Poynting–Robertson effect

Robertson considered dust motion in a beam of radiation emanating from a point source. A. W. Guess later considered the problem for a spherical source of radiation and found that for particles far from the source the resultant forces are in agreement with those concluded by Poynting.[1]

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

Radiation from a star (S) and thermal radiation from a particle seen (a) from an observer moving with the particle and (b) from an observer at rest with respect to the star.

From the perspective of the grain of dust circling a star (panel (a) of the figure), the star'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 star (panel (b) of the figure), the dust grain absorbs sunlight entirely in a radial direction, thus the grain's angular momentum is not affected by it. But the re-emission of photons, which is isotropic in the frame of the grain (a), is no longer isotropic in the frame of the star (b). This anisotropic emission causes the photons to carry away angular momentum from the dust grain.

Note that this anisotropic emission does not imply that an isolated radiating body in motion would decelerate (which would violate the principle of relativity). In this case, there would still be a net deceleration force (i.e. a decrease in momentum over time), but since the body's mass decreases as energy is radiated away, its velocity can remain constant.

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. While the dust grain thus spirals slowly into the star, its orbital speed increases continuously.

The Poynting–Robertson force is equal to:

${\displaystyle F_{\rm {PR}}={\frac {v}{c^{2}}}W={\frac {r^{2}L_{\rm {s}}}{4c^{2}}}{\sqrt {\frac {GM_{\rm {s}}}{R^{5}}}}}$

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

The Poynting–Robertson effect is more pronounced for smaller objects. Gravitational force varies with mass, which is ${\displaystyle \propto r^{3}}$ (where ${\displaystyle r}$ is the radius of the dust), while the power it receives and radiates varies with surface area (${\displaystyle \propto r^{2}}$). So for large objects the effect is negligible.

The effect is also stronger closer to the sun. Gravity varies as ${\displaystyle {\frac {1}{R^{2}}}}$ (where R is the radius of the orbit) whereas the Poynting–Robertson force varies as ${\displaystyle {\frac {1}{R^{2.5}}}}$, so the effect also gets relatively stronger as the object approaches the Sun. This tends to reduce the eccentricity of the object's orbit in addition to dragging it in.

In addition, as the size of the particle increases, the surface temperature is no longer approximately constant, and the radiation pressure is no longer isotropic in the particle's reference frame. If the particle rotates slowly, the radiation pressure may contribute to the change in angular momentum, either positively or negatively.

Radiation pressure affects the effective force of gravity on the particle: it is felt more strongly by smaller particles, and blows very small particles away from the Sun. It is characterized by the dimensionless dust parameter ${\displaystyle \beta }$, the ratio of the force due to radiation pressure to the force of gravity on the particle:

${\displaystyle \beta ={F_{\rm {r}} \over F_{\rm {g}}}={3LQ_{\rm {PR}} \over {16\pi GMc\rho s}}}$

where ${\displaystyle Q_{\rm {PR}}}$ is the Mie scattering coefficient, and ${\displaystyle \rho }$ is the density and ${\displaystyle s}$ is the size (the radius) of the dust grain.[2]

• Differential Doppler effect
• Radiation pressure
• Yarkovsky effect
• Speed of gravity