# Composition of collateral rotations of a planet

1. Jan 22, 2014

### erore

A body is orbiting the sun and rotates about its axis (z). My coordinate system is co-rotating with the body. I need to determine how does a vector that points to the sun change after a certain period of time. Initially the sun vector lies in the xz plane. Basically I need to find the rotation matrix that I can apply to the vector. I have been googling for over a day without any result. Can you point me to an article or book with a good explanation and derivation or explain how this is done? Thank you.

2. Jan 22, 2014

### maajdl

It is probably a composition of several rotation matrices.
You need to rephrase your question in terms of the various rotations involved.
Doing so you will probably see how to calculate your overall rotation matrix.
Eventually, this might help us too.

3. Jan 22, 2014

### D H

Staff Emeritus
4. Jan 22, 2014

### erore

I will make an example. The Sun-Earth system - the Earth rotates about its own axis and at the same time it also orbits the Sun. I choose a coordinate system that corotates with the Earth (z axis is the spin axis) and I need to find how the direction vector to the Sun evolves with time. Unlike Earth - Sun system I presume circular orbit.

For example at t0 = 0, the non-normalized Sun direction vector is (1,0,1) and I want to know what is this vector at some time t (given the ω rotation angular speed and Ω orbiting angular speed).

(The same problem can be viewed from another coordinate system - Sun centric, where the z axis is perpendicular to the orbital plane and x axis is the direction to the Earth at t0. In this case I need to know the time evolution of a normal to the Earth surface given at t0.)
The Earth - Sun system is just an illustration of the rotations involved. I need this for a general case of a body orbiting another body while rotating about its axis.

This cannot be done as composition of rotation matrices, they do not commute and the results depend on what rotation I do first. I believe the proper derivation must start with the equations of motion.

Last edited: Jan 22, 2014
5. Jan 22, 2014

### D H

Staff Emeritus
Composition of reference frames does not necessarily commute in three dimensions. Matrix multiplication is not commutative: A*B ≠ B*A in general.