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Comets orbiting the sun have a tail that points away from the Sun, due to the solar wind.

(Strictly speaking, a comet can have both a type I tail - ion tail, and a type II tail - a dust tail. This question is about the type II tail - the dust tail).

Now I understand that some of the source of the dust is volatile materials in the comet oxidising due to the comet's proximity to the sun. But surely some of these comets have been orbiting for hundreds of thousands of years (if not millions). Why do they still have unspent fuel? Why are they still dusty? Hasn't all the dust blown off by now?

My question is: Why do comets appear to have a continuous supply of dust?

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The dust ejected is a small fraction of the overall comets mass, and the proportion of a time a comet spends close to the sun is low compared to the duration of their orbits, so they don't always appear so "dusty". – Dean Mar 7 at 12:27
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My outside take on it is that once they run out of exposed surface volatiles to supply a tail or coma in strong sunlight, it is simply no longer classified as a comet. This as a matter of our conventional historical accounting rather than as a matter of relevantly describing nature as it is understood today. And a comet turns off when most of the volatiles on its surface has sublimated. Like snow plowed up at a street corner. In early spring the surface snow melts, which increases the concentration of sand and dirt on its surface, blocking sunlight and slowing down further melting. – LocalFluff Mar 7 at 15:03
    
Don't know enough to call this an answer, but when the comet is inactive, wont it "gather" a little "dust" as it's essentially the only mass around. Not a lot of dust, but If it "burns" up 100kg of dust per orbit and gains 2kg of dust per orbit this would stretch out it's lifespan. – coteyr Mar 7 at 16:42
    
It's a vacuum, no oxygen- does it actually burn? – Spotdog Mar 7 at 20:40
up vote 12 down vote accepted

The rate of loss of mass from a comet is perhaps surprisingly low. A paper The calculation of $Af\rho$ and mass loss rate for comets gives a rate for a "typical" comet at 1.29AU as 153kg/s. A typical comet has a mass of about $10^{13}kg$. Even if the comet is constantly active (and they are not) it would have enough material for over 2000 years.

In fact comets are only active when the are near perihelion, and they are only near perihelion for a small fraction of their orbit. Put it together and you have the potential for an active life of hundreds of thousands of years.

Eventually, however, comets do run out of volatile substances, at which point they become inactive, and appear as small dark asteroids. An example of which is the object 2015 TB145, which passed the Earth on Halloween 2015.

Note: not "oxidising", but "vaporising", the ices in the comet are changing into gas, but there is no burning.

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To be fair, 2000 years is basically nothing on astronomic timescales. I think more math is called for to work out the typical amount of time a comet is active per orbit versus the typical duration of the comet's complete orbit, and from that the typical number of orbits that a comet might be expected to attain before becoming inactive. The result, as compared to the total age of the solar system, might be interesting. Especially if it works out to be a number of less than a few billion years, as that would imply that new comets are created over the life of the solar system. – aroth Mar 7 at 15:41
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@aroth: The usual explanation is that most comets (or "comet precursors"?) spend their entire life in very distant orbits where they never become active, making up the hypothetical Oort cloud. Every so often one of them will be perturbed and fall towards the sun. A small minority of those will become further perturbed when they pass through the inner solar system and thereby lose enough energy to end up with human-scale orbital periods. So the population of "Halley-like" comets is indeed continually replenished. – Henning Makholm Mar 7 at 18:41
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Halley's comet, maybe the most active known, can be "visible" for maybe two months of each 74-79 year period. 2000 years of activity in 2-month chunks then means about 12000 orbits. So, it could easily have been close to its current orbital characteristics for a million years if "typical" (which it apparently isn't). – user2338816 Mar 7 at 23:51

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