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When a comet loses all his volatile material after many orbits around the Sun, it becomes a "dead" comet (like an asteroid, a centaur or whatever), see for example 2015 TB145.

Is there any way to determine the lifetime of a comet (expressed as the number of orbits before it becomes a dead comet)?

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If you don't mind an oversimplified answer. There is a relatively straight forward relationship. The life of a comet (in orbits not years) depends on 3 factors. It's size, it's composition and how close it passes to the sun.

Comets are often referred to as dirty snowballs but the ratio of different types of ice and how much "dirt" or silicate material varies from comet to comet. Average density is about 0.6 g/cc. That variation is a factor in how much light gets reflected right back off the comet, and how easily the surface melts, but to make this answer somewhat easier, I'm going to ignore composition and make a simplified assumption that all comets are made of basically ice.

With that in mind, how much a comet loses per pass around the sun can be simplified into loss of radius. Think of a block of ice sitting in space, facing the sun. The rate of sublimation would simply be a product of how close the ice was to the sun. It's not far off to oversimplify melt rate in terms of loss of radius over a pass near the sun.

Halley's comet, for example has a fairly close pass to the sun, at about 0.587 astronomical units (closer than Venus). The closer a comet passes to the sun the faster it sublimates. Halley's is expected to last another 100-125 orits. Source.

As mentioned above, there's some variation in terms of the density and different materials that make up a comet, but the primary factors are Perihelion and size of the comet.

Halley's has a diameter of about 8 KM. (roughly speaking, it's 16x8x8 km, ellipsoid shape). 8,000 meters over 100-125 passes suggests it loses about 60-80 meters per pass around the sun and if we estimate that the "dead comet" after 100 or so orbits retains 1-3 KM of rocky material, the ballmark melt-rate of Halley's is somewhere in the 30-60 meters per pass around the sun, or a meter every couple-few days by sublimation. A comet with a perihelion at .8 AU or 1 AU would lose less radius per pass.

A comet that passes within 0.3 AU instead of 0.6 might lose about 4 times as much per day as it's hit by 4 times the solar energy per square meter, but it spends fewer days due to a faster orbit near the sun. Somebody might be able to run the math based on highly elliptical orbits, perihelion and solar energy per square meter over the entire pass. I'm afraid to try myself, but it would be something less than the square of the distance due to the faster orbital speed for smaller perihelion.

Wit this estimate in mind, Wikipedia says a sungrazing comet with a perihelion of about 0.055 AU and an initial radius of about 2-3 km could survive that pass and be about 1 km. That would be a loss of 1,000-2,000 KM over the pass, but that's a very close pass.

I think that's the start of an answer to this question. The closer to the sun, the more meters of radius the comet will lose per orbit and calculating the radius losing so many meters every pass gives a rough estimate of the life of the comet. Sublimation can begin as far away as the orbit of Saturn, but at that distance the sublimation is very slow.

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