# Function for orbits based on time.

• RandomMystery
In summary, the conversation discusses methods for finding a formula for the position of two bodies in space based on initial velocity and time. The use of calculus II and above, specifically "quadrature," is suggested to solve the problem. The conversation also mentions the use of elliptic integrals, which cannot be integrated in terms of elementary functions.
RandomMystery
Okay, I am trying to find a formula for the position of two bodies (or more) in space based on the initial velocity and time.

I'm trying to integrate the equation for gravitational acceleration to find the velocity equation, however, the radius is changing, and the degree (theta) is also changing:

Edit:Does anyone know any calculus II and above method to solve this? I can't so far with Calc I

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You will be able to reduce that a bit by using "quadrature". Let v= dy/dt. Then
$$\frac{d^2y}{dt^2}= \frac{d}{dt}\left(\frac{dy}{dt}\right)= \frac{dv}{dt}$$
and, by the chain rule,
$$\frac{dv}{dt}= \frac{dv}{dy}\frac{dy}{dt}= v\frac{dv}{dt}$$

so you have
$$\frac{dv}{dy}= \frac{Gmy}{\sqrt[3/2]{x^2+ y^2}}$$
Of course, you will have to have something of the same form in terms of x, probably, letting
u= dx/dt
$$\frac{du}{dx}= \frac{Gmx}{\sqrt[3/2]{x^2+ y^2}}$$

I suspect that, at best, you will be able to reduce that to an elliptic integral (so named because they arise in calculating the elliptic orbits of planets) which cannot be integrated in terms of elementary functions.

Last edited by a moderator:

## 1. What is a function for orbits based on time?

A function for orbits based on time is a mathematical equation that describes the path of an object as it orbits around a central body, such as a planet or star. It takes into account the position and velocity of the object at a specific time and can be used to predict its future location.

## 2. How is the function for orbits based on time calculated?

The function for orbits based on time is typically calculated using Kepler's laws of planetary motion and Newton's laws of motion. These laws describe the relationship between an object's position, velocity, and acceleration, and can be used to derive the equations for the orbital path of an object.

## 3. What factors can affect the function for orbits based on time?

The function for orbits based on time can be affected by a number of factors, including the mass and distance of the central body, the mass and velocity of the orbiting object, and the presence of other objects in the system. These factors can alter the shape, size, and speed of the orbit.

## 4. How is the function for orbits based on time used in space exploration?

The function for orbits based on time is an essential tool for space exploration missions. It is used to calculate the trajectory of spacecraft, predict the timing of flybys and rendezvous, and plan orbital maneuvers. It is also used to monitor and control the orbits of satellites and other objects in space.

## 5. Can the function for orbits based on time be applied to objects other than planets and satellites?

Yes, the function for orbits based on time can be applied to any object that follows a predictable path around a central body. This includes comets, asteroids, and even artificial satellites launched by humans. The function can also be used to study the motion of celestial bodies, such as stars and galaxies.

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