Formula to locate a scratch on a CD

I know this is a school boy mathematics problem but I'm lazy - has anyone come across the formula to locate a scratch on a CD?

I have a CD that skips about 1 minute in on track 5 and can see 1 or 2 scratches. I would like to polish a small area where the scratch is.

Since CD data starts at the centre so measuring the distance from the start of the data to where it ends, indicated by plain silvering, and knowing total play time and time where it skips, it should possible to calculate the distance from the start of data where the scratch is. The only complication is that unlike a vinyl record that plays at constant RPM, time is not constant with distance, and that's the part that needs thought.

Anyone up for the challenge?
 
There is an allowable variation between how close the spiral paths are next to each other. Meaning no estimate would be accurate enough to determine the exact spot.

If the scratch is on the top of the disc, the reflective layer could be damaged.

The CD could also have disc rot.
While not common for factory made audio CDs, I have a few examples that have visible evidence of rot damage.
 
Here are some pictures of a CD I have that has some easily visible disc rot.
 

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If the scratch is along the track, the laser will skip; if the scratch is a bit longer, the laser might even fail to track. This happened to more than one CD in my collection - usually by operating the tray too soon, before the disc had spun down.
However small scratches do polish out, unless the metal layer has been harmed. There are even "professional" CD descratching machines available...
Dirty lens or weak laser or bad adjustment might be an issue, too. As well as "different laser, different tracking ability".
 
Yes to all the above... but none of it answers the original question.... no matter, I figured it out and will post it up, don't have access to the mathematical symbols required.

There were 2 scratches inline with the tracks, the formula were within 2mm, I expected it to be closer but I am not sure the starting radius is measured correctly. I have polished the area around each and the CD no longer skips.