Why Poynting–Robertson drag takes away orbital energy

Keep the first velocity correction

Use heliocentric grain velocity $\boldsymbol v$ and its radial component $v_r=\boldsymbol v\cdot\hat{\boldsymbol r}$ . To first order in speed divided by light speed, the radiation force adds a velocity-dependent correction to the reduced-gravity equation. The radial correction and the part opposite the full velocity both matter; retaining only a hand-drawn backwards arrow loses the radial term.

The expression describes a small grain with fixed pressure efficiency, in the usual isotropic-rest-frame reradiation approximation. It does not require a warmer afternoon hemisphere. Absorbed solar momentum and reradiation must be transformed consistently between frames; treating isotropic emission in the grain's rest frame as zero momentum loss in the solar frame misses the effect.

\ddot{\boldsymbol r}=-\mu(1-\beta)\frac{\boldsymbol r}{r^3}-\frac{\beta\mu}{cr^2}\left(v_r\hat{\boldsymbol r}+\boldsymbol v\right)

The first term is the radial force used in the short-time stage. The second is the PR correction. All terms are accelerations. For a circular one-AU orbit, speed divided by light speed is of order one ten-thousandth, so the correction is much smaller than radial radiation pressure.

Reading: Burns · JPL Solar System Dynamics astronomical parameters

Check the sign with energy and angular momentum

Define specific orbital energy using the reduced gravitational parameter $\mu'=\mu(1-\beta)$ . Differentiate the kinetic and potential terms. The central-force contributions cancel, leaving velocity dotted into the PR acceleration. Every remaining squared speed has a nonnegative sign inside a negative factor.

The torque gives a similarly useful check. Radial forces produce no torque about the Sun. The term opposite the velocity reduces the magnitude of the orbital angular momentum. These identities test an integrator independently of whether its plotted spiral looks convincing.

\varepsilon=\frac{v^2}{2}-\frac{\mu'}r,\qquad \frac{d\varepsilon}{dt}=-\frac{\beta\mu}{cr^2}(v_r^2+v^2)\le0

This is the orbital energy of a grain with constant beta. The complete energy accounting includes the radiation field. Negative orbital power does not mean every initially unbound grain becomes bound before it escapes.

\boldsymbol h=\boldsymbol r\times\boldsymbol v,\qquad \frac{d\boldsymbol h}{dt}=-\frac{\beta\mu}{cr^2}\boldsymbol h

In this approximation the direction of angular momentum stays fixed while its magnitude decreases. A purely central radiation-pressure model must conserve angular momentum instead.

Reading: Burns · Finson & Probstein

Derive the bound circular clock

For a slowly evolving nearly circular orbit, replace radius by semimajor axis and use the reduced-gravity circular speed. The specific orbital energy is negative reduced gravity divided by twice the semimajor axis. Differentiate with respect to semimajor axis and equate the result to the power above. The reduced-gravity factor cancels, leaving an especially simple inward drift.

Multiply the drift equation by twice the semimajor axis and integrate. The square of the orbital size decreases linearly with time. This is an orbit-averaged approximation for a bound orbit, distinct from a freshly released grain that retained the parent body's faster speed.

\frac{da}{dt}=-\frac{2\beta\mu}{ca},\qquad a(t)^2=a_0^2-\frac{4\beta\mu}{c}t

The circular approximation changes orbital size slowly compared with one revolution. The formal time to zero radius is a convenient scale, not a physical prediction through sublimation or the solar surface.

t_{\rm PR,formal}=\frac{ca_0^2}{4\beta\mu}=\frac{400.494}{\beta}\left(\frac{a_0}{\mathrm{AU}}\right)^2\mathrm{yr}

The coefficient uses a Julian year of 365.25 days and the frozen solar parameter. Reducing the starting orbit from one AU to half an AU reduces this reference time by a factor of four.

Reading: Burns · JPL Solar System Dynamics astronomical parameters

Calculate what can happen in 120 days

For the ten-micrometre teaching grain beta is about 0.0574, so the formal one-AU circular time is about 6,974 years. Over 120 days the fractional orbital-size change is only about twenty-four millionths. A large visual tail made in that time comes from the changed release orbit and launch velocities, not from rapid PR inspiral.

The table follows the same circular law, density and efficiency as chapter one. It is a saved analytic reference; the Lab never increases the drag coefficient to make the effect visible over months. The separate graph changes its time axis to actual years. A direct ten-year integration of the physical PR force checks the orbit-averaged calculation independently.

Radius (micrometres)	Beta	Formal circular one-AU time (years)
1	0.574	697
3	0.191	2090
10	0.0574	6970
100	0.00574	69700

Reading: Burns · JPL Solar System Dynamics astronomical parameters

A thermal lag asks a further question

A rotating solid can emit more thermal radiation from its afternoon side than from its morning side. The resulting recoil depends on a temperature map, rotation direction, obliquity, thermal conductivity and shape. This is the starting point for the diurnal Yarkovsky effect. A seasonal temperature pattern gives another contribution. Neither can be determined from beta alone.

The Lab therefore stops at central pressure and the defined PR reference. The reading guide follows a pinned paper through its thermal variables and astrometric evidence. For a comet, an empirical nongravitational acceleration can also describe asymmetric outgassing. Its fitted transverse component needs a physical interpretation that distinguishes outgassing from thermal recoil.

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

Try the observation

Read the two clocks separately

Compare the moving dust fan with the lower circular-reference graph. Read the graph's year axis before comparing distances.
Increase grain radius by a factor of ten. The assumed beta falls tenfold and the circular PR reference time grows tenfold.

Exercises and explained answers

A programmer implements a PR correction that increases orbital energy during some outward-moving intervals. Which algebraic check should fail?

For constant beta the power must equal $-\beta\mu(v_r^2+v^2)/(cr^2)$ . It is nonpositive for either sign of radial velocity. Check the radial sign, the solar frame of velocity and whether the central potential was differentiated with the same reduced parameter.

How long does the ten-micrometre circular reference take to shrink from one AU to half an AU?

The square-size law uses seventy-five percent of the formal time: about 5,230 years. This assumes fixed grain properties and continued survival. An escaping grain from an eccentric comet instead requires its own release state and subsequent trajectory.

An asteroid has a transverse orbital acceleration. What additional information would support a diurnal thermal-recoil interpretation?

A measured spin direction and obliquity, size and thermal properties should predict the sign and approximate magnitude; a thermal model should explain the temperature lag. Competing outgassing or radiation models must also be considered. The coefficient describes an acceleration; the additional observations test its physical cause.

Sources and calculation scope

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.
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.
JPL Horizons API and observer/vector conventions
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.
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.
Vokrouhlický et al., The Yarkovsky and YORP Effects, arXiv: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.