will be counter -balanced by a different date, e.g. for the winter solstice – which would have to be larger (i.e. Dec. 23 >   Dec. 21), to compensate for the lesser r.

Again, the error is in presuming fixed dates to mark the termination and initiation points of your orbital (seasonal)   intervals.