An Engineer's View
In the following my opinion is given on different topics at irregular times.
6-th August 2019:
A formula which should be well-known and widely-known!
In many scientific articles on many different subjects the sentence This formula is well-known
appears and apparently means that one can find the formula in textbooks on the subject.
The sentence This formula is widely-known goes a bit further. It means that all experts of
the field know the formula. My impression is that for a widely-known formula one does not have to bother
to give any reference to where it can be found.
In this note, I want to present a formula which should be well-known and widely-known.
It concerns a subject which belongs to the most studied in science, namely the movement of
celestial bodies.
Since Kepler found out we know that planets move on elliptical orbits around the sun and Newton
showed that such orbits result from an inverse square force law of gravitation with the sun
being in one ot the foci of the ellipse.
Given the semi-major axis a of the ellipse and its excentricity ε one can easily compute the
time-averaged distance from the focus where the central body (sun) is to the object which moves
around it on an ellipse.
All one needs to know is Kepler's equation (which is of course well- and widely-known) the relationship
between true and excentric anomaly (also well- and widely-known parameters), and some very basic analytical skills.
The very simple result for the time-averaged mean distance a is given by
a=a (1+ε2/2)
To check the correctness of this formula one can take the earth-moon data from precision
measurements done by lasers for the time averaged distance.
From the major semi axis and the time averaged distance one obtains the excentricity for the
ellipse of the moon which agrees quite good with the excentricity obtained by other means.
I am sure that this simple and useful formula has been obtained long ago (although I have not yet been able
to locate it). But in current textbooks one does not find it.
© 2019 Klaus Huber