Composition of collateral rotations of a planet

In summary, the conversation touches on the topic of determining the rotation matrix needed to calculate the change in direction vector to the sun for a body orbiting it and rotating about its own axis. The question arises about how to approach this calculation and whether it can be done solely through composition of rotation matrices. The speaker mentions needing to rephrase the question in terms of the various rotations involved and suggests consulting relevant resources for a better understanding of the derivation and explanation.
  • #1
erore
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0
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.
 
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  • #2
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.
 
  • #4
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.
 
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  • #5
Composition of reference frames does not necessarily commute in three dimensions. Matrix multiplication is not commutative: A*B ≠ B*A in general.
 

1. What is the composition of collateral rotations of a planet?

The composition of collateral rotations of a planet refers to the materials or substances that make up the surface and interior of a planet, as well as the movement and distribution of these materials due to the planet's rotation.

2. How do the materials in a planet's collateral rotations affect its overall composition?

The materials in a planet's collateral rotations play a crucial role in determining its overall composition. For example, the presence of heavy elements such as iron and nickel in the planet's interior can result in a denser composition, while the presence of lighter elements such as carbon and oxygen can result in a less dense composition.

3. Are there different types of collateral rotations in planets?

Yes, there are different types of collateral rotations in planets. For example, terrestrial planets have solid rotations, meaning their entire surface and interior rotate together. Gas giants, on the other hand, have differential rotations, where their outer layers rotate at different speeds than their inner layers.

4. How does a planet's rotation affect its composition?

A planet's rotation can affect its composition in various ways. For instance, a faster rotation can result in a bulging at the equator and a flattening at the poles, leading to a slightly oblate shape. This can also affect the distribution of materials and the formation of geological features on the planet's surface.

5. Can the composition of collateral rotations change over time?

Yes, the composition of collateral rotations can change over time due to various factors such as geological processes, impacts from other celestial bodies, and atmospheric changes. These changes can result in alterations to the planet's surface features and overall composition.

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