Voyager 2 course corrections

[morrisganderson]

Thank you for your videos on YouTube. I also appreciate that you’ve made the source code and the SolarSystemSimulatorTechnicalNotes.pdf document available on GitHub. The accuracy of your code is truly impressive.

Could you please explain how you modeled the course corrections for Voyager 2? I assume you must have incorporated some corrections; otherwise, I wouldn’t expect the results to align so well with the data.

I’m just a beginner programmer, so I haven’t tried installing your code on my MacBook yet to give it a try.

I’ve developed a simple N-body simulation code in MATLAB using a method I derived for modeling motion in a gravitational field. I’ve published a preprint to document this code ( https://www.researchgate.net/publication/389211533_N-body_Gravity_Simulation_by_Curvature_of_Wave_Propagation ) . I’d like to update this preprint to include the following information about your code.

Dr. Nico Kuijpers has developed a Solar System Simulator (available at https://github.com/nicokuijpers/SolarSystemSimulator) that allows the application of the PPN methodology to simulate general relativity and enhance accuracy. However, the PPN equations significantly increase solution time.

Please let me know if you would like any changes.

Thank you for your help!

Morris Anderson

[nicokuijpers]

Dear Morris,

Thank your for your interest in my Solar System Simulator and accompanying videos!

Regarding the course corrections for Voyager 2. You are right, I included course corrections shortly after each Trajectory Correction Maneuver (TCM). I extended the simulator with a scheduler to reinitialize position and velocity for given orbital paramaters at a certain time. Usually, I re-initialize position and velocity at 0.00 UTC the next day. Corresponding orbital parameters were obtained from JPL's Horizons System, from which position and velocity were calculated. Date/times for the TCMs were obtained from:

https://web.archive.org/web/20110723095624/http://www.dmuller.net/spaceflight/index.php

However, this list of TCMs is not complete. I also found one more TCM shortly before Neptune arrival (4 days and 5 million km till Neptune) in the book of Robin Kerrod entitled The Journeys of Voyager: NASA reaches for the planets, ISBN 0-792-45382-4. In addition, I have found some more between Uranus and Neptune by comparing the simulated velocity to the velocity computed from Horizons orbital parameters. In between the simulated corrections, Newton mechanics is used as the added value of General Relativity is limited (but it is possible to use GR as you mentioned). The results are pretty accurate.

The source code to simulate the trajectory of the Voyager 2 spacecraft can be found here:

https://github.com/nicokuijpers/SolarSystemSimulator/blob/master/src/main/java/spacecraft/VoyagerTwo.java

Date/times and orbital parameters are defined as constants. A SpacecraftTrajectory is defined by start time, stop time, orbital parameters and a center body. A SpacecraftEvent is a date/time at which the spacecraft and/or moon systems are re-initialized by the scheduler. I hope this clarifies it.

I think my method for PPN is the same as the one you compare your method with. I have used Equation 27 of W.M. Folkner et al., The Planetary and Lunar Ephemerides DE430 and DE431, IPN Progress Report 42-196, February 15, 2014:

https://ipnpr.jpl.nasa.gov/progress_report/42-196/196C.pdf.

For the implementation of the PPN method in my Solar System Simulator, see:
https://github.com/nicokuijpers/SolarSystemSimulator/blob/master/src/main/java/particlesystem/Particle.java

I would appreciate it if you mention my work the way you propose in your paper. May be you can mention the results I found wrt Mercury precession for comparison.

I will study your matlab code. It may be worthwhile to try your approach in my simulator for comparison, and perhaps speed up GR computations without giving in on accuracy.

Kind regards, Nico Kuijpers

[morrisganderson]

Dear Nico Kuijpers,

Thank you for explaining the course corrections you made for Voyager 2. I also appreciate the link to the Voyager JAVA code. Your documentation is excellent! I didn’t know about the TCM link. Thank you for pointing that out as well.

Your modeling of the Solar System and the creation of these videos truly bring these space probe missions to life.

I was also unable to identify all the course corrections for the Pioneer 10, 11, Voyager 1, and 2 spacecraft. Consequently, I included sample corrections in my paper that would yield the desired trajectories. I have posted some videos at https://www.youtube.com/@SimulatingGravity. I plan to create a few more in the future.

I intend to submit an updated version of my paper to ResearchGate sometime this fall. I will include the information about your SolarSystemSimulator, along with a note stating that it closely matches the NASA PPN prediction for the total precession of Mercury’s orbit.

Based on my understanding, I calculate the comparison as follows:

The calculated total precession of Mercury with the SolarSystemSimulator is 576.49 ’’/century
(From SolarSystemSimulatorTechnicalNotes.pdf Table 13).

This simulation started at Julian Year 2000 and extended for 1000 years; therefore, it is centered on Julian year 2500.

Applying the drift calculated by Park, Ryan S. et al. (See 2017 AJ 153 121 https://doi.org/10.3847/1538-3881/aa5be2 page 4), the corresponding calculated precession centered at Julian Year 2000 is then equal to 576.49 + (2000-2500)*(-0.0448/100) = 576.714 ’’/century.

Assuming the SolarSystemSimulator calculation is for the in-plane precession, we can calculate the expected observed value as:

( 0.998 * 576.714 ’’/century ) = 575.56 ’’/century.
See https://www.researchgate.net/publication/389211533_N-body_Gravity_Simulation_by_Curvature_of_Wave_Propagation Figure 8 and Appendix A.

This agrees very closely with the 575.31’’/century estimate calculated by Park, Ryan S. et al. (2017) with a PPN simulation.

Therefore, if you agree, I will update Figure 8 of my paper to include a SolarSystemSimulator precession rate for Mercury of 575.56 ’’/century.

If you find any errors in this comparison, please let me know.

Thank you for considering my approach as one of the options in your SolarSystemSimulator. I’m eager to know if you find any advantages in using it. Since it differs from general relativity, I suggest retaining the PPN approximation option for comparison.

I’m quite interested in knowing if you can enhance the numerical solution efficiency of my method by using an ODE solver or any other approach. I believe a crucial aspect of this is to accurately determine the change in beta (v/c) for each body along the direction of motion for each iteration. Perhaps curve fit information from previous time steps could be utilized for the current time step. I attempted this and managed to reduce the computational time by approximately 60% compared to the original without compromising on accuracy. However, it also reduced solution stability, so I have retained the simple approach outlined in my paper for now.

Please let me know if you have any questions about the equations.

Best regards,

Morris Anderson

[mhhbrasse]

Dear Morris Anderson,

I have been working together with Nico on the SolarSystemSimulator, and I am currently studying the TMG paper (great work!).
The SolarSystemSimulator software uses the Runge-Kutta 4th-order integration method (as ODE solver), and for some cases the Adams-Bashforth-Moulton (ABM) method. The easiest way, for us, to add/test your approach in the simulator would be to (only) use [TMG] Postulate 6 and formula (4-10), and integrate these formulas into our time-based simulation integration scheme (with the proper adaptations for the N-body simulation). Would this be a valid, or proper, approach to test the method in the SolarSystemSimulator ? Or am I overlooking something ?

In addition, I have another question.
I've been able to execute the Fortran code (listed in TMG, Appendix A), and I can reproduce the exact results listed in Appendix B (using the gfortran compiler). My question is about the listing in Appendix B on the Precession of Mercury. It reads:
rr, The radius from m for cr. [...]
.149600000000000E+12
This value for rr is about ~1 AU, which seems unrelated to the position of Mercury to the Sun. Can you please confirm that this is the intended value for rr for this case, and how accurate should this value be for the intended accuracy of the simulation ?

Best regards,
Marco Brasse'

[morrisganderson]

Dear Marco Brasse’

Thank you for your compliment and questions!

For this discussion, I will refer to my book, Time, Matter, and Gravity, as TMG.

Before attempting to answer your questions, I would like to address the constancy of the speed of light:

In the paper FOURTH TEST OF GENERAL RELATIVITY: PRELIMINARY RESULTS - see PHYSICAL REVIEW LETTERS (Shapiro 1968) https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.20.1265

I. I. Shapiro makes the following statement:

... The proposed experiment was designed to verify the prediction that the speed of propagation of a light ray decreases as it passes through a region of increasing gravitational potential.

This statement by Shapiro can be confusing and difficult to comprehend. If, as Einstein proposed, the speed of light is a constant, how can it also be a variable?

The answer lies in the definition of the speed of light. Einstein postulated that the measured speed of light is always the same regardless of velocity or position in a gravitational field when evaluated with a standard clock and ruler (Einstein). This is possible if the motion and gravitational influence on the clock and ruler cause the ruler’s length to change and the clock’s natural frequency to change by the necessary amounts. If so, despite the speed of light being a function of position within a gravitational field, the time-of-flight method consistently yields a value of 299792458 m/s.

This is true for the derived method documented in TMG.

Einstein, A., et al., with notes by Sommerfeld, A. (1952)
The principle of relativity : a collection of original memoirs on the special and general theory of relativity page 117
https://archive.org/details/principleofrelat00lore_0/page/116/mode/2up

Now, let’s address your questions.

Your question:
The easiest way, for us, to add/test your approach in the simulator would be to (only) use [TMG] Postulate 6 and formula (4-10), and integrate these formulas into our time-based simulation integration scheme (with the proper adaptations for the N-body simulation). Would this be a valid, or proper, approach to test the method in the SolarSystemSimulator ? Or am I overlooking something ?

My response:
I don’t believe using only TMG Postulate 6 and equation 4-10 will yield the most accurate results. When we examine TMG Postulate 6, we observe that gravitational force is directly proportional to rest mass. However, a photon of light possesses zero rest mass. Consequently, employing this approach will not enable us to calculate the bending or deflection of light such as near an apparent black hole (see example illustrating a light wave front slowing down and bending around the apparent black hole of M87). Therefore, I think you will need to apply the full method the get accurate results.

Your question:
This value for rr is about ~1 AU, which seems unrelated to the position of Mercury to the Sun. Can you please confirm that this is the intended value for rr for this case, and how accurate should this value be for the intended accuracy of the simulation ?

My response
I use equation 4-10 to calculate the speed of light for each position in the calculation of the planet’s orbit. On page 105 of the TMG, cr represents the speed of light away from the Earth’s influence at a distance of rr from the Sun. I used the semimajor axis of the Earth’s orbit for this calculation (see page 105 of the TMG). My thought process is as follows:

Begin with the standard speed of light on Earth at its standard location:
cs = 299792458 m/s

Use equation 4-11 to calculate the speed of light away from the influence of the Earth. This will be the approximate value of the speed of light away from the Sun at a distance equal to the Earth’s semimajor orbital distance. I must have calculated this value outside of the program using equation 4-11, which yielded the listed values for rr and cr. These values are then used to calculate the speed of light at the planet’s position from the Sun using equation 4-10.

If you use a slightly different value for rr, it won’t make a significant difference.

The MATLAB N-body code calculates cr from rr. Here’s some code for Example 1.

Note, in this example rr = reoa

% Calculate c away from the earth in the solar system. Also calculate c away from the solar system.
re = 6371008.7714; % Earth's IUGG mean radius (R1) see https://en.wikipedia.org/wiki/Earth_radius
reoa = 149.598e9; % Earth's Semi-major orbital axis, see https://nssdc.gsfc.nasa.gov/planetary/factsheet/earthfact.html
c_inf = csexp((2sum(mb(4))g/cs/cs)/re); % Calculate c away from the earth with TMG eq 4-11.
disp(['c = ' num2str(c_inf,15) ' m/s, away from Earth at ' num2str(re,8) ' meters from the Sun or the Earth’s average orbital radius'])
c_inf = c_infexp((2*(sum(mb)-mb(4))*g/cs/cs)/reoa); % Calculate c away from the solar system TMG eq 4-11.
disp(['c = ' num2str(c_inf,15) ' m/s, away from the Solar Systems influence' ])

Please be aware that the Fortran code provided in the TMG book is designed for a fixed central body. This simplifies the equations of motion for the orbiting body significantly. In contrast, the MATLAB n-body code is intended for an n-body simulation, which is considerably more complex. This is because the bodies in the simulation can exchange energy, altering their orbits. Therefore, the MATLAB code incorporates a few additional equations to accommodate this capability.

Additional questions and comments:

Could you compare the runtime of Example 6 from https://www.mathworks.com/matlabcentral/fileexchange/178949-n-body-gravity-simulation-by-curvature-of-wave-propagation with a similar SolarSystemSymulation run? This comparison should offer some insight into the potential advantages of the wave propagation method. Note that, although the MATLAB n-body simulation code doesn’t include an ODE, it does integrate the orbit by rotation steps, which allows for a variable time step for increased efficiency.

Could you run the MATLAB code as an external routine during a SolarSystemSimulation run?

If you have any further questions, please let me know. We can continue our correspondence through GitHub or feel free to contact me directly via email (listed in the n-body simulation paper). Alternatively, we can have a web meeting using Google Meet if you prefer.

Best Regards,

Morris Anderson

[mhhbrasse]

Dear Morris Anderson,

Thank you for your quick response. Your explanation is really helpful.
I will respond to the additional question and comments by e-mail, but please allow some time due to other activities.

About the following question:

Could you run the MATLAB code as an external routine during a SolarSystemSimulation run?

We have not experimented yet with such option. Alternatively, it could be beneficial to port the relevant code to Java.
Let's consider the options during our follow-up.

Best regards
Marco Brasse'

[morrisganderson]

Dear Marco Brasse’

Thank you!

I think the Java option would work. The MATLAB code is quite simple and should be easy to convert.

I look forward to your email to discuss options when you have time.

Best regards,

Morris Anderson