AC11 · Grains inherit an orbit, light changes it

Comet Dust Tails and Radiation Pressure

Why does a comet's tail bend away from its path? Released dust keeps the comet's orbital velocity. Sunlight then reduces the inward force on small grains more than on large ones, so their new orbits spread into a curved tail. An ion tail follows a different process driven by the solar wind and magnetic field.

Drag the nucleus around the Sun or change a grain's size. Watch released dust form a three-dimensional fan, then separate months of radiation pressure from millennia of Poynting–Robertson drift.

Radiation pressure / solar gravity $0.191$

Nucleus–Sun distance $1.27\,\mathrm{AU}$

Farthest grain from nucleus $0.711\,\mathrm{AU}$

Circular one-AU PR reference time $2090\,\mathrm{yr}$

Light sorts the grains

Nucleus orbital position

65

f\;[{}^\circ]

Central grain radius

3

s\;[\mu\mathrm m]

Oldest release

120

\tau_{\max}\;[\mathrm d]

Launch-speed scale

120

v_0\;[\mathrm{m\,s^{-1}}]

Drag the gray nucleus along its orbit. Drag the cyan reference grain to change its radius; the camera follows the full dust fan.

What the model assumes

The cyan grain has zero launch velocity and was released sixty-five percent of the oldest age ago. Other grains sample four three-dimensional launch directions, seven logarithmic radii within a factor of two of the central radius (clipped at one and one hundred micrometres), and twenty-eight release ages. The burst uses one age; the single-size preset uses one radius.

v_{\mathrm{ej}}=v_0\sqrt{s_0/s}\,(r_{\mathrm{rel}}/\mathrm{AU})^{-1/2}

This is an assumed post-gas-drag launch law. All particles have equal visual weight; brightness is not a photometric prediction. The clock advances four model days per second. PR drag and thermal recoil do not shape this short-time stage. No ion tail is simulated.

Read the course and project real HST pixels

Model assumptions

The stage is a synthetic test-particle model, not an observation of a particular comet. The nucleus follows a Kepler ellipse with semimajor axis 2.8 AU and eccentricity 0.65. Spherical grains have density one thousand kilograms per cubic metre and fixed radiation-pressure efficiency unity; grain radii span one to one hundred micrometres. These assumptions give an inverse-size force ratio, which real compositions need not follow. Dust inherits the nucleus velocity plus a stated post-gas-drag launch law. Solar gravity and radial radiation pressure are integrated with a conic propagator; gas acceleration, collisions, fragmentation, sublimation, planets and ion tails are omitted. Reference curves omit launch velocity. The separate PR plot is an orbit-averaged, initially circular, bound one-AU reference with the physical light speed; it is not the future of every released grain. Thermal recoil is not modeled. Sizes are enlarged; positions within the current camera view share one distance scale. The companion tutorial projects a separately frozen real HST image using JPL geometry and an explicitly limited solar relative-release approximation.

Release history and slow energy loss answer different questions

Ideal orbital-plane offsets from the nucleus

Gold is the synthetic cloud; coloured curves are ideal zero-launch-velocity release families. The orbital-plane axes share one scale.

Read the release families

Gold: teaching cloud. White arrow: antisolar direction, with schematic length. Violet, cyan and pink: zero-launch-velocity synchrones released thirty, sixty-five and one hundred percent of the oldest age ago. Both axes use AU and the same scale; they rotate with the model orbit, not the sky.

\Delta y_{\mathrm{orb}}\;[\mathrm{AU}]

\Delta x_{\mathrm{orb}}\;[\mathrm{AU}]

A different experiment: bound circular PR drift

Bound circular-orbit reference with the physical speed of light. Near-Sun survival is outside this model.

Circular reference assumptions

A hypothetical grain starts on a circular orbit under reduced gravity at one AU. Its radius follows the orbit-averaged law at the physical speed of light; the formal zero-radius endpoint lies outside the near-Sun survival model. This is not the fate of every escaping comet grain.

a(t)\;[\mathrm{AU}]

t\;[\mathrm{yr}]

The observed ISON image belongs to the separate tutorial. Its sectors are measured from real pixels. The synthetic interactive cloud explores release geometry; the tutorial carries out the separate observed-direction comparison.

Physics tutorial

Follow the dust after it leaves the comet

BackgroundDust leaves a comet with almost the same orbital velocity as the nucleus. Solar gravity bends both trajectories; radiation pressure makes the grain's inward acceleration smaller. Different grain sizes and release times spread these trajectories into a tail.

Why it mattersA comet's dust tail records its recent release history and our viewing angle. Its direction can differ from both the antisolar direction and the direction opposite its velocity.

Start with the essentials

Focus question: Which changes come from grain size, release age and projection?
One-sentence intuition: A grain inherits velocity before light changes its acceleration. Reduced inward attraction can produce a wide or escaping orbit without any outward net central force.

Core mathematical model

Size sets the adopted force ratio

\beta=\frac{3L_\odot Q_{\rm pr}}{16\pi cGM_\odot\rho s}

The stage fixes spherical-grain density at one thousand kilograms per cubic metre and pressure efficiency at one. The inverse-size scaling follows from these assumptions; real material efficiencies can vary with size and wavelength.

Preserve the release velocity

\boldsymbol v_d=\boldsymbol v_c+\boldsymbol v_{\rm ej},\qquad \ddot{\boldsymbol r}_d=-GM_\odot(1-\beta)\frac{\boldsymbol r_d}{r_d^3}

The nucleus and grain are propagated from the same release event to the same observing time. With zero beta and zero launch velocity they coincide. The cyan reference grain always has zero launch velocity.

Read the slow circular reference

a(t)^2=a_0^2-\frac{4\beta GM_\odot}{c}t

This orbit-averaged PR relation describes a bound, initially circular reference at one AU with physical light speed. Escaping comet grains can have a different future. The stage treats neutral dust with central pressure, and the separate curve treats this defined bound PR reference.

Common difficulties

Separating from the nucleus does not require outward force

Typical misconceptionIf dust falls behind or outside the nucleus, sunlight must exceed gravity.

Better mental modelBoth objects can accelerate inward at different rates. An inherited velocity also allows escape under an inward force. Check release energy as well as beta.

A curve match can have several histories

Typical misconceptionMatching one projected ridge uniquely gives its grain age and size.

Better mental modelDifferent release ages and sizes can project almost together. The real-image tutorial shows actual overlapping alternatives; brightness and additional epochs would add information.

Run the experiment

01
Change one reference grain
Pause and drag the cyan grain. Predict whether a smaller radius increases or decreases the separation before checking beta.
What to observe: A smaller grain has a larger force ratio under the adopted material assumptions. Its inherited motion bends less sharply inward.
02
Separate age from size
Compare the one-burst and one-size presets. Then increase the launch-speed scale and rotate the camera.
What to observe: Release history changes the family curve; launch velocities give the dust real depth. The lower colored synchrones omit launch velocities for comparison.
03
Test the zero-force limit and the two clocks
Choose the zero-force preset, reset, then compare the fan with the circular PR reference's year axis.
What to observe: With no pressure or launch velocity the grains coincide with the nucleus. Restoring physical pressure makes a tail in months; the separate PR circular clock spans far longer times.