## The planet Mercury

### Question 1

What are the Geocentric (Earth centered) co-ordinates of the planet Mercury, when Mercury is at Aphelion, in the last 100 years, based on actual observations ?

### Question 2

What are the Heliocentric (Sun centered) co-ordinates of the planet Mercury at those same moments ?

### Question 3

What are the Geocentric & Heliocentric co-ordinates of the other planets at those same moments ?

### Question 4

What are the Heliocentric co-ordinates of the planet Mercury for one complete revolution of Aphelion ?

### Question 5

What is the value of the angle, explained by the Relativity Theory, during one complete revolution of Aphelion ?

### Question 6

What is the reason that the Heliocentric co-ordinates are a function of the precession of the Earth axis ?

### Purpose

The author is involved in a project to simulate the movement of the planets, in particular the planet Mercury.
The results of three of those simulations are shown in: Simulation of the forward movement of Mercury
The purpose of the question 1, 2 and 3 is to compare the results of the simulations with actual observations.
The purpose of question 4 is to see if the third influence (43 arc sec angle) is constant or not.

### Description

The planet Mercury moves in an ellipse around the Sun.
The shortest distance is called Perihelion (P).
The furthest distance from the Sun is called Aphelion (A).
The Line between the Aphelion and Perihelion is called the Longest Axis (LA).
The Elongation is the angle between Mercury, Earth and Sun.
```
. . .
.            .        M1   .
.                   .                   .
.                      .                      .
.                       .                       .                  2
P.........S.....................................A                3 E1 1
.                       .                       .                  4
.                      .                      M2
.                  .                   .
.            .             .
. . .

3
4 E2 2
1
```
The above figure shows that in a simplified way. The Earth is indicated at two positions: E1 and E2.

There are two reasons that make observation of Mercury difficult:

1. Observation of Mercury can only be done during a small time frame per day.
2. The position when Mercury is at Aphelion does not coincide when Mercury is: "highest in the sky". This means elongation is at maximum.
In order to explain the first the position of the observer around the Earth is indicated at 4 different moments: 1, 2, 3 and 4.
1. When the observer is at 1 you are in the middle of the night. You can neither see the Sun nor Mercury.
2. When the observer is at 3 you are in the middle of the day. You can see the Sun, but you can not see Mercury.
3. When you are before 2 (Sun rise) or just after 4 (Sun set) it becomes possible to observe Mercury.

In order to explain the second (maximum elongation) the earth is indicated at two positions:E1 and E2

When the Earth is at E1 maximum elongation is when Mercury is at M1.
When the Earth is at E2 maximum elongation is when Mercury is at M2.
Both positions of Mercury are not when Mercury is at Aphelion.

The above picture shows the plane of the ecliptic. The picture shows as if the plane of Mercury and the plane of the equator are all the same. This is too simplistic. In reality the plane of Mercury makes an angle with the plane of the ecliptic. The same is true for the equator.

For Europe the best visible maximum elongation dates for Mercury are

``` 23 April    1996 8 Hour elongation 20.2354
3 October  1996 6 Hour elongation 17.9191
```

The line LA does not always point in the same direction, but slowly rotates in space.

There are three causes for this behavior:
1. The precession of the Earth Axis: 5029 arc sec per century
2. The influence of the other planets: 531 arc sec per century
3. Relativity Theory: 43 arc sec per century
All three causes influence both the Heliocentric (or Sun centered) co-ordinates of Mercury and the Geocentric (or Earth centered) co-ordinates.
the reason why the precession of the Earth Axis influences the Heliocentric co-ordinates is not clear to the author.

The answer for Geocentric co-ordinates is the following table:
```Day Mth Year Hr Min  Ecliptic       Ecliptic     Apparent RA  Apparent dec
Long D,M,S     Lat D,M,S     H,M,S          D,M,S
23  Mar 1994 12   9  335 24 17.25  -1 36 54.09  22 31 17.21  -11  1 53.85
19  Jun 1994 11  26   96 49 49.42  -2 48 10.86   6 29  7.79   20 27 47.79
15  Sep 1994 10  40  196 15  1.13  -1 22 58.42  12 57 44.39   -7 40  0.69
12  Dec 1994  9  57  259 11 24.25  -1  5 34.56  17 12 34.82  -24  5 16.48
10  Mar 1995  9  17  323 49 34.39  -1 27 59.53  21 46 34.68  -14 57 45.07
6  Jun 1995  8  35   73 29 47.71  -2 53  6.46   4 49 54.21   19 33 18.85
2  Sep 1995  7  50  185 24 15.44  -1 30 23.82  12 17 26.78   -3 31 47.14
29  Nov 1995  7   4  249 58 49.64  -1  5 44.38  16 32 39.42  -23  1 46.50
25  Feb 1996  6  22  313  1  9.71  -1 21  4.55  21  3 32.90  -18 12  9.38
23  May 1996  5  40   50 23 57.25  -2 45 29.76   3 14 52.47   15 11 13.38
```
The above table is calculated based on the book Astronomy with your Personal Computer

The answer for question 2 Heliocentric co-ordinates is the following table:
```Day Mth Yr Hr Min  Ecliptic        Ecliptic      Radius    Elonga    Angle
Long D,M,S      Lat D,M,S  vector AU    tion    arc sec
23  Mar 94 12   9  257 10 48.35   -3 24  0.72  .4666956     27.276       0
19  Jun 94 11  26  257 11  3.75   -3 24  1.21  .4667039      9.299    6396
15  Sep 94 10  40  257 11  9.75   -3 24  0.34  .4667047     23.92     4444
12  Dec 94  9  57  257 11 31.60   -3 24  1.61  .4666996      1.453    5985
10  Mar 95  9  17  257 12 10.10   -3 24  5.15  .4666993     25.55     8486
6  Jun 95  8  35  257 12 39.42   -3 24  7.41  .4667066      3.349    9223
2  Sep 95  7  50  257 12 47.11   -3 24  6.70  .4667117     26.054    8218
29  Nov 95  7   4  257 12 53.10   -3 24  5.79  .4667076      3.57     7400
25  Feb 96  6  22  257 13 17.30   -3 24  7.41  .4667014     22.955    7730
23  May 96  5  40  257 13 47.43   -3 24  9.91  .4667025     12.292    8262
19  Aug 96  4  56  257 13 58.31   -3 24  9.82  .4667083     27.266    7887
15  Nov 96  4   8  257 13 51.70   -3 24  7.40  .4667078      7.47     6920

6  Feb 04  5  59  257 20  5.96   -3 24 14.67  .4666887     18.212    5646
5  Mar 94 18  52  258 43 57.10   -3 26 25.14  .4667044     24.6423   5591
```
The above table is calculated based on the book Astronomy with your Personal Computer
Small values in the column Elongation mean that Mercury is very close to the Sun and "invisible" even during daytime.
```Day Mth Yr Hr Min  Ecliptic        Ecliptic      Radius    Elonga    Angle
Long D,M,S      Lat D,M,S  vector AU    tion    arc sec
23  Mar 94 12  10  257 11  1.37   -2 46 13.47  .4666974                  0
19  Jun 94 11  26  257 11 14.04   -2 46 13.66  .4666994               5259
15  Sep 94 10  41  257 11 21.28   -2 46 13.44  .4666996               4133
12  Dec 94  9  57  257 11 34.92   -2 46 13.97  .4666985               4642
10  Mar 95  9  15  257 11 58.38   -2 46 15.27  .4666994               5917
6  Jun 95  8  32  257 12 21.94   -2 46 16.24  .4667022               6690
2  Sep 95  7  48  257 12 34.65   -2 46 16.35  .4667031               6455
29  Nov 95  7   3  257 12 44.01   -2 46 16.32  .4667007               6087
25  Feb 96  6  19  257 12 57.91   -2 46 16.89  .4666985               6048
23  May 96  5  36  257 13 16.10   -2 46 17.72  .4666999               6215
19  Aug 96  4  51  257 13 33.02   -2 46 18.02  .4667028               6296
15  Nov 96  4   6  257 13 41.81   -2 46 17.89  .4667024               6055

6   Feb 04  6   6  257 20 16.83   -2 46 27.97  .4666942               5624
5   Mar 94 18  53  258 44  9.88   -2 46 13.47  .4667044               5591
```
The above table is calculated based on the book Astronomical Algorithms
I expect that the above values are not "good" enough to calculate the angle of 43 arc sec per century, explained by the Relativity Theory.
In in year 2094 angle = 5591 arc sec.
This angle - precession 5591 - 5029 = 562 arc sec
General relativity angle = 562 - 531 = 31 arc sec

None

``` Year  Precession +574       EL       R   angl      EL       R   angl
0                      226 19 .466540         226 17 .466534
2000    5029     5603     288 29 .466862 5596    288 29 .466855 5598
6000    5117     5691     351 40 .467134 5686    351 36 .467122 5679
10000    5206     5780      55 51 .467376 5775     55 10 .467322 5721
14000    5295     5899     121  2 .467590 5866    118 17 .467436 5681
18000    5383     5957     187 10 .467746 5952    179 24 .467455 5500
22000    5471     6045     254 12 .467875 6032    236 12 .467379 5111
26000    5560     6134     322  7 .467974 6113    285 31 .467120 4437
30000    5648     6222      30 55 .468009 6191    323 23 .466570 3407
34000                      100 36 .468039 6272    345  7 .466552 1956
```
• Column 1 shows the year in increments of 4000 years.
• Column 2 shows the precession of the Earth Axis
See Supplement Astronomical Almanac page 103 & 104 for more details.
• Column 3 shows the precession + 531 + 43
• Column 4 to 7 are based on the book Astronomy with your Personal Computer The date of the values is 2000 years higher than in column 1.
• Column 8 to 11 are based on the book Astronomical Algorithms The date of the values is 2000 years higher than in column 1.
• Column 4 and 5 (8 and 9) show the Ecliptic Longitude in degrees and minutes.
• Column 6 (10) Shows the distance to the Sun in AU
• Column 7 (11) Shows the forward angle in arc sec per century. This value is calculated based over a period of 4000 years.