How to estimate the temperature of an interplanetary dust grain?

Question

I'm trying to solve this task:

Estimate the equilibrium temperature of an interplanetary spherical dust grain if it is located at a distance of 3 AU. from the Sun, and the substance of the dust particle reflects 30% of the incident radiation.

I think that this formula is the right one, but I am not sure:

$$T=(1 - A)^{1/4} T_{star} \left(\frac{r_{star}}{2 d} \right)^{1/2} $$

where $d$ is the distance from the Sun to the grain. This is the same as the last equation in Wikipedia's Planetary equilibrium temperature; Calculation for extrasolar planets where $A$ is the planet's bond albedo.

According to the formula the temperature of the dust is about 145.6 K.

Is it the right formula to use? If not then how should I calculate the temperature?

Answer 1

Partial answer so far

This is a really interesting question!

I've just started looking at Inferring the interplanetary dust properties from remote observations and simulations and surprisingly the power law fit to observed solar system dust temperature between 0.5 and 1.5 AU as shown in Eq. 2 varies as $d^{-1/3}$ instead of $d^{-1/2}$ in your equation.

It turns out

that we shouldn't use a blackbody (actually graybody since $A \ne 0$) model because dust particles are so thin they can't be treated as opaque, so more detailed optical simulations of the interaction of light with the particles is necessary, and that means models of the size, index of refraction, shapes and surfaces of dust particles is necessary.

Then (for population studies) you have to look at how those change with distance from the Sun.

Albedo alone is insufficient

But even for a single particle, albedo alone seems insufficient as a way to characterize a dust particle, unless it's really an opaque, smooth sphere.

Like a said, this is a partial answer so far, now reading section "5. Equilibrium temperature of the particles"

The thin solid line is $d^{-1/2}$ for a blackbody model and we can see that for some types of "rocky" (silicates, pyroxenes) particles there is a close match to the trend, with offsets due to albedo greater than zero. But lines with smaller slopes are those associated with organics or at least amorphous carbon.

I think...

but can't say for sure yet) that the reason these temperature distributions deviate from $d^{-1/2}$ is that the particles change their nature closer or farther from the Sun, probably due to temperature.

As they warm, their structure and chemical composition changes and therefore their optical properties and abilities to absorb and radiate light. So they are not the same at 1.5 and 0.5 AU from the Sun.

Therefore

For a "rocky" dust grain that doesn't change its properties as it warms or cools, the $d^{-1/2}$ variation is correct, but albedo alone is insufficient to get more than a trend. The plot below shows some simulated rocky particles are hotter and some are cooler than a blackbody model would suggest.

For those that contain organics, there will be no simple formula because the particles change their nature over time depending on their temperature history which in turn depends on their distance history. Some dust particles move towards the Sun, others move away.

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note: I'm only showing the top plot here.

Fig. 6. Logarithmic plot of the dust temperature in the symmetry surface as a function of the solar distance. Temperature inferred from observations (thick solid line) as compared to black body temperature (thin solid line) and temperature computed for spheres (top), spheroids (middle) and BCCA-BPCA aggregates (bottom) of equivalent diameter 1.5 µm. A satisfactory trend for the variation of the temperature with the solar distance is obtained in the case of absorbing organics or carbon material.