The eccentricity constant of solar objects

Peter Riedt wrote:

On Friday, January 5, 2018 at 3:20:27 AM UTC+8, Anders Eklöf wrote:
Peter Riedt wrote:

On Wednesday, January 3, 2018 at 4:23:40 PM UTC+8, Libor 'Poutnik' St?í?
wrote:
Dne 03/01/2018 v 09:12 Libor 'Poutnik' St?í? napsal(a):
Dne 03/01/2018 v 04:21 Peter Riedt napsal(a):
The eccentricity constant of solar objects

The eccentricity constant X of solar objects can be calculated by the
formula .5*sqrt(4-3(a-b)^2/(a+b)^2) where a = the semi major axis and
b = the semi minor axis. The eccentricity constant X of nine planets
is equal to 1.0 as follows:

I'm curious:

Where does that formula come from?
What *is* the eccentricity constant in the first place)
How does it relate to e = sqrt(1-(b/a)^2) ?

You listed the semi minor axes (b) of the planets with 11 to 13
significant digits. Impressive!. Where did you get those numbers?
(I have a hunch...)

Even 8 digits for the semi major axes is quite a feat.

Why do you round off to one decimal? Just to prove a point?
Isn't it more interesting to explore the differences? 3 decimals?


So you say a circle has the eccentricity constant 1.0.
Interesting.

.5*sqrt(4-3(r-r)^2/(r+r)^2) = .5*sqrt(4) = 1

Similarly, any ellipse similar enough to a circle
like those of planets has this constant 1.0,
if rounded to 1 decimal place.

Actually, the eccentricity of a circle is 0 (zero).
Poutnik should maybe have pointed that out to you...


I have improved the formula to read 1-3(a-b)^2/(a+b)^2).

How does .5*sqrt(4-3(a-b)^2/(a+b)^2) simplfy to 1-3(a-b)^2/(a+b)^2) ?

Let's do the algebra. Oh bummer!

.5*sqrt(4) = 1 !!! Trust me, you'd fail any high school math test with
that one (pun not intended).

Orbits are subject to the Law of X.

What's the Law of X.?


Satellites ecc 1-3(a-b)^2/(a+b)^2)
Moon 0.0549 0.99999978624
Io 0.0041 0.99999999999
Europa 0.0090 0.99999999985
Ganymed 0.0013 1.00000000000
Calli 0.0074 0.99999999993
Mimas 0.0202 0.99999999607
Encela 0.0047 0.99999999999
Tethys 0.0200 0.99999999624
Dione 0.0020 1.00000000000
Rhea 0.0010 1.00000000000
Comets
Halley 0.9670 0.85758592313
Encke 0.8470 0.96428781552
Tempel1 0.5190 0.99769836444
Planets
MER 0.2056 0.99995625439
VEN 0.0068 0.99999999995
EAR 0.0167 0.99999999817
MAR 0.0934 0.99999819976
JUP 0.0484 0.99999987116
SAT 0.0542 0.99999979788
URA 0.0472 0.99999988373
NEP 0.0086 0.99999999987
PLU 0.2488 0.99990429852
Asteroid
Pallas 0.2313 0.99992918251

I repeat:

Where does that formula (.5*sqrt(4-3(a-b)^2/(a+b)^2))come from?
What *is* the eccentricity constant (X) in the first place?
Where do your 11-13 digit values for the semi minor axis com from?
How does .5*sqrt(4-3(a-b)^2/(a+b)^2) simplfy to 1-3(a-b)^2/(a+b)^2) ?

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