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Sandbox Physics
Comet dust, from light pressure to real images

L1 · Comet dust dynamics

From photon momentum to a grain's reduced solar gravity

Sandbox Physics ·

A dust grain can move away from a comet even when its acceleration points toward the Sun. To see why, first compare the two central forces. Their ratio gives a simpler starting point than either force alone.

Choose units and the direction of the force

Measure the vector r\boldsymbol r from the Sun to the grain, and write its length as rr. An outward radial unit vector is r^\hat{\boldsymbol r}. The solar luminosity is energy emitted per second. Spread it over a sphere of radius r to obtain the energy flux at the grain.

For this first derivation use metres, kilograms and seconds. We freeze solar luminosity at 3.828 times ten to the twenty-sixth watts and solar gravitational parameter at 1.3271244004193938 times ten to the twentieth cubic metres per square second. The simulation later changes units to astronomical units and days without changing the physics.

S(r)=L⊙4πr2,Fg=−GM⊙mr2r^S(r)=\frac{L_\odot}{4\pi r^2},\qquad \boldsymbol F_g=-\frac{GM_\odot m}{r^2}\hat{\boldsymbol r}
Energy flux has units watts per square metre. The negative sign in gravity means inward. Solar distance will cancel from the force ratio because both flux and gravity decrease with inverse distance squared.

Reading: Burns · JPL Solar System Dynamics astronomical parameters

Turn intercepted light into momentum

Light carrying energy E carries momentum E/cE/c. A grain of radius s intercepts light over area πs2\pi s^2. Multiply intercepted power by a dimensionless pressure efficiency QprQ_{\rm pr} and divide by the speed of light to obtain the outward force. The efficiency includes the momentum consequences of absorption and scattering; it is not simply the fraction of photons absorbed.

For a sphere, mass is density times volume. Substituting the mass and dividing radiation force by gravity cancels the grain's cross-sectional area, solar distance and several factors of pi. One power of grain radius survives in the denominator.

Frad=L⊙Qprs24cr2,m=4π3ρs3,β≡FradFg=3L⊙Qpr16πcGM⊙ρsF_{\rm rad}=\frac{L_\odot Q_{\rm pr}s^2}{4cr^2},\qquad m=\frac{4\pi}{3}\rho s^3,\qquad \beta\equiv\frac{F_{\rm rad}}{F_g}=\frac{3L_\odot Q_{\rm pr}}{16\pi cGM_\odot\rho s}
Here the denominator Fg means the magnitude of gravity. Beta is dimensionless. It is independent of solar distance only when the adopted grain properties and spectrum-averaged pressure efficiency stay fixed.

Reading: Burns · Finson & Probstein

Work an example before integrating an orbit

Set ρ=1000 kg m−3\rho=1000\,\mathrm{kg\,m^{-3}} and Qpr=1Q_{\rm pr}=1. A three-micrometre radius is three millionths of a metre. Substitution gives beta about 0.191. The outward radiation force therefore cancels about nineteen percent of solar gravity, leaving roughly eighty-one percent of the original inward acceleration.

The following table uses the same frozen constants, spherical density and efficiency. It can be reproduced by direct substitution, without opening the Lab. Doubling grain radius halves beta; doubling density does the same. A real grain's efficiency can depend on size and wavelength, so extending the inverse-size table to molecular sizes would be unjustified.

β=0.574236761 Qpr(1000 kg m−3ρ)(1 μms)\beta=0.574236761\,Q_{\rm pr}\left(\frac{1000\,\mathrm{kg\,m^{-3}}}{\rho}\right)\left(\frac{1\,\mu\mathrm m}{s}\right)
The numerical coefficient follows from the stated luminosity, gravitational parameter and exact speed of light. It is a model conversion, not a measured ISON grain density.
Radius (micrometres)BetaInward fraction of gravity
10.5740.426
30.1910.809
100.05740.943
1000.005740.994

Reading: Burns · JPL Solar System Dynamics astronomical parameters

Combine forces without changing the inherited velocity

The two radial forces combine into a Kepler problem with a smaller gravitational parameter. This replacement changes acceleration, not the velocity the grain already had at release. A useful first comparison sets the gas launch velocity to zero: the grain and nucleus begin at the same position and velocity, then their accelerations diverge.

Beta below one guarantees inward central acceleration in this simplified model. It does not guarantee a bound orbit. Orbital energy also depends on the inherited speed. The next chapter derives the escape threshold and shows why the familiar one-half threshold only applies to release from a circular parent orbit.

r¨=−GM⊙(1−β)rr3≡−μ′rr3\ddot{\boldsymbol r}=-GM_\odot(1-\beta)\frac{\boldsymbol r}{r^3}\equiv-\mu'\frac{\boldsymbol r}{r^3}
This short-time model assumes constant beta, collisionless grains and an already completed gas-acceleration phase. It omits PR terms of relative order speed divided by light speed. Our interactive radii give beta below one, allowing attractive elliptic, parabolic-limit and hyperbolic conics.

Reading: Finson & Probstein · Burns

Try the observation

Use size to change the force ratio

  1. Pause the default orbit and drag the cyan grain toward the fine-grain end. Compare beta and its separation from the nucleus.
  2. Select the zero-force preset. With no launch velocity, every grain returns to the same nucleus trajectory.

Exercises and explained answers

A ten-micrometre grain has twice the teaching density. What beta should you use, and does solar distance change it?

Halving the area-to-mass ratio halves beta from about 0.0574 to 0.0287. Solar distance cancels from this ratio under fixed pressure efficiency. Both individual forces still become weaker farther from the Sun.

A student sees a grain separating from the nucleus and concludes its net force must point outward. What observation is missing?

Separation compares two trajectories, not the direction of either acceleration. The nucleus turns inward more strongly. Check the grain's curvature in a solar frame or compute its acceleration; beta below one still gives an inward acceleration. Its outward displacement relative to the nucleus is compatible with that fact.

Sources and calculation scope

  1. Finson & Probstein, A Theory of Dust Comets I (1968)

    ApJ 154, 327–352, DOI 10.1086/149761. Sections I–III, especially pp. 329 and 334–335 / Figure 4: post-gas-drag dust motion, reduced solar gravity, synchrones and syndynes. We add an explicitly assumed launch law for the teaching scene.

  2. Burns, Lamy & Soter, Radiation Forces on Small Particles (1979)

    Icarus 40, 1–48, DOI 10.1016/0019-1035(79)90050-2. Sections III–IV derive the first-order radiation force; section VI, equation (19) / Figure 7 discuss pressure efficiency; sections VII–VIII, equations (47)–(50) give reduced-gravity orbits and orbit-averaged PR evolution. The paper was read in the author-hosted 48-page PDF.

  3. JPL Solar System Dynamics astronomical parameters

    Exact AU, day and speed of light; planetary-system GM values for the sampled relative-force check. Solar gravitational parameter 132712440041.93938 cubic kilometres per square second and luminosity 3.828 times ten to the twenty-sixth watts are frozen from the Horizons Sun metadata. The teaching density and pressure efficiency are assumptions, not measured ISON grain properties.

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