USING THE PERIHELION ADVANCE OF MERCURY

TO VERIFY GENERAL RELATIVITY

by

Joseph Harris

A senior thesis submitted to the faculty of

Brigham Young University - Idaho

in partial fulfillment of the requirements for the degree of

Bachelor of Science

Department of Physics

Brigham Young University - Idaho

April 2020

Copyright c© 2020 Joseph Harris

All Rights Reserved

BRIGHAM YOUNG UNIVERSITY - IDAHO

DEPARTMENT APPROVAL

of a senior thesis submitted by

Joseph Harris

This thesis has been reviewed by the research committee, senior thesis coor-dinator, and department chair and has been found to be satisfactory.

Date David Oliphant, Advisor

Date David Oliphant, Senior Thesis Coordinator

Date Stephen McNeil, Committee Member

Date Brian Tonks, Committee Member

Date Todd Lines, Chair

ABSTRACT

USING THE PERIHELION ADVANCE OF MERCURY

TO VERIFY GENERAL RELATIVITY

Joseph Harris

Department of Physics

Bachelor of Science

The perihelion advance of Mercury is one of the celestial occurrences that

can provide evidence for general relativity. The code provided can be used

to predict the future locations of Mercury, and predict the time of the next

transit. If general relativity was not affecting Mercury as Einstein suggested

then the 2019 transit of Mercury would have occurred 17.973±11.596 seconds

later than predicted.

ACKNOWLEDGMENTS

Thank you, to Nonnie Woodruff for presenting me with the idea that this

work is based on, and for the opportunity to push myself.

David Oliphant, Dr. Brian Tonks, and Dr. Lance Nelson for their patience

and assistance in the different aspects of my work.

Jesica Seeley-Williams for her assistance in quality control.

Katie Harris for believing in me and supporting me in many aspects during

the time of my work.

Contents

Table of Contents xi

List of Figures xiii

1 Introduction 11.1 History of General Relativity . . . . . . . . . . . . . . . . . . . . . . . 11.2 What is Perihelion Advance . . . . . . . . . . . . . . . . . . . . . . . 31.3 Why do We Care . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Methods 72.1 Finding a Celestial Body in Three Dimensions . . . . . . . . . . . . . 72.2 Predicting the Future Location of a Celestial Body . . . . . . . . . . 102.3 The Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Results 13

4 Conclusion 154.1 What This Means for General Relativity . . . . . . . . . . . . . . . . 154.2 Other Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.3 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Bibliography 18

A Code for Mercury 21

B Code for Earth 27

xi

List of Figures

1.1 Perihelion Advance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Example of Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1 Anomalies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.1 Motion After Each Orbit . . . . . . . . . . . . . . . . . . . . . . . . . 14

xiii

Chapter 1

Introduction

1.1 History of General Relativity

Albert Einstein was one of the most notable and iconic minds of his time. He spent

many years of his life figuring out mathematics and applying theories that have shaped

the way we understand the universe. One of the theories that he came up with was

the idea of general relativity, which he used to explain the force of gravity. He realized

that if one were in a box accelerating at 9.8ms2

no experiment that they performed

could prove whether they were in fact on Earth or accelerating through space. This

came with some interesting implications, from the way light worked to the existence

of black holes. However, the implication we are interested in today is how it helps

explain what Mercury does. An ideal way of describing how Einstein thought of the

universe is like a sheet of stretchy fabric with some mass in the middle and another

object being rolled past it. The second object will begin to orbit the larger central

one, and for most objects, that’s the end of the thought experiment. However, for

Mercury, there are some additional effects that I will cover.

The image of space being some form of stretchy fabric where everything warped

1

2 Chapter 1 Introduction

said fabric causing the effect we call gravity was a theory that not many people wanted

to agree with. It seemed to resemble the idea of the aether, which had already been

rejected. However, Einstein postulated that one could verify his theory by waiting

for a solar eclipse. If you plot the location of the visible stars then during the eclipse

plot them again they will appear to have moved. Arthur Eddington and his team

actually did this and Einstein’s predictions came out correct. This all but confirmed

his theories; however, nothing in science is ever proven and as such more evidence is

always a must.

During the time that Einstein was working on his theory, people were trying to

figure out why Mercury moved the way it did. Mercury had a tendency to appear

earlier than they predicted based on the Newtonian gravitational force of the planets.

It seemed to be moving about 42.98 additional arcsec/ cent (arc seconds per cen-

tury). Many people were confused by this until Einstein applied his theory of general

relativity. The full derivation can be found in “Explanation of the Perihelion Motion

of Mercury from General Relativity Theory” [9]. The conclusion comes down to one

equation

ε = 24π3 a2

T 2c2(1 − e2)(1.1)

with ε being the missing angle, a the semi-major axis, e the eccentricity, T the

period, and c is the speed of light. This leads to the missing angle of almost exactly

the 43 arcsec/cent they were missing. I decided that it would be beneficial to use this

lesser-known method to help verify general relativity, as Mercury would be transiting

this year.

1.2 What is Perihelion Advance 3

Figure 1.1 Perihelion Advance

1.2 What is Perihelion Advance

The perihelion is the point in a planet’s orbit that is closest to its sun. A perihelion

advance is when that point also moves around its sun. Many factors contribute to

perihelion advance including, the gravitational force of the other planets and the

movement of the sun, as the sun does have a slight wobble that can affect nearby

planets. However, in some cases, there is still a missing part and that is where general

relativity comes in.

Mercury has the largest eccentricity or in other words the most elliptical orbit of

all the planets in our solar system. Also being as close to the Sun as it is means

that its orbit can have some interesting effects. The orbit seems to have an orbit of

its own this is known as a perihelion advance. It is slow and hardly noticeable but

there is about 574.1 arcsec /cent in total movement. The other planets and Sun i.e.

Newtonian mechanics can account for 532.805 arcs/cent leaving about 43 arcs/cent.

As already demonstrated in equation 1.1 Einstein predicted that the effects of gravity

would change given Mercury’s proximity to the Sun and that change comes out to

the 43 arcs/cent that Newtonian mechanics was missing.

To be clear, perihelion advance can be seen in any orbit. However, most orbits

4 Chapter 1 Introduction

in our solar system are quite circular and their small advance is easily explained by

Newtonian mechanics i.e. the pull from other planets. I believe that being able to

predict locations of planets and how much effect general relativity has on them can

be useful in understanding extra-solar planets. If we have the necessary information,

we can make better estimates and predictions of a solar system’s makeup, which we

discuss later. For now we are focused on Mercury and its odd orbit.

1.3 Why do We Care

The implications of general relativity reaches much further than simply being able to

predict the location of Mercury at any given time. Additionally, it would allow us

to use similar methods in predicting the orbital mechanics of planets in other solar

systems. This also leads to being able to think of gravity more accurately. Since the

time of Sir Isaac Newton, people have thought of gravity as a force, for many aspects

it acts like one, but if Einstein’s theory is right it’s not so much a force that is pulling

objects together but rather the object falling into a “dip” or warp in the fabric of

space time.

Figure 1.2 Example of Gravity

In addition, knowing about general relativity and how much it affects the space in

which we live can help us avoid misunderstandings of what is or is not out there. When

1.3 Why do We Care 5

observers first started trying to discover what was causing Mercury’s odd behavior,

a scientist by the name of Le Verriere [4] believed that a 3rd inner planet or rather

a series of small objects was to blame. Named Vulcan, this invisible object needed

to be approximately 10% of Venus’s mass and be within the orbit of Mercury. It was

determined that such an object would not only affect Mercury but Venus and Earth

as well. This was not found and other options where needed to explanation of this

phenomenon.

There are a few implications with the verification of general relativity and the

accompanying theories. The great thing about testing general relativity is that anyone

can do it thanks to transits. A transit is when a planet moves between the observer

and a star. In the case of Earth, only two planets can transit between us and the

Sun; Venus and Mercury. My goal is to allow any amateur observer to be able to

predict Mercury’s location. Using stringing information provided by NASA [6] for

the location of Mercury during its last transit. As a result, we could predict when

Mercury would transit again if we include verses don’t include Einstein’s theory of

general relativity.

6 Chapter 1 Introduction

Chapter 2

Methods

2.1 Finding a Celestial Body in Three Dimensions

When people think of the motion of the solar system many will often think of Kepler’s

heliocentric model with the Sun in the center and the slightly elliptical orbits of the

planets. The problem with this mental image is that it is two dimensional. Many

people will depict the motions of the planets as two dimensional movements and

though for many situations this simplification works well. For my experiment, we are

trying to find a difference of approximately 0.04 arcsec per month. This requires that

we consider not only the orbit but the tilt, or inclination to the ecliptic, that each

orbit has. Thankfully, we can simply place Earth’s inclination at 0 and only consider

the inclination of Mercury 7.0047◦.

To do this first we need to know the semi-major axis a, the eccentricity e, and the

mean anomaly M . Eccentricity is the variable that helps us determine how elliptical

an orbit is, and semi-major axis is the length from the center of the orbit to the

nearest edge. The mean anomaly is a variable used to make the geometry a little

easier; briefly, it is the angle that a planet would have moved if it had a circular orbit

7

8 Chapter 2 Methods

Figure 2.1 Anomalies

i.e. its eccentricity was 0.

The current mean anomaly can be calculated with the following equation [8].

M = M(0) + 2π(t/T ) (2.1)

M(0) the initial mean anomaly, t the current time, T the orbital period of the

planet.

We also need to find the eccentric anomaly E, which is the angle forming a right

triangle with a perpendicular line going from the semi-major axis intersecting the

planet and ending at the imaginary circle that was drawn when finding the mean

anomaly [8]. This has to be done with a recursive method as it cannot be calculated

analytically.

E = M + (180◦/π)esin(E) (2.2)

Once the eccentric anomaly is found the radius and the true anomaly θ (which

is the angle that the planet has traveled from perihelion) can be found at any given

time using [8]

r = a ∗ (1 − e ∗ cos(E)) (2.3)

2.1 Finding a Celestial Body in Three Dimensions 9

θ = arccos(1/e ∗ ((a ∗ (1 − e2))/r − 1)) (2.4)

or [5]

θ = arccos((e− cos(E))/(e ∗ cos(E) − 1)) (2.5)

Thinking about the 3rd dimension one can give the planet tilt. This will give

φ or its position above or below the solar ecliptic, it was found that converting to

Cartesian for a moment gives the accuracy needed [3]. This is done by using the

fallowing formulas

x = r ∗ (cos(N) ∗ cos(θ + ω) − sin(N) ∗ sin(θ + ω) ∗ cos(i)) (2.6)

y = r ∗ (sin(N) ∗ cos(θ + ω) + cos(N) ∗ sin(θ + ω) ∗ cos(i)) (2.7)

z = r ∗ (sin(θ + ω) ∗ sin(i)) (2.8)

φ = arccos(z/(sqrt(x ∗ ∗2 + y ∗ ∗2 + z ∗ ∗2) (2.9)

x, y, z being the planets location in space, N being the longitude of ascending

node, which is the angle the planet would be at as it crossed the plane of reference,

in this case the ecliptic. ω being the argument of perihelion, which is the angle from

the reference plane, the ecliptic, to its perihelion [3]. Then repeat the same steps

for Earth and compare the two as time moves on. When both θ for Earth and θ

for Mercury align in addition to φ for Earth and φ for Mercury then the planets are

aligned with the Sun and thus a transit is in progress.

In summary, find the mean anomaly at any given time so that the eccentric

anomaly and then the 2 angles φ and θ can be found. When these values are known

for each planet and they are within a certain degree of each other. This means that

10 Chapter 2 Methods

Mercury is passing between Earth and the Sun or is transiting. Now if general relativ-

ity is taken away there should be an observable difference in location and by extension

the next transiting time. Using the variable M(0) in equation 2.1, I added the effects

of gravity from the other planets and the effects of general relativity separately. I

then took out the effect from general relativity and compared the final location.

2.2 Predicting the Future Location of a Celestial

Body

To know where you’re going you need to have a starting point, NASA keeps track of

planetary locations and other useful data [6]. This gives us a starting location with

which we can calculate the eccentric anomaly and mean anomaly using

E =e+ cos(θ)

1 + ecos(θ)(2.10)

and equation 2.1 setting t as 0 [5]. For this experiment I will use the last time

Mercury transited as this was the last time that anyone with a solar telescope would

have been able to verify where the planet was in reference to Earth and the Sun.

After that I will have my program from appendix A step in time; the calculations

being used are designed in such a way there is no need to worry about velocity of the

planet and can focus only on time past.

With this application the program can run for as long as it is needed; however

all of this does have a purpose. For this experiment we are going to find the next

time that Mercury will be visible to anyone with a solar telescope. The difference

between general relativity’s effect and the lack thereof is only a matter of seconds,

so for simplicity’s sake, I will use the starting time of a transit as the beginning and

2.3 The Calculation 11

end. If a few seconds is not enough to be convincing of the validity of the experiment

it can always run longer or have a different starting point further back in history.

Scientists’ have done a few different thought/ mathematical experiments for the

motion of the planets, each one with varying degrees of accuracy. Burchell [1] explains

many of these methods and explains the positives and negatives of them. For purposes

of this experiment, I will be accepting the generally used 531.63 arcs/cent effect due

to Newtonian mechanics, as well as the 0.0254 arcs/cent due to oblateness from the

Sun. The only question is whether or not general relativity’s effect will give me a

proper prediction.

2.3 The Calculation

Mercury’s most recent transit before my experiment happened was on May 09, 2016

second contact at 11:12:18. Mercury’s position in AU (astronomical units) was x =

−2.9835 ∗ 10−1, y = −3.3989 ∗ 10−1, z = −4.0108 ∗ 10−4 with Earth’s being x =

−6.6195 ∗ 10−1, y = −7.6244 ∗ 10−1, z = 3.3869 ∗ 10−5. That doesn’t mean much until

the location is converted to spherical coordinates, and one can see that θ for Earth

is 3.9973 rads, and for Mercury is 3.9918 rads. this gives us a θ difference of about

0.0055 rads. φ for Earth is 1.570796 rads and Mercury is 1.581504 rads giving us a

difference of about 0.0107 rads. This will be our threshold to indicate that Mercury

is transiting. According to NASA, the next transit was to happen on November 11,

2019, at 12:35:16.

The first thing I had to do was make a rough estimate of how big of a difference

general relativity was going to make. I did this by breaking the advance into two parts,

one being the advance due to all known effects, i.e. gravity from the other planets,

and oblateness of the Sun, the second being the calculated effect of general relativity.

12 Chapter 2 Methods

I then broke the effects down from arcs/century to arcs/month and multiplied by the

number of months it was going to take. This gave me 1.24 total arcs for the 31 month

difference. By using the small angle formula [2];

R = 206265d/D (2.11)

R being the missing angle in arcs (1.24), d being the distance between the planet

and its new predicted location, and D being the average distance of the planet from

the Sun. I am able to determine that the planet will be approximately 348.077 Km

behind if we don’t include the effects of general relativity. Given the average velocity

of Mercury 47.4 Km per second, I estimated that if general relativity was not in

play then the transit would occur 7.34 seconds after NASA’s prediction. Hardly a

difference, I know but it’s enough for a camera to catch and that is the important

part.

Now that I have shown my experiment is measurably possible, I had to write a

code that would give me a much more accurate result. The code shown in appendix 1

is broken into two parts; one shows the movement of Mercury while the other shows

the moment of the Earth. Each of these is broken into three parts one to calculate

the eccentric anomaly E the others to calculate the φ and θ. Now knowing that the

transits will be close together no matter if we use or don’t use general relativity. I

had the code run stepping in time, one minute at a time, until the full 31 months

had past. After which I compared the two φ angles of each planet, and the two θ

angles of each planet, once with general relativity and once without to see how they

compared.

Chapter 3

Results

The first result that I was interested in was what the location of Mercury would be

relative to the Earth from the perspective of the Sun, if I do not include general

relativity. To find this I had to adjust the M(0) seen in equation 2.1 with the effects

of Newtonian gravity but excluding the effects of General Relativity. This gave me a

final θ angle of 7.1836650 rads and a φ angle of 1.5539418 rads for Mercury compared

to Earth’s θ 7.1874917 rads and φ 1.5707963 rads giving us a difference of 0.0038267

rads in θ and 0.0168545 rads in φ which means it will be transiting. Now I just need

to find how much of a difference general relativity will make on Mercury’s location.

After adding in the effects of general relativity in to the M(0) variable in equation

2.1 the new θ angle is 7.1836741 rads and the new φ angle is 1.5539260 rads. This

gives us a difference in θ from the general relativity and θ from the lack of general

relativity to be 0.0000091 rads and a difference in φ of -0.0000158 rads. These angles

are so small, I converted them to arcs and applied the small angle formula. Once I

had the distance between these points, I found it was about 581.236 Km and using

Mercury’s average speed it would take about 17.973 ± 11.596 seconds to cross that

distance, meaning that unless we include general relativity the transit will happen

13

14 Chapter 3 Results

Figure 3.1 Motion After Each Orbit

about 18 seconds after NASA’s current prediction.

The transit happened exactly on time and as such, it would seem that Einstein was

right and that general relativity can be used to predict planetary location. This would

suggest that the other parts of the theory are valid and can be taken as accurate,

at least up to our ability to calculate and measure the effects. The transit itself

took about five and a half hours and according to my estimates there would be no

measurable difference in the transit time unless, we had more sensitive equipment.

Chapter 4

Conclusion

4.1 What This Means for General Relativity

Specific conditions are generally required to provide evidence for general relativity

as such it’s normally just a thought experiment, so it’s nice to provide some hard

evidence. The difference is very small and I cannot deny that this could be a simple

calculation error, however, we are talking about 43 arc seconds per century that’s

less than 12

an arc second per year. Burchell [1] voiced some disagreement and has

concluded that general relativity may not be as conclusive as it seems. Given how

small of a difference it made in my experiment, he may still have a point; however,

the experiment would need to be done with more precision in order to say for sure.

Once I included general relativity it made up the difference I needed. I am confident

in concluding that general relativity is the best explanation for the odd movement of

Mercury.

This lends itself to a lot of suggested evidences, namely that Einstein was right

in his way of thinking about the universe. This is not something that is universally

disputed but now anyone can test and see that it’s not just a thought experiment.

15

16 Chapter 4 Conclusion

With the understanding that general relativity is verifiable, we can conclude that

space-time does in fact act like an elastic fabric. It also suggests that if an object as

small as our Sun is able to create a measurable effect on a planet, then we can only

imagine what something that is ten or even a hundred solar masses can do.

Of course let’s not forget one of the more recent discoveries, gravitational waves.

Though my experiment does not directly provide evidence for the existence of gravi-

tational waves it does help validate the idea. Simply put, if one prediction of general

relativity can be verified then other parts of the theory are likely to be true as well.

And this may even include Time Travel as explained in the book ”Albert Einstein,

philosopher-scientist” [7]. Kurt Godel, explains in in a section titled ”A remark about

the relationship between relativity theory and idealistic philosophy”, that under very

extreme conditions time travel may be possible, he attributes the foundations of his

theory to General Relativity.

4.2 Other Methods

There are a few things that I would have liked to do differently and I think that

each has their positives and their negatives. Namely, when it comes to the math

of the gravitational effects of the other planets I used the calculations presented by

Burchell [1]. Now, I did this because I wanted to prove him wrong as he was trying

to prove that general relativity did not account for what we understand. However,

if I were to repeat the experiment I would prefer to calculate the effects of each

planet myself as this would provide a more accurate calculation. In addition, NASA

gives all their information in Cartesian for planetary location [6]. I then converted it

into spherical to help with the comparison to Earth. This may have caused a lot of

unnecessary math, it may be prudent to see if there would have been a better way.

4.3 Application 17

Using the perihelion advance of Mercury is not the only experiment one can per-

form. The most historically notable experiment for providing evidence for general

relativity is gravitational lensing. This is what happens when light passes a large

object like a distant galaxy or black hole. The only time someone without a NASA

telescope can see this effect is during a solar eclipse. Just like Eddingtons eclipse

experiment I referenced earlier, if you know the path of the eclipse you can chart the

stars that will be in the aria of the Sun. Then when the eclipse happens, chart them

again and you will see that the locations have changed. This would also show that

gravity has an effect on space time.

There are many other ways that one could provide evidence or verification for the

theory of general relativity. The method, I went with provides a level of simplicity

but I believe at the cost of some accuracy. Though many times in physics we give up

accuracy for simplicity and for most cases it comes out to be the same I would still

like to see how this could be perfected. The next transit will not be until 2032, which

would provide for a wonderful opportunity as there will be more time between it and

this last transit giving a bigger difference to look for.

4.3 Application

For this experiment and code you could simply change the initial conditions and look

at a different planet as I tried to keep it as general as possible. The one that comes to

mind first is Venus as it is the only other planet within Earth’s orbit. There are a few

problems with using Venus to prove general relativity. The main problems are that

the transits for Venus happen in pairs of 8 years apart, but each pair is separated by

121 years. The next transit for Vinus is not predicted to occur until the year 2125.

In addition one would need to find what effect general relativity has on Venus’s orbit

18 Chapter 4 Conclusion

if any.

This application could be taken to extra-solar planets as well; as I mentioned

I tried to keep it general though I had to make some assumptions for this specific

system. As long as you know the mass of the star a planet is orbiting, and the

observed period you could determine transit times, as well as if it is being affected

by general relativity. This could help in verifying mass of a planet or distance to

its star. Once you have verified that general relativity is affecting a planet you can

find the Newtonian effects as well and by extension make some estimates about the

organization of said solar system.

Finally we can see that the proper method to accurately predict the location of

Mercury is by including general relativity as explained by Einstein. By verifying that

his Mercury prediction is correct we are providing validity for the rest of his related

theories. Using these ideas we can infer information about massive objects or possibly

time travel. Though I believe more accuracy may be in order, I am convinced that

Einstein was right again and now anyone with a solar telescope can verify that.

Bibliography

[1] Bernard Burchell. Mercury’s perihelion advance, 2016.

http://alternativephysics.org/book/index.htm.

[2] University of Iowa. Imaging the universe, small-angle formula. Web Page, 2017.

http://astro.physics.uiowa.edu/ITU/glossary/small-angle-formula/.

[3] Sweden Paul Schlyter, Stockholm. how to compute planitary positions.

https://stjarnhimlen.se/comp/ppcomp.html, 2019.

[4] Chris Pollock. Mercury’s perihelion. http :

//www.math.toronto.edu/ colliand/42603/Papers03/CPollock.pdf , 2003.

[5] Jerry E. White Roger R. Bate, Donald D. Mueller. Fundamentals of Astrodynam-

ics. Dover Publications, Inc., 1971.

[6] Alan B. Chamberlin Ryan S. Park. Solar system dynamics.

https://ssd.jpl.nasa.gov/horizons.cgi, January 2020.

[7] & Einstein Albert Schilpp, P. A. Albert Einstein, philosopher-scientist. Library

of Living Philosophrs, 1949.

[8] Adam Szabo. How orbital motion is calculated. https://www-

spof.gsfc.nasa.gov/stargaze/Smotion.htm, 2016.

19

20 BIBLIOGRAPHY

[9] Anatoli Vankov. Einstein’s paper:“explanation of the perihelion motion of

mercury from general relativity theory”. Koniglich Preußische Akademie der

Wissenchaften, 01 2011. origonaly published 1915 Translated by Professor Roger

Rydin, University of Verginia.

Appendix A

Code for Mercury

1 # -*- coding: utf -8 -*-

2 """

3 Created on Tue Oct 1 14:04:50 2019

4

5 @author: user 1

6 """

7

8 from numpy import pi , sin , cos , sqrt , arctan , arccos

9

10 #######################################################

11 # using a recursive method to find the eccentric anomaly

12 #######################################################

13 def EcentricAnomaly(E,em ,M):

14 from math import sin

15 Enew = M + em*sin(E)

16 while abs(E - Enew) > 1e-8:

17 E = Enew

18 Enew = M + em*sin(E)

19

21

22 Chapter A Code for Mercury

20 return(Enew)

21

22 ########################################################

23 # calculate phi i.e. location above or below the ecliptic

24 #######################################################

25

26 def phiCalc(r,theta ,t):

27 from numpy import arccos , sqrt , pi

28 #staring information y = year , m = month , D = Day of month

29 y = 2016

30 m = 5

31 D = 9

32 d1 = 367*y-7*(y+(m+9) /12) /4+275*m/9+D -730530

33 UT = t/24 # a variable to keep a

consistent time

34 d = d1 + UT/24

35 #adjustment to phi with all mechanics considered

36 N = (48.3313+4.70935e-5*d)*(pi/180) #longitude of

ascending node

37 w = (29.1241+1.01444e-5*d)*(pi/180) #argument of

perihelion

38 i = (7.0047+5.00e-8*d)*(pi/180) #inclination to the

ecliptic

39

40 #adjustments to the phi angle with only Newtonian mechanics

considered

41 #N = (48.3313+((4.70935e-5*d)*0.92521))*(pi/180)

42 #w = (29.1241+((1.01444e-5*d)*0.92521))*(pi/180)

43 #i = (7.0047+((5.00e-8*d)*0.92521))*(pi/180)

44

45 x = r*(cos(N)*cos(theta+w)-sin(N)*sin(theta+w)*cos(i))

23

46 y = r*(sin(N)*cos(theta+w)+cos(N)*sin(theta+w)*cos(i))

47 z = r*(sin(theta+w)*sin(i))

48

49 J = z/(sqrt(x**2+y**2+z**2))

50 phi = arccos(J)

51

52 return phi

53

54 em = 0.20563069 #eccentricity of Mercury

55 a = 57909050.0 #semi -major axis in Km

56 T = 126675.36 #orbit of Mercury in Min

57

58 G = 6.67408e-11 #m^3/Kg s^2

59 Ms = 1.989 e30 #mass of the Sun in Kg

60

61 #Mercury ’s starting location

62 Mx = -2.98352231805042e-01 * 1.496e8

63 My = -3.39887939596431e-01 * 1.496e8

64 Mz = -4.01080274306904e-04 * 1.496e8

65 thetaT = arctan(My/Mx)+pi #total initial theta

66

67 #starting Mercury perihelion location

68 Px = 9.05492e-02 * 1.496e8

69 Py = 2.93617e-01 * 1.496e8

70 Pz = 1.56845e-02 * 1.496e8

71 thetaMp = arctan(Py/Px) #initial theta for perihelion

72 Mp = arccos ((em+cos (0))/(1+em*cos (0)))

73

74 theta1 = (thetaT - thetaMp) #initial true

anomaly

75 E = arccos ((cos(theta1)+em)/(1+em*cos(theta1))) #initial eccentric

24 Chapter A Code for Mercury

anomaly

76 n = sqrt((G*Ms)/((a*1000) **3))

77 Tp = -((Mp -em*sin(Mp))/n) #time of perihelion

passage

78 t = 0

79 t0 = (((E-em*sin(E))/n)+Tp)/60

80 Mo = 2*pi*(t0/T)+((13.6*( pi /(180*3600))))+((1.24*( pi /(180*3600))))

81 phi1 = arctan ((sqrt(Mx**2+My**2))/Mz)

82

83 x=[]

84 y=[]

85 z=[]

86 from numpy import mod

87 while t <= 1844362.63:

88 M = 2*pi*(t/T) + Mo #mean anomaly

89

90 Ef = EcentricAnomaly(E,em ,M) #replace old anomaly with new

91

92 r = a*(1-em*cos(Ef)) #solve for radius at time t

93

94 theta = arccos (1/em*((a*(1-em**2))/r-1)) #true anomaly at time

t

95

96 #keep the planet going in a circle

97 #correction statement to compensate for arccos limitation after

180

98 if pi < mod(Ef ,2*pi) < 2*pi:

99 theta = (2*pi) + (-1*theta)

100

101 phi = phiCalc(r,theta ,t)

102 phi1 = phi

25

103

104 xa = r*cos(theta)*sin(phi)

105 ya = r*sin(theta)*sin(phi)

106 za = r*cos(phi)

107

108 x.append(xa)

109 y.append(ya)

110 z.append(za)

111

112 E=Ef #new eccentric anomaly

113 t+= 1 #time step

114 last_r = r

115

116 print("phi",phi ,"Radius R", (r/10**6) ,"theta", (theta+thetaMp))

117

118 #########################################

119 #print a graph

120 #########################################

121

122 #from mpl_toolkits import mplot3d

123 #import numpy as np

124 #import matplotlib.pyplot as plt

125

126 #fig = plt.figure ()

127 #ax = plt.axes(projection ="3d")

128

129 #ax.scatter3D(x, y, z)

130 #ax.set_xlabel(’X ’)

131 #ax.set_ylabel(’Y ’)

132 #ax.set_zlabel(’Z ’)

133

26 Chapter A Code for Mercury

134 #ax.set_xlim (3.2*10**7 ,3.5*10**7)

135 #ax.set_ylim (3.2*10**7 ,3.5*10**7)

136 #ax.set_zlim (0.2*10**7 ,0.8*10**7)

137 #ax.set_aspect ("equal ")

138 #plt.show()

Appendix B

Code for Earth

1 # -*- coding: utf -8 -*-

2 """

3 Created on Mon Feb 10 16:29:33 2020

4

5 @author: user 1

6 """

7

8 from numpy import pi , sin , arccos , cos , sqrt , arctan

9

10 #######################################################

11 #using a recursive method to find the eccentric anomaly

12 #######################################################

13 def EcentricAnomalyE(E,ee ,M,Edif):

14 from math import sin

15 Enew = M + ee*sin(E)

16 while abs(E - Enew) > 1e-8:

17 E = Enew

18 Enew = M + ee*sin(E)

19 return(Enew)

27

28 Chapter B Code for Earth

20

21 ########################################################

22 #calculate phi i.e. location above or below the ecliptic

23 #######################################################

24

25 def phiCalcE(rE ,thetaE ,t):

26 from numpy import arccos , sqrt , pi

27 N = ( -11.26064) *(pi/180) #longitude of ascending node

28 w = (114.20703) *(pi/180) #argument of perihelion

29 i = 0 #inclination to the ecliptic

30

31 x = rE*(cos(N)*cos(thetaE+w)-sin(N)*sin(thetaE+w)*cos(i))

32 y = rE*(sin(N)*cos(thetaE+w)+cos(N)*sin(thetaE+w)*cos(i))

33 z = rE*(sin(thetaE+w)*sin(i))

34

35 J = z/(sqrt(x**2+y**2+z**2))

36 phiE = arccos(J)

37 return phiE

38

39 ee = 0.003353 #eccentricity of Earth

40 Ea = 149.60 e6 #semi -major axis in Km

41 T = 525948.768 #orbit of Earth in Min

42

43 G = 6.67408e-11 #m^3/Kg s^2

44 Ms = 1.989 e30 #mass of the Sun in Kg

45 m = 5.972 e24 #mass of Earth in kg

46

47 #Earth starting location

48 Ex = -6.619487581588294e-01 * 1.496e8

49 Ey = -7.62436937394947e-01 * 1.496e8

50 Ez = 3.386984851213855e-05 * 1.496e8

29

51 thetaT = arctan(Ey/Ex)+pi #total initial theta

52

53 #starting Earth perihelion location

54 Px = -2.00278e-01 * 1.496e8

55 Py = 9.62692e-01 * 1.496e8

56 Pz = -3.50150e-05 * 1.496e8

57 thetaEp = arctan(Py/Px)+pi #initial theta for perihelion

58 Ep = arccos ((ee+cos (0))/(1+ee*cos (0)))

59

60 thetaE = thetaT - thetaEp #initial true

anomaly

61 E = arccos ((ee+cos(thetaE))/(1+ee*cos(thetaE))) #initial eccentric

anomaly

62 n = sqrt((G*(Ms+m))/((Ea *1000) **3))

63 TpE = -((Ep -ee*sin(Ep))/n) #time of perihelion

passage

64 t = 0

65 t0 = (((E-ee*sin(E))/n)+TpE)/60

66 Mo = 2*pi*(t0/T)

67 phi1 = arccos(Ez/(sqrt(Ex**2+Ey**2+Ez**2)))

68

69 x=[]

70 y=[]

71 z=[]

72 from numpy import mod

73 while t <= 1845768.63:

74 M = 2*pi*(t/T) + Mo #mean anomaly

75

76 Edif = 1 #starting difference

77 Ef = EcentricAnomalyE(E,ee ,M,Edif) #replace old anomaly with

new

30 Chapter B Code for Earth

78

79 rE = Ea*(1-ee*cos(Ef)) #solve for radius at time

t

80

81 thetaE= arccos ((((Ea*(1-ee**2))/rE) -1)/ee) #true anomaly at

time t

82

83 #keep the planet going in a circle

84 #correction statement to compensate for arccos limitation after

180

85 if pi < mod(Ef ,2*pi) < 2*pi:

86 thetaE = (2*pi) + (-1* thetaE)

87

88 phiE = phiCalcE(rE ,thetaE ,t)

89 phi1 = phiE

90

91 xa = rE*cos(thetaE)*sin(phiE)

92 ya = rE*sin(thetaE)*sin(phiE)

93 za = rE*cos(phiE)

94

95 x.append(xa)

96 y.append(ya)

97 z.append(za)

98

99 E=Ef #new eccentric anomaly

100 t+= 1 #time step

101

102 print("phi", phiE ,"Radius R", (rE /10**6) ,"theta", (thetaE+thetaEp))

103

104 #########################################

105 #print a graph

31

106 #########################################

107

108 #from mpl_toolkits import mplot3d

109 #import numpy as np

110 #import matplotlib.pyplot as plt

111

112 #fig = plt.figure ()

113 #ax = plt.axes(projection ="3d")

114

115

116 #ax.scatter3D(x, y, z)

117 #ax.set_xlabel(’X ’)

118 #ax.set_ylabel(’Y ’)

119 #ax.set_zlabel(’Z ’)

120

121 #ax.set_xlim ( -10*10**7 ,10*10**7)

122 #ax.set_ylim ( -10*10**7 ,10*10**7)

123 #ax.set_zlim ( -10*10**7 ,10*10**7)

124 #ax.set_aspect ("equal ")

125 #plt.show()

32 Chapter B Code for Earth