Poynting–Robertson effect

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. Poynting gave a description of the effect in 1903 based on the "luminiferous aether" theory, which was superseded by the theories of relativity in 1905–1915. In 1937 Robertson described the effect in terms of general relativity.

Explanation

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

Radiation from the Sun (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 Sun.

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:

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

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.

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 ${\frac {1}{R^{2}}}$ whereas the Poynting–Robertson force varies as ${\frac {1}{R^{2.5}}}$, the Poynting–Robertson effect also 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 µm in diameter, the radiation pressure 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.

Robertson considered dust motion in a beam of radiation emanating from a point source. Guess also considered the problem but 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 Robertson.

The dimensionless dust parameter $\beta$ is the ratio of the force due to radiation pressure to the force of gravity on the particle:

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

where $Q_{\rm {PR}}$ is the Mie scattering coefficient, and $\rho$ is the density and $s$ is the size of the dust grain.

The Equations of Motion for the dust grain are expressed by

$m{\operatorname {d^{2}} {\vec {x}} \over \operatorname {d} t^{2}}=-GMm{\Big (}1-\beta {\Big )}{{\vec {x}} \over r^{3}}+GMm\beta {\Bigg \{}{-{{{\vec {x}}\cdot {\vec {v}}} \over {cr}}{{\vec {x}} \over r^{3}}-{{\vec {v}} \over {cr^{2}}}+{R_{\rm {s}}^{2} \over {cr^{4}}}{\Big (}{\vec {\omega }}\times {\vec {x}}{\Big )}}{\Bigg \}}$

where $R_{\rm {s}}$ is the stellar radius.

See also

• Yarkovsky effect
• Differential Doppler effect

References

• Poynting, J. H. (1904). "Radiation in the Solar System: its Effect on Temperature and its Pressure on Small Bodies" (PDF). Philosophical Transactions of the Royal Society of London, Series A. 202 (346–358). Royal Society of London: 525–552. doi:10.1098/rsta.1904.0012.

• Poynting, J. H. (1903). "Radiation in the solar system: its Effect on Temperature and its Pressure on Small Bodies". Monthly Notices of the Royal Astronomical Society. 64 (Appendix). Royal Astronomical Society: 1–5. Bibcode:1903MNRAS..64A...1P. {{cite journal}}: Unknown parameter |month= ignored (help) (Abstract of Philosophical Transactions paper)

• Robertson, H. P. (1937). "Dynamical effects of radiation in the solar system". Monthly Notices of the Royal Astronomical Society. 97. Royal Astronomical Society: 423–438. Bibcode:1937MNRAS..97..423R. {{cite journal}}: Unknown parameter |month= ignored (help)