Runge-Kutta on Elliptical Orbits

In summary, the conversation discusses running a numerical analysis on elliptical orbits using the standard Runge-Kutta method and the difficulties of finding examples online. The equations from the Euler-Lagrange method in cartesian x,y-coordinates are provided, but the individual is unsure of how to define the velocity function. The suggestion is made to do a web search on "2D Runge-Kutta" and to convert the second order equations into a system of coupled first order equations.
  • #1
DrDress
3
0

Homework Statement


Run a numerical analysis on elliptical orbits using the standard Runka-Kutta method. I already have the equations from Euler-Lagrange method in cartesian x,y-coordinates.

d2x/dt2 = -K x (x2 + y2 )-3/2
d2y/dt2 = -K y (x2 + y2 )-3/2

Homework Equations


I find it a little to get started. Most examples online are first order differential equations and/or single variable. So I'm not sure how to define my f(x,y,t) funktion. It's supposed to be the velocity dx/dt, right? But I "only" have the acceleration, so do I just multiply dt to get the velocity or what?

The Attempt at a Solution

 
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  • #2
Do a web search on "2D Runge-Kutta"
 
  • #3
You need to convert your system of second order equations into a system of coupled first order equations. That means introducing two new dependent variables, representing the first derivatives of x and y.
 

1. What is Runge-Kutta method used for in elliptical orbits?

The Runge-Kutta method is a numerical method used for solving differential equations, which are mathematical equations that describe how one variable changes in relation to another variable. In the case of elliptical orbits, the Runge-Kutta method can be used to approximate the position and velocity of an object as it moves along its elliptical path.

2. How does the Runge-Kutta method work for elliptical orbits?

The Runge-Kutta method works by breaking down the orbit into small time intervals and calculating the position and velocity at each interval using a set of equations. These equations take into account the gravitational force between the object and the central body, as well as any other external forces acting on the object.

3. What are the advantages of using the Runge-Kutta method for elliptical orbits?

One of the main advantages of the Runge-Kutta method is its accuracy. By using smaller time intervals, the method can provide more precise results compared to other numerical methods. Additionally, the method can handle complex systems with multiple interacting bodies, making it useful for studying celestial bodies in our solar system.

4. Are there any limitations to using the Runge-Kutta method for elliptical orbits?

While the Runge-Kutta method is a powerful tool for approximating the motion of objects in elliptical orbits, it does have some limitations. For example, the method assumes that the forces acting on the object remain constant throughout the entire orbit, which may not always be the case in real-life scenarios.

5. How does the order of the Runge-Kutta method affect its accuracy for elliptical orbits?

The order of the Runge-Kutta method refers to the number of equations used to calculate the position and velocity at each time interval. Generally, a higher order method will provide more accurate results, but it also requires more computational power. For elliptical orbits, a fourth-order Runge-Kutta method is commonly used for a good balance between accuracy and efficiency.

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