I've been looking for examples at how mass-loss rates are determined.

I'm studying a circumstellar dust shell ejected from a Wolf-Rayet star. I have some parameters like, expansion velocity of the shell (60km/s), the dust mass of the shell (0.1 M_sun), the radii of the shell (R_in=10000 AU and R_out=60000) and its age (T=26000 yrs).

I was wondering if there's a formula relating those parameters and if they are sufficient to determine the dust mass-loss rate or its dependent on other factors.

Could anyone help?

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    $\begingroup$ Hi, Sarah, welcome to Astronomy! I recall seeing your earlier post of this on Physics Stack Exchange. Cross-posting on Stack Exchange sites is frowned upon; if you want to get more attention for a question, you can put a bounty on it (once you have enough rep). If you want your question migrated, flag for moderator attention and request migration. Thanks! $\endgroup$
    – HDE 226868
    Jan 8 '16 at 22:38

Roughly: the mass loss rate for a spherically symmetric wind is given by $$\dot{M} = 4\pi r^2 \rho v_w,$$ where $\rho$ is the density in the wind, $r$ is the radius at which that density is measured and $v_w$ is the wind velocity.

Here you have $v_w$ and have the mass of material within two concentric radii - so you can work out the average density at some average radius in the shell.

The age of the shell is not relevant to the mass-loss rate, but assuming the wind parameters are unchanged over that time, they do tell you ho much mass has been lost in total $\simeq \dot{M} \tau$.


There is a lot of additional information you would need to get a mass-loss rate. What you have is a detection of some dust mass, and a time interval over which that mass was lost, so you could get something that looks like a mass-loss rate by taking the mass and dividing by the time over which it was lost, so 0.1 M_sun divided by 26,000 years. That's already a mass-loss rate of .0004 M_sun/year, which is quite high, but there are several problems that could lead to this being either an overestimate or an underestimate:

1) You presumably want an average mass-loss rate over very long time intervals, not just 26,000 years. So you need to know if this represents a steady process, or a kind of special event. Stars related to Wolf-Rayet stars, called "luminous blue variables," are often observed to lose mass in discrete episodes that happen an unpredictable intervals. So is the dust surrounding your star a typical feature that should be expected to always be there, or was the star selected because it has such a feature? If the latter, then the 26,000 year timescale is not a characteristic time for that object. A related issue is that 60 km/s is a very slow speed for a Wolf-Rayet wind, so that also suggests you had some kind of mass release incident when the star was very puffed out, not a more steady kind of wind.

2) An even worse problem is that all you know about is the mass in dust. That is probably only a small fraction of the total mass lost, because depending on the evolutionary state of the Wolf-Rayet star, there could be a lot of hydrogen, and very likely a lot of helium, that do not participate in dust-making. Also, the reason dust is not detected inside 10000 AU could be that the dust hasn't formed yet when it is so close to the ionizing radiation from the Wolf-Rayet star. If so, you cannot even use the dust mass to characterize the mass of the material that does form dust, without also knowing the fraction that has made dust at each particular radius.

3) Also, if the outer radius of the feature is at 60000 AU, and the feature has taken 26,000 years to form, then the average speed of the outer edge of the feature is the ratio of those numbers, yielding 2.3 AU/year. That translates into 11 km/s. So if the dust is moving at 60 km/s, but the outer edge of the feature is only moving at 11 km/s, that means the wind is piling into material in the interstellar medium. So it's not at all clear that the dust mass you are seeing is all from the Wolf-Rayet wind, some of it could be forming out of gas that was previously in the interstellar medium because something like 5/6 of the gas mass in that feature will have to have come from the interstellar medium. The Wolf-Rayet wind will be enriched in dust-forming material, but if it's only 5 times more so enriched, there will still be dust coming from the ISM. So that also depends on the evolutionary state of the Wolf-Rayet star. (The fact that you see so much dust suggests that this might be a "WC" type star, which are very enriched in carbon, so this last issue might not be a concern.)

So it would take some more thinking and more research on the object, with additional quantitative constraints, to get a longer term mass-loss rate estimate. But at least you have .0004 M_sun/yr as a kind of benchmark to begin that conversation.

  • $\begingroup$ Good point about the dust/gas ratio - I missed that. As regards the wind speed etc. This is a shell far from the star, so I don't think it's too slow. Also you expect the wind speed to fall as $(\rho r^2)^{-1}$, so no reason for it to be constant when $r$ changes by a factor 6. I guess I was thinking it was more of a homework problem than a "real" problem. $\endgroup$
    – ProfRob
    Dec 30 '16 at 0:20
  • $\begingroup$ You don't normally expect the wind speed to fall that way, you expect the wind speed to stay the same and the density to fall. To make the wind speed fall requires piling into something, but then the density is mostly not from the wind, it's from what is being piled into. The dust might be another matter, however. $\endgroup$
    – Ken G
    Dec 30 '16 at 12:59
  • $\begingroup$ The main point is, this gas is so slow it sounds like it was burped out by the star in a short-lived episode when the star was very puffed out by some kind of internal instability, perhaps a "failed supernova," as LBV eruptions are sometimes described. If so, the effect on the long-term mass-loss rate can only be known by knowing how often such episodes occur. $\endgroup$
    – Ken G
    Dec 30 '16 at 13:15

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