Calculating the radial, in-track and cross-track distances

Question

I'm computing the relative distances between satellites. The following picture shows the distances, which are considered as risky.

1. What do Radial, In-track and Cross-track mean and why are they used?
2. How are they calculated?
3. Which one is the most important for decision making?

Answer 1

It looks like you're working from the JSpOC Spaceflight Safety Handbook for Operators (https://www.space-track.org/documents/JSpOC_Spaceflight_Safety_Handbook_For_Operators.pdf). In this case they define the RIC frame as identical to what is often called the UVW frame (https://www.space-track.org/documents/JSpOC_Pc_4Aug16.pdf pg 3).

This frame is defined such that:

Radial (R or U) is in the direction of the position vector

Cross-track (C or W) is in the direction of the angular momentum vector (P cross V)

In-track (I or V) is W cross U

The in-track vector will be coincident with the velocity vector for a perfectly circular orbit.

To calculate this for a conjunction scenario, calculate the relative position vector in ECI coordinates between your primary and secondary objects. Then multiply it by the ECI->UVW transformation matrix [T] for the primary.

      {u} = |{P}|    
[T] = {w} = |{P}x{V}|
      {v} = |{w}x{u}|

Where {P}, {V} are the ECI position and velocity vectors of the primary object. || indicates taking the unit vector.

{u}, {v}, {w} are then the rows of the transformation matrix [T]

So, to get the relative position vector in the RIC frame... Start by calculating the relative position in the ECI frame

{Prel} = {P} - {Psecondary}

where {P} is the ECI position of the primary object and {Psecondary} is the ECI position of the secondary object.

Calculate the transformation matrix [T] as described above using the ECI position & velocity of the primary object ({P}, {V})

Then the relative position in the RIC frame, {Pric} is:

{Pric} = [T]{Prel}