C-Gravity:

Gravity visualization simulation in C using OpenGL version 4.2. This project is entrirely in C, no CPP, please ignore github reporting it is 39% CPP, this is a bug.

This simulation implements the Runge-Kutta-Fhelberg method for approximating solutions to Ordinary Differential Equations (Newton's Law of Graviational Attraction being an ODE) Implenetation is dervivied from Low-Order Classical Runge-Kutta Formulas with Stepsize Control and Their Application to Some Heat Transfer Problems by Erwin Fhelberg (NASA TR R-315, 1969) and can be found at:

https://ntrs.nasa.gov/api/citations/19690021375/downloads/19690021375.pdf

Coefficients used in the algorithm can also be found in the above, though one may find it easier to locate these references at https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta%E2%80%93Fehlberg_method

Special thank you to user kavan010 https://github.com/kavan010/gravity_sim/tree/main for teaching me how the heck to draw a sphere in OpenGL (https://www.youtube.com/watch?v=_YbGWoUaZg0), this project was very stuck until I found that.

Build:

Prerequisites

Before building, you'll need to install the required development libraries:

Ubuntu/Debian:

sudo apt update
sudo apt install build-essential libglfw3-dev libgl1-mesa-dev libglu1-mesa-dev

Fedora/RHEL/CentOS:

sudo dnf install gcc make glfw-devel mesa-libGL-devel mesa-libGLU-devel

Arch Linux:

sudo pacman -S gcc make glfw-x11 mesa glu

openSUSE:

sudo zypper install gcc make libglfw3-devel Mesa-libGL-devel glu-devel

Windows:

On Windows it is more complicatied to build this, but possible (I originally built this on Windows):
You will need to setup http://glfw.org/download.html .
If you do not know what you are doing, I reccomend visiting https://learnopengl.com/Getting-started/Creating-a-window as the instructions here are better than I can provide.

Mac:

Sorry, I do not currently have a Mac to build on.

Compile/Run

Once you have the dependencies installed, build the project with:

gcc -o solar_system *.c -lglfw -lGL -lGLU -lm -ldl
./solar_system

Troubleshooting

If you get "command not found" for gcc:

• Install build tools: sudo apt install build-essential (Ubuntu/Debian)

If you get library not found errors:

• You might need libglfw3-dev instead of glfw-devel

Examples:

First crack at textures (work in progress):

Recording.2025-08-08.100037.mp4

Recording.2025-08-07.221549.mp4

Older examples from before textures:

Please excuse the choppy lighting in the below examples for now, it is much smoother when I am running it versus how it comes out in the recordings, but I plan to redo lighting and add textures in the future which will hopefully make better videos.

Recording.2025-08-02.225341.mp4

Overhead shot:

Recording.2025-08-02.225254.mp4

Zoom in to show there are moon orbits:

Recording.2025-08-02.225521.mp4

A note on Low-Order Classical Runge-Kutta Formulas with Stepsize Control and Their Application to Some Heat Transfer Problems:

For those who are curious, a draft of the below is presented which is part of a larger work related to this project:

A Correction to the Truncation Error Coefficient for the Runge-Kutta-Fehlberg (RKF45) Method in Low-Order Classical Runge-Kutta Formulas with Stepsize Control and their Application to Some Heat Transfer Problems (NASA TR R-315, 1969)

Author: chudleyj

Date: August 2025

Abstract

In his 1969 NASA Technical Report R-315, Erwin Fehlberg presented the now widely used Runge-Kutta-Fehlberg (RKF45) method for solving ordinary differential equations (ODE). A typographical error in the leading term of the local truncation error is identified and clarified here. While this error does not propagate to other numerical results or alter the reliability of the method in practice, its documentation is important for pedagogical clarity and historical accuracy.

Introduction

The RKF45 method remains a cornerstone of numerical ODE solvers. Fehlberg's 1969 NASA report, Low-Order Classical Runge-Kutta Formulas with Stepsize Control and their Application to Some Heat Transfer Problems, is often referenced in both software implementations and academic texts. On page 12 of the report, Fehlberg explains that the leading term in the local truncation error can be found by subtracting the coefficients in the last two columns of Table III. These correspond to the weights of the 5th- and 4th-order Runge-Kutta methods, respectively. However, on page 14, a small but significant error appears in the result of that subtraction.

The Reported Error

Fehlberg's Table III (page 13) presents the coefficients for the 4th- and 5th-order formulas. At K=3 in that table are:

• $C_K = \frac{2197}{4104}$
• $C_K^{\hat{}} = \frac{28561}{56430}$

Fehlberg then states (page 14) that the difference between these values—used to compute the coefficient of the leading term in the local truncation error—is:

$$C_{T,K} = C_K - C_K^{\hat{}} = \frac{2187}{75240}$$

This result is incorrect. The correct difference is:

$$\frac{2197}{4104} - \frac{28561}{56430} = \frac{2197}{75240}$$

not $\frac{2187}{75240}$ as printed. This appears to be a typographical error where "2197" was mistyped as "2187".

Significance of the Error

This coefficient appears in an illustrative formula for the local truncation error and does not affect the main numerical experiments or the design of the RKF45 algorithm. Nonetheless, this value has occasionally been repeated in derivative materials, and its correction may assist students and developers in verifying the RKF45 coefficients themselves.

Has the Error Propagated?

We have not found evidence that this typographical error propagated into software implementations. Widely used sources such as Hairer's Solving Ordinary Differential Equations I list the correct coefficients for both the 4th- and 5th-order formulas. Additionally, established Fortran libraries such as those on Netlib (e.g., rkf45.f) implement the correct Fehlberg pair and do not include the subtraction error.

Impact

This error:

• Does not affect the core RKF45 method, which uses the original coefficients in Table III
• Is not propagated into the later sections of the report, which rely on numerical experiments, not symbolic expressions
• May cause confusion for readers attempting to independently verify the derivation of the local truncation error term or re-derive the RK pair
• Prompted debate on the Wikipedia article on the Runge–Kutta–Fehlberg method, where contributors disagreed about whether the stated coefficient was a typo or a valid form. Clarifying this helps stabilize open knowledge sources and prevents future re-edits based on the original misprint

Verification

To verify this correction yourself, calculate:

2197/4104 = 0.535331384015594...
28561/56430 = 0.506097560975610...
Difference = 0.029233823039984... = 2197/75240

The incorrect value would give:

2187/75240 = 0.029100478468900...

Conclusion

Although minor, this error in the original RKF45 paper deserves correction for historical accuracy and pedagogical clarity. It offers a useful example in courses or documentation that involve deriving or verifying Runge–Kutta coefficients. By clearly presenting the origin and nature of the error, we aim to provide a definitive reference—one that can help resolve future disputes in open knowledge sources and assist students and developers in confirming the correct coefficients with confidence.

References

1. Fehlberg, E. (1969). Low-order classical Runge-Kutta formulas with step-size control and their application to some heat transfer problems. NASA Technical Report R-315. Link

2. Hairer, E., Nørsett, S. P., & Wanner, G. (1993). Solving Ordinary Differential Equations I: Nonstiff Problems. Springer.

3. Dormand, J. R., & Prince, P. J. (1980). A family of embedded Runge-Kutta formulae. Journal of Computational and Applied Mathematics, 6(1), 19–26.

4. SLATEC Common Mathematical Library (1993). Subroutine RKF45. Netlib Repository. Link

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This correction is provided in the interest of mathematical accuracy and educational clarity.