## 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 Duffett-Smith 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) co-ordinates 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:
1. The planets are not stable.
2. The Ephemerides calculations are not accurate.
3. 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 well-known 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 Duffett-Smith 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 ice-ages 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.