---
title: "Inherited velocity turns light pressure into a dust tail"
description: "Derive the grain's energy and escape threshold, construct synchrones and syndynes, and test a three-dimensional release model."
canonical_url: "https://sandboxphysics.com/stories/comet-dust/inherited-orbits-and-release-times/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/comet-dust/inherited-orbits-and-release-times/"
last_modified: "2026-10-28"
---

[Comet dust, from light pressure to real images](https://sandboxphysics.com/stories/comet-dust/)

L2 · Comet dust dynamics

# Inherited velocity turns light pressure into a dust tail

Sandbox Physics · 2026-10-28

The force ratio alone cannot tell us a grain's orbit. We also need its release position and velocity. Holding those initial conditions fixed reveals why even an inward force can produce an escaping grain, and why a tail remembers when the dust left.

## Before you begin

Chapter one defines beta and reduced gravity. This chapter uses vectors, orbital energy per unit mass and the vis-viva relation. The integral of acceleration determines position and velocity through time.

## Specify the release event

Let the observation time be t and the release age be $\tau$. Evaluate the nucleus at the earlier time $t-\tau$. A new grain starts there with the nucleus velocity plus its post-gas-drag launch velocity. Propagate the grain under reduced gravity for the full age; propagate the nucleus under ordinary solar gravity to the same observation time.

Subtract the two final positions only after both trajectories reach the same epoch. Starting dust at today's nucleus and evolving it for a past release age gives the wrong geometry. Likewise, subtracting a fully perturbed JPL nucleus from an unrelated solar-only parent orbit can create a false dust offset even when beta is zero. The real-image tutorial handles that mismatch explicitly.

$$
\boldsymbol r_d(t-\tau)=\boldsymbol r_c(t-\tau),\quad \boldsymbol v_d(t-\tau)=\boldsymbol v_c(t-\tau)+\boldsymbol v_{\rm ej},\quad \Delta\boldsymbol r(t)=\boldsymbol r_d(t)-\boldsymbol r_c(t)
$$

Subscripts d and c mean dust and comet. All three-dimensional vectors share the same solar origin, frame and time units. Setting both beta and launch velocity to zero must recover the nucleus trajectory; this is a decisive model check.

Reading: Finson & Probstein · JPL Horizons API and observer/vector conventions

## Derive the escape threshold

Temporarily set launch velocity to zero. A parent ellipse with semimajor axis $a_c$ has speed squared $\mu(2/r-1/a_c)$, where $\mu=GM_\odot$. The dust inherits that speed but has potential energy from the smaller parameter $\mu(1-\beta)$. Subtract the potential from the kinetic energy and simplify.

A bound conic has negative specific energy; zero is the parabolic limit and positive energy gives a hyperbola. The threshold therefore depends on the release distance along the parent ellipse. The same grain size can be bound when released far out and unbound when released near perihelion. A launch velocity changes the kinetic term and shifts the threshold further.

$$
\varepsilon_d=\frac{v_c^2}{2}-\frac{\mu(1-\beta)}{r}=\mu\left(\frac{\beta}{r}-\frac{1}{2a_c}\right),\qquad \beta_{\rm escape}=\frac{r_{\rm rel}}{2a_c}
$$

This threshold assumes zero launch speed and a bound Kepler parent. It is not the beta-equals-one condition for outward acceleration. On the teaching ellipse, eccentricity 0.65 gives thresholds 0.175 at perihelion and 0.825 at aphelion.

Reading: Burns · Finson & Probstein

## Solve a circular example completely

Release a grain from a circular one-AU orbit with beta 0.1 and no added launch speed. The parent speed squared is the solar parameter divided by one AU, and the grain keeps that speed. Its specific angular momentum also stays the same at release. Use energy to find the new semimajor axis and angular momentum to find its eccentricity.

The new orbit has semimajor axis 1.125 AU and eccentricity one ninth. Release occurs at its perihelion, still one AU. Its aphelion is 1.25 AU. At beta 0.5 the inherited circular speed becomes exactly the reduced-gravity escape speed. Above that value the grain escapes, although its central acceleration still points inward until beta reaches one.

$$
a_d=\frac{1-\beta}{1-2\beta}r_{\rm rel},\qquad e_d=\frac{\beta}{1-\beta}\quad(0\le\beta<1/2)
$$

These expressions apply only to a circular parent with zero ejection velocity. Derive them from specific energy and the angular-momentum identity. Applying them to an eccentric comet without checking the release state is a common error.

Reading: Burns · Finson & Probstein

## Turn individual conics into families

A synchrone holds the release time fixed and varies beta. A syndyne holds beta fixed and varies release time. With zero launch velocity and fixed grain properties, these families expose the two principal dimensions of a simple dust image. Their sky projection can make separate spatial curves cross or nearly coincide.

The stage adds an assumed release speed after gas drag has ended. It scales inversely with the square root of grain radius and heliocentric release distance. Four directions on a sphere provide a sparse three-dimensional sampling. The dots all have equal visual weight, so their displayed density is not a surface-brightness calculation. Predicting brightness would also require a production history, scattering phase function and grain-size distribution.

$$
v_{\rm ej}=v_0\sqrt{\frac{s_0}{s}}\left(\frac{r_{\rm rel}}{\mathrm{AU}}\right)^{-1/2}
$$

This is an adopted teaching law, not a fitted ISON gas model. The cyan reference grain uses zero added velocity, so dragging its size isolates the radiation-pressure response. The lower family diagram also uses zero launch velocity for its colored reference lines.

Reading: Finson & Probstein · Li et al.

## Save a result and challenge the geometry

At orbital position sixty-five degrees, the teaching ellipse has a nucleus–Sun distance about 1.27 AU. With releases extending back 120 days, central radius three micrometres and launch scale 120 metres per second, the largest sampled nucleus–grain separation is about 0.711 AU. This table records representative outputs at entry, before autoplay advances the clock. Values are rounded from the stated discrete particle family.

To reproduce this discrete example, use parent semimajor axis 2.8 AU, eccentricity 0.65, inclination 35 degrees, ascending node 35 degrees and argument of perihelion 25 degrees. The default ages have indices one through twenty-eight; the radius index runs zero through six. Four launch directions use the sphere construction below in the teaching inertial axes. Convert launch speed to AU per day before adding it to the orbital velocity.

A universal-variable conic solver propagates these grains through bound and unbound cases. Independent Runge–Kutta integrations test circular, near-parabolic and hyperbolic examples; energy, angular momentum and backward propagation provide further checks. Camera framing is a separate geometric task: it includes the Sun and the full particle family, retaining one distance scale in either image direction.

$$
\tau_j=\frac{120j}{28}\,\mathrm d,\qquad s_k=3\,2^{2k/6-1}\,\mu\mathrm m
$$

These default nodes use j from 1 to 28 and k from 0 to 6. Outside this example, radii are clipped to the allowed one-to-one-hundred-micrometre range.

$$
z_\ell=1-\frac{2(\ell+1/2)}4,\quad\phi_\ell=\ell\pi(3-\sqrt5),\quad\hat{\boldsymbol u}_\ell=(\sqrt{1-z_\ell^2}\cos\phi_\ell,\sqrt{1-z_\ell^2}\sin\phi_\ell,z_\ell)
$$

Use direction index zero through three and multiply each unit vector by the adopted launch-speed law. The default therefore contains 784 particles. A zero launch speed needs only one direction; repeated direction samples would represent the same trajectory.

| Case | Central beta | Release history |
| --- | --- | --- |
| Default | 0.191 | 28 ages, 7 radii, 4 launch directions |
| Fine grains | 0.574 | same ages; smaller grains; stronger separation |
| Heavy grains | 0.00574 | 180 days; launch scale 5 metres per second |
| No pressure or launch speed | 0 | all grains coincide with the nucleus |

Reading: Finson & Probstein · Burns

## Try the observation

### [Compare one age with one size](https://sandboxphysics.com/labs/comet-dust-tails/)

1. Pause One burst, many sizes and note its narrow family. Choose One size, many ages and compare its curvature.
2. Increase the launch-speed scale from zero. Rotate near the orbital plane to see the extra thickness.
3. Move the nucleus close to perihelion, then far from it. Check how the release energy changes at the same beta.

## Exercises and explained answers

Would a beta-0.2 grain released with zero launch speed at the teaching orbit's perihelion escape? What about aphelion?

The thresholds are (1−0.65)/2 = 0.175 and (1+0.65)/2 = 0.825. Beta 0.2 exceeds the perihelion threshold and gives positive energy there. It falls below the aphelion threshold and remains bound there. In both cases beta stays below one, so the force is inward.

Your beta-zero, zero-launch model draws a tail when registered to a JPL nucleus. Which part should you inspect first?

Inspect whether dust and the reference parent used the same forces, origin and epoch. A solar-only dust trajectory compared directly with a fully perturbed JPL comet need not coincide. First make the solar-only dust and its matching solar-only parent agree at beta zero; then add both their relative offset to a declared common reference, or integrate the missing forces consistently.

A projected ridge matches one synchrone. Does that uniquely date an outburst?

Not from the ridge alone. Other beta-age combinations, launch speeds and lines of sight may overlap it. Compare several image directions or epochs and a brightness model before interpreting a curve match as a dated production event.

[Previous: From photon momentum to a grain's reduced solar gravity](https://sandboxphysics.com/stories/comet-dust/light-pushes-small-grains/) [Next: Why Poynting–Robertson drag takes away orbital energy](https://sandboxphysics.com/stories/comet-dust/slow-loss-of-orbital-energy/)

## Sources and calculation scope

1. [Finson & Probstein, A Theory of Dust Comets I (1968)](https://adsabs.harvard.edu/pdf/1968ApJ...154..327F)

   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)](https://ntrs.nasa.gov/citations/19790070469)

   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. [Li et al., HST observations of Comet ISON, arXiv:1311.0826v1](https://arxiv.org/abs/1311.0826v1)

   Section 2 gives the April 2013 HST observation sequence and original reduction; the sunward inner jet and extended tail motivate excluding the inner two arcseconds in our coarse direction diagnostic. Our 2026 archive single-frame reduction is different from the paper's combined images.

4. [JPL Horizons API and observer/vector conventions](https://ssd.jpl.nasa.gov/horizons/manual.html)

   API response version 1.2; API documentation version 1.3. Frozen comet record 90004196, solution JPL#57 dated 2025-Nov-21\_13:42:48, DE441. Read TIME\_TYPE, TLIST, ICRF/FRAME vector conventions and observer quantities 1, 19, 20, 21, 24, 27, 30. The saved requests and responses preserve the adopted time scale and geometry.