---
title: "Read PR drag and thermal recoil without mixing their clocks"
description: "Read the fixed Burns 1979 and Yarkovsky review equations and figures, define a thermal model, and separate orbital fits from force measurements."
canonical_url: "https://sandboxphysics.com/stories/comet-dust/reading-slow-radiation-forces/"
language: "en"
page_type: "story"
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  zh-CN: "https://sandboxphysics.com/zh/stories/comet-dust/reading-slow-radiation-forces/"
last_modified: "2026-10-28"
---

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

L4 · Comet dust dynamics

# Read PR drag and thermal recoil without mixing their clocks

Sandbox Physics · 2026-10-28

Open Burns, Lamy and Soter's 1979 paper beside Vokrouhlický et al., arXiv:1502.01249v1, dated 4 February 2015. Follow the former's energy-loss equations and the latter's thermal variables and Figures 2–3. The goal is to tell which physical model an orbital drift tests. This is a guide to those fixed sources; it does not reproduce their astrometric fits or inventory later detections.

## Before you begin

The PR chapter supplies the energy and circular time-scale derivation. Here thermal inertia means a material's resistance to temperature change over a heating cycle; obliquity is the angle between spin and orbital angular momentum.

## Burns section VIII: identify the orbit being averaged

Read equations (47)–(50) with the preceding definition of beta and the first-order radiation force. Mark the assumed bound orbit, its semimajor axis and eccentricity, and the averaging over one revolution. An instantaneous acceleration and an orbit-averaged drift have different units and roles. Recover the circular limit from chapter three before using an eccentric-orbit result.

A useful calculation beside the paper is the ten-micrometre, one-AU example: beta about 0.0574 and a formal circular time about 6,974 years. This clock belongs to a defined bound grain. Escaping ISON grains require their own initial states and energy evolution. Use physical light speed when comparing these clocks.

Reading: Burns · JPL Solar System Dynamics astronomical parameters

## Define a temperature model before assigning thermal thrust

For a homogeneous solid with constant density, specific heat and conductivity, temperature evolves by heat diffusion. Let the outward surface normal be n. At the surface, absorbed sunlight is split between emitted thermal power and heat conducted into the interior. Choose a spin state and periodic interior boundary condition before solving. This is a minimal illustrative model, not an implemented asteroid reconstruction.

Here C is heat capacity per unit mass, k is conductivity, A is Bond albedo, emissivity is dimensionless and S is local solar energy flux. Shadowing, roughness, changing material properties and sublimating gas would require additional terms. Once the temperature map is known, a Lambertian-emission assumption gives the net recoil by integrating the outward emitted momentum over the surface.

$$
\rho C\frac{\partial T}{\partial t}=k\nabla^2T,\qquad k\frac{\partial T}{\partial n}=(1-A)S\max(\hat{\boldsymbol n}\cdot\hat{\boldsymbol s},0)-\epsilon\sigma T^4
$$

The Sun direction is outward from the surface toward the light source. Each boundary term has units watts per square metre. The chosen sign makes a hotter surface conduct heat inward.

$$
\boldsymbol F_{\rm th}=-\frac{2}{3c}\int_{\rm surface}\epsilon\sigma T^4\hat{\boldsymbol n}\,dA
$$

The two-thirds factor comes from integrating Lambertian photon emission over an outward hemisphere. Uniform temperature on a sphere gives zero rest-frame net force because the surface normals cancel. The separate first-order PR frame correction still acts.

Reading: Vokrouhlický et al. · Burns

## Review section 2.1: read the sign before the magnitude

In equations (1)–(3), track obliquity $\gamma$ and the thermal response W. For the paper's classical assumptions, the diurnal contribution can change sign with spin orientation, whereas the seasonal contribution decreases semimajor axis. The parameter definitions following equation (3) connect rotation and thermal diffusion; changing density alone does not specify a temperature lag.

A direct check is to reverse the spin pole in the simple diurnal model. Its orbital drift should reverse. Repeat with only the seasonal part and the same sign remains. PR drag provides a different comparison: its energy loss does not depend on a hot afternoon hemisphere.

$$
\Gamma=\sqrt{k\rho C},\qquad \ell_\nu=\sqrt{\frac{k}{\rho C\nu}},\qquad \Theta_\nu=\frac{\Gamma\sqrt\nu}{\epsilon\sigma T_*^3}
$$

Gamma is thermal inertia, the length is a diffusion scale, and Theta compares a heating frequency with radiative cooling. Use rotation frequency for the diurnal response and orbital frequency for the seasonal response. These definitions follow the fixed review's convention.

Reading: Vokrouhlický et al. · Burns

## Figures 2 and 3: find the actual observable

Figure 2 places Golevka's orbital predictions in radar range and range-rate coordinates. Locate the gravity-only prediction and the radar measurement, then compare the prediction that includes thermal forces. The plotted uncertainty concerns an orbital prediction, not a photograph of emitted heat.

Figure 3 compares 1992 BF's old photographic trails with backward-predicted sky positions. Read the right-ascension and declination directions before comparing the modeled trails. Both figures use accumulated positional consequences to test a force model. Neither measures semimajor-axis drift by looking at a single orbit snapshot.

Reading: Vokrouhlický et al. · JPL Horizons API and observer/vector conventions

## Section 2.2 and table 1: ask what would change the inference

Section 2.2 motivates more detailed thermal and shape calculations. Table 1's orbital-fit uncertainties answer a statistical question conditional on adopted observations and dynamics. A physical interpretation also needs plausible size, density, spin and thermal response. Read the observation arc and radar coverage before comparing apparent precision between objects.

For the next paper, write down its dated version, measured quantities, adopted force law, material/spin information and alternative accelerations. Keep fitted transverse acceleration separate from its proposed cause. A comet's Marsden outgassing parameters and an asteroid's thermal-drift fit can share a transverse direction while describing different physics. The ISON tutorial tests the nucleus's small sampled A1/A2 outgassing contribution separately from thermal physics.

Reading: Vokrouhlický et al. · JPL Horizons API and observer/vector conventions · Li et al.

## Try the observation

### [Use the Lab as the pressure and PR reference](https://sandboxphysics.com/labs/comet-dust-tails/)

1. Choose a grain radius and calculate its circular time from chapter three; compare the lower graph's actual year axis.
2. Identify the stage's declared omissions before comparing it with either paper's thermal model.

## Exercises and explained answers

A model uses only beta and a switch called thermal recoil. What physical ingredients are still missing?

A temperature evolution law, surface boundary condition, spin state, shape, thermal properties and an emission-momentum law are needed to predict a recoil vector. A beta value specifies pressure relative to gravity, not a temperature map. Without those ingredients the switch has no defined thermal prediction.

Which result has actually been reproduced in this course?

The companion tutorial recomputes the fixed HST direction diagnostic, JPL projection and declared solar relative-release geometry. The PR chapter reproduces a circular analytic reference and checks it against direct integration. The review's asteroid detections and thermal-shape inversions are read here, without rerunning their original fits.

[Previous: Project frozen JPL dust geometry onto a real HST image of ISON](https://sandboxphysics.com/stories/comet-dust/ison-image-projection-tutorial/)

## Sources and calculation scope

1. [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.

2. [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.

3. [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.

4. [JPL Solar System Dynamics astronomical parameters](https://ssd.jpl.nasa.gov/astro_par.html)

   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.

5. [Vokrouhlický et al., The Yarkovsky and YORP Effects, arXiv:1502.01249v1](https://arxiv.org/abs/1502.01249v1)

   Fixed 2015 version, 24 pages. Section 2.1, equations (1)–(3), treats diurnal/seasonal semimajor-axis drift; section 2.2 explains thermal/shape refinements; section 3.1 and Figures 2–3 discuss astrometric detections including Golevka and 1992 BF. This is a source-reading guide, not a new thermal simulation or a claim about the latest detections.