Definitions

# Poynting-Robertson effect

[poin-ting-rob-ert-suhn]
The Poynting-Robertson effect, also known as Poynting-Robertson drag, named after John Henry Poynting and Howard Percy Robertson, is a process by which solar radiation causes a dust grain in the solar system to slowly spiral inward. The drag is essentially a component of radiation pressure tangential to the grain's motion. The first description of this effect, given by Poynting in 1903, was still "luminiferous aether"-based. Later, in 1937, Robertson described the effect correctly in terms of general relativity.

## Explanation

The effect can be understood in two ways, depending on the reference frame chosen.

From the perspective of the grain of dust circling the Sun (panel (a) of the figure), the Sun's radiation appears to be coming from a slightly forward direction (aberration of light). Therefore the absorption of this radiation leads to a force with a component against the direction of movement. (The angle of aberration is extremely small since the radiation is moving at the speed of light while the dust grain is moving many orders of magnitude slower than that.)

From the perspective of the solar system as a whole (panel (b) of the figure), the dust grain absorbs sunlight entirely in a radial direction, thus the grain's angular momentum remains unchanged. However, in absorbing photons, the dust acquires added mass via mass-energy equivalence. In order to conserve angular momentum (which is proportional to mass), the dust grain must drop into a lower orbit.

Note that the re-emission of photons, which is isotropic in the frame of the grain (a), does not affect the dust particle's orbital motion. However, in the frame of the solar system (b), the emission is beamed anisotropically, and hence the photons carry away angular momentum from the dust grain. It is somewhat counter-intuitive that angular momentum is lost while the orbital motion of the grain is unchanged, but this is an immediate consequence of the dust grain shedding mass during emission and that angular momentum is proportional to mass.

The Poynting-Robertson drag can be understood as an effective force opposite the direction of the dust grain's orbital motion, leading to a drop in the grain's angular momentum. It should be mentioned that while the dust grain thus spirals slowly into the Sun, its orbital speed increases continuously.

The Poynting-Robertson force is equal to:

$F_\left\{PR\right\} = frac\left\{Wv\right\}\left\{c^2\right\} = frac\left\{r^2\right\}\left\{4 c^2\right\}sqrt\left\{frac\left\{G M_s \left\{L_s\right\}^2\right\}\left\{R^5\right\}\right\}$

where W is the power of the incoming radiation, v is the grain's velocity, c is the speed of light, r the object's radius, G is the universal gravitational constant, Ms the Sun's mass, Ls is the solar luminosity and R the object's orbital radius.

Since the gravitational force goes as the cube of the object's radius (being a function of its volume) whilst the power it receives and radiates goes as the square of that same radius (being a function of its surface), the Poynting-Robertson effect is more pronounced for smaller objects. Also, since the Sun's gravity varies as one over R2 whereas the Poynting-Robertson force varies as one over R2.5, the latter gets relatively stronger as the object approaches the Sun, which tends to reduce the eccentricity of the object's orbit in addition to dragging it in.

Rocky dust particles sized a few micrometers need a few thousand years to get from 1 AU distance to distances where they evaporate.

For particles much smaller than this, radiation pressure, which makes them spiral outwards from the sun, is stronger than the Poynting-Robertson effect that makes them spiral inward. For rocky particles about half a micrometer µm in diameter, the radiation presure equals gravity, and they will be always blown out of the solar system even though the Poynting Robertson effect still affects them Particles of intermediate size will either spiral inwards or outwards depending on their size and their initial velocity vector.

## References

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