Average Distance of Planets
question
What is the Average Distance of Planets over the next 100 of Years ?
Purpose
The purpose of the question is to establish, based on observations, if the planets are moving away or are approaching the Sun.
Calculation
In order to answer this question I used two programs to calculate the maximum distance in 10 increments of 1000 years over a period of 10000 years.
The programs used is one by Peter DuffettSmith and one by Jean Meeus.
See Litterature for more information about those programs.
The methods used in both programs in order to calculate the Heliocentric (Sun centred) coordinates is quite different, however the results are identical.
Here are the results at aphelion:
1 10000 d in AU d in Km in %
Mercury .466537 .467232 .0006949 10 .148% 17/3/2001
Venus .728965 .726133 .0028328 42 .388% 14/6/2001
Earth 1.017501 1.012642 .0048586 72 .477% 4/7/2001
Mars 1.663228 1.676124 .012896 192 .775% 21/9/2002
Jupiter 5.439173 5.503232 .064059 958 1.17 % 14/4/2005
Saturn 10.12808 9.815793 .31229 4671 3.08 % 17/4/2018
Uranus 20.0770 20.04552 .0315 471 27/2/2009
Neptune 30.3323 30.33958 .00723 108 27/6/2049
Column 2 shows the distance in AU at aphelion of the planet after year 1.
Column 3 shows the distance in AU at aphelion of the planet after year 10000.
Column 4 shows the difference between 2 and 3 in AU
Column 5 shows average increase per year in km (column 4 * 14960)
Column 6 is (column 4 divided by column 2) * 100
Column 7 shows the first aphelion after the date 1/1/2001
In reality I did this calculation in increments of 1000 years.
For the first 6 planets the increase or decrease is linear.
For Jupiter the values are:
5.439, 5.446, 5.456, 5.462, 5.471, 5.477, 5.480, 5.491, 5.492,
5.502, 5.503
For Uranus the values are:
20.077, 20.064, 20.099, 20.120, 20.066, 20.033, 20.067, 20.103,
20.060, 20.022, 20.045
Reflection 1
The results are amazing.
First some of the values are positive and some negative, why one should expect all positive.
The values in column 5 (increase in km) are very large.
In order to evaluate those values I did a simulation of all the planets using Newton's Law and with a speed of gravity propagation equal to c.
Venus:
0 1 2 3
10799999.88 108000000.67 108000001.46 108000002.27
average increase 0.80
Earth:
14549999.985 149500001.109 149500002.244 149500003.360
average increase 1.12
Mars:
227999999.9 228000000.12 228000000.27 228000000.51
average increase 0.18
For Jupiter:
778299999 778300809 778301618 778302427
Average increase 809
For Saturn:
1429999999 1430000329 1430000659 1400000988
Average increase 329
For Uranus:
2874999999 2875000072 2875000143 2875000214
Average increase 72
In order to understand the above better I did the following changes to the Earth configuration.
Mass * 2
0 1 2 3
14549999.985 149500002.203 149500004.436 149500006.669
Mass * 4
0 1 2 3
14549999.985 149500004.455 149500008.9357 149500013.423
Distance * 2 = 2AU
0 2.83 5.66 8.49
14549999.985 149500001.484 149500003.069 149500004.655
Distance * 4 = 4AU
0 8 16 24
14549999.985 149500002.27 149500004.88 149500007.126
The above shows:
Increase in distance is strictly linear with mass.
Increase in distance is a function of the square root of distance in AU
Revolution time is a function of distance in AU to the power 3/2
Example:
Jupiter distance to sun 5.20 AU
Mass Jupiter = 317 Mass Earth = 1
Square root 5.20 = 2.28
Revolution time = 2.28 * 2.28 * 2.28 = 11.85 years
Increase in distance 1.12 * 317 * 2.28 = 809
Example:
Uranus distance to sun 19.182 AU
Mass Uranus = 14.54 Mass Earth = 1
Square root 19.182 = 4.3797
Revolution time = 4.38 * 4.38 * 4.38 = 84.011 years
Increase in distance 1.12 * 14.54 * 4.3797 = 71.32
Reflection 2
Only the simulated value of Jupiter is in accordance to observation.
In all the other values there is a great difference
between (calculated) observations and a simulation using Newton's Law.
I have the greatest doubts in the programs which calculate the ephemeris of the planets.
As a on feedback received I also calculated the shortest distance to the Sun
Here are the results at perihelion :
1 10000 d in AU d in Km in %
Mercury .307661 .307009 .0006517 9.8 p
Venus .717728 .720551 .0028234 42 p
Earth .982904 .987410 .0048706 72.8 p
Mars 1.384290 1.371451 .012839 192 p
Jupiter 4.965379 4.905077 .060302 902 p
Saturn 8.95600 9.255181 .2991807 4475 p
Uranus 18.32855 18.34819 .01963 294 p
Reflection 3
What the above shows is that when in aphelion is positive with a certain amount that then the distance at perihelion is negative with the same amount. In short for each planet the total distance between aphelion and perihelion is constant.
However the average position relative to the Sun changes.
Three explanations are possible:
 The planets are not stable.
 The Ephemerides calculations are not accurate.
 There is something else wrong with the Ephemerides.
The last reason is the most likely.
Feedback

16 April 1997

I'm a little surprised by some of the signs for a single planet
orbiting the Sun, the effect of a time delay in Newtonian gravity
should always be to increase energy and enlarge the orbit, but
not by the magnitudes. See if you can
get a copy of the book "Problem Book in Relativity and Gravitation"
by Lightman et al., and look at the solution to problem 12.4,
which gives a simple (approximate) analytic calculation of this
effect.

12 June 1997

I quote from the wellknown
history of astronomy by Pannekoek: "Lagrange [proved] in 1776, in a general
way, that the mutual attractions of the planets could not produce any secular
progressive changes in the mean distances to the sun...; they were subject to
periodic variations only."
Although I am not a specialist in celestial mechanics, I suspect your
results could be explained by a combination of two things:
 The formulae given by Meeus and by DuffettSmith were probably not intended
for use over such long time intervals. I suspect they are truncations of
longer series, so that the missing terms would become significant at such
long intervals.
 There are periodic cycles in the orbital parameters of the planets on time
scales that range from centuries up to hundreds of thousands of years, and
you might be picking up some of these effects. The famous Milankovitch
theory of the earth's iceages depends on cycles of this kind.


23 June 1997

You've done a careful job of accumulating data, which is why I'm taking
the time to respond. I'm honestly interested in helping, and I'm
assuming that you are honestly interested in learning.


26 June 1997

You ask a question about the average distances of the planets and then
proceed to compare aphelion distances. That is not a good idea, because
aphelion distances change much more rapidly than average distances.

Last modified: 1 September 1997
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