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

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? • See this answer, particularly the "Addendum" in one of the answers. – Dave Nov 30 '18 at 18:54

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}

• Raw distance is a bad metric for collision risk because all distances aren't created equal. This is due to the position covariance (aka position uncertainty) of the objects. For most objects, particularly at some significant time after elset epoch, the in-track position uncertainty is much larger then the radial and cross-track uncertainty. So when assessing collision risk, a predicted radial offset reduces risk much more then the same offset in the in-track direction. – CoAstroGeek Nov 30 '18 at 18:58
• You would be well served to study the second document that I linked above - it's a decent overview of how collision risk is assessed. – CoAstroGeek Nov 30 '18 at 19:00
• As above, radial uncertainty is much smaller. These values are approximations arrived at through analysis. They can be used for simple "keep out" screening (what the JSpOC calls ellipsoid screening, or as a pre-screen for probability of collision (Pc) based screening. If the objects involved are well tracked and characterized, you should go with the Pc screening over the generic ellipsoid screening for exactly the reason you state. – CoAstroGeek Nov 30 '18 at 19:05
• I'm editing the answer above to make your last question clearer... – CoAstroGeek Nov 30 '18 at 19:06
• Yes ... and 12 more characters so I can submit – CoAstroGeek Nov 30 '18 at 19:26