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Exploring the Two-Body Problem: A Python Approach to Gravity

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Introduction to the Two-Body Problem

In an inertial reference frame, two masses experience a gravitational attraction towards each other. The aim of the two-body problem is to analyze how these two objects move in relation to one another due to this gravitational force. Below is a simulation of the two-body problem that will be discussed throughout this piece. For a deeper understanding, refer to Example 2.2 in the latest edition of "Orbital Mechanics for Engineering Students."

Understanding the Two-Body System

The diagram presented illustrates the two-body problem, featuring masses m₁ and m₂ within an inertial reference frame, which remains unaccelerated. This system is unaffected by external forces, with gravity being the sole interaction between the two bodies.

Diagram of Two Masses in Inertial Space

The essential parameters highlighted include:

  • R₁: the position vector from the origin of the inertial frame to m₁, represented in Equation 1.
  • R₂: the position vector from the origin to m₂, as defined in Equation 2.
  • r: the position vector from m₁ to m₂, derived from Equation 3.
  • G: the center of mass (COM) of the system, which shifts closer to m₁ if m₁ is greater than m₂.
  • RG: the position vector from the inertial frame's origin to the COM, calculated using Equation 4.

Deriving Motion Equations

Applying Newton's Law of Universal Gravitation alongside Newton's Second Law yields the equations of motion (EOM). With the EOM and the initial position and velocity vectors known, the system's state can be evolved over time.

Equation 5 illustrates Newton's Law of Universal Gravitation, which defines the attractive force F between two masses. Equation 6 expresses Newton's second law, correlating an object's acceleration with the net forces acting upon it. In the context of this two-body system, gravity is the only force at play. Thus, the force F₁₂ acting from m₁ to m₂ is synthesized into the two-body EOM as shown in Equation 7.

Newton's third law asserts that the force exerted by m₂ towards m₁ equals the force exerted by m₁ towards m₂. Equation 8 provides the unit vector from object 1 to object 2. By substituting this unit vector into Equation 7, we arrive at the complete form of the motion equations.

The results of these equations, represented in Equations 9 and 10, describe the inertial motion of both bodies. The vector expression for acceleration encompasses components in the X, Y, and Z directions, leading to Equations 11 to 16, which break these relationships down further.

Numerical Integration of Motion Equations

To forecast the positions and velocities of the masses over time, numerical integration is employed. The odeint function from the SciPy Python library serves as an effective integral solver. This method, akin to Runge-Kutta, requires the conversion of second-order ordinary differential equations (ODEs) into first-order ones, achieved by introducing auxiliary variables as indicated in Equations 17 to 29.

The initial conditions define both objects' positions and velocities at the start of the simulation, as demonstrated in Equations 42 to 45, which are referenced from the orbital mechanics textbook utilized in this analysis.

Python Implementation of the Two-Body Problem

Gist 1 showcases the Python code that sets up the simulation parameters, including the masses of both bodies and their initial conditions encapsulated in the state vector y₀.

The two-body problem is articulated in Python as a Numpy array, as seen in Gist 2. Ensure these equations align with Equations 11 through 16. The propagation of initial conditions over time to solve the two-body problem is succinctly executed with the following Python command:

# solve two-body problem

y = odeint(two_body_eqm, y0, time, args=(G, m1, m2))

Here, y captures the history of the state vectors, documenting the positions and velocities of m₁ and m₂ from time t₀ = 0 to t = 480 seconds. Specifically:

  • y[0:3]: the X, Y, and Z components of R₁ at time tᵢ
  • y[3:6]: the position of m₂ in relation to the inertial frame

Gist 3 illustrates the Python code responsible for computing additional position vectors relative to significant points of interest, enabling visualization of motion from various perspectives, such as from the center of mass or one of the bodies.

Simulation Results and Analysis

Figure 2(a) presents the simulation animation from two distinct perspectives. The trajectories of m₂ and the center of mass appear elliptical when viewed from body 1. Conversely, both bodies' orbits are also elliptical from the perspective of an observer positioned at the center of mass.

The simulation results provide insight into the periodic spiral motion of the two-body system as it orbits around the straight-line trajectory of the center of mass within the inertial frame.

Conclusion

Within the inertial frame, the center of mass travels in a straight line at a constant speed, affirming the absence of external forces on the system. This perpetual motion continues indefinitely.

This article illustrates a methodology for deriving and numerically integrating the equations governing the motion of two bodies solely influenced by their mutual gravitational force. The approach discussed can be simplified into a single-body problem. For further analysis, refer to the sources listed below.

Find the Python code for solving the two-body equations of motion in Gist 4.

References

[1] Orbital Mechanics for Engineering Students

[2] Simplifying the Two-Body Problem to a single-body scenario

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