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Sandbox Physics
Debris disks from grain collisions to thermal evidence

AC16 · L1

From a collision to a new dust orbit

Sandbox Physics ·

A fragment can inherit its parent’s velocity and still acquire a different orbit. The missing step is the change in force per unit mass.

Choose a grain before choosing a picture

Take a spherical grain of radius s and material density rho. Its mass grows with the third power of radius, while its projected absorbing area grows with the second power. At a fixed stellar distance, dividing the radiation force by the gravitational force removes the common inverse-square distance dependence. The remaining dimensionless ratio is beta.

The optical factor is an average over the incident stellar spectrum. This course uses an absorption-only teaching law and a blackbody star, so the momentum efficiency equals the absorption efficiency. Real scattering changes that relation. Keep this choice visible: beta is not a universal table determined by radius alone. The density here is 2700 kilograms per cubic metre, and the default star has solar mass and luminosity.

m(s)=4π3ρs3,Frad=L⋆πs2⟨Qpr⟩4πr2cm(s)=\frac{4\pi}{3}\rho s^3,\qquad F_{\rm rad}=\frac{L_\star\pi s^2\langle Q_{\rm pr}\rangle}{4\pi r^2c}
Absorbed photon momentum gives the force; gravity acts on the full grain mass.
β(s)=FradFg=3L⋆⟨Qpr⟩16πcGM⋆ρs\beta(s)=\frac{F_{\rm rad}}{F_g}=\frac{3L_\star\langle Q_{\rm pr}\rangle}{16\pi cGM_\star\rho s}
The inverse-square factors cancel. The average efficiency must be computed for the stated star and optical law.

Reading: REBOUNDx 5.1.0 · radiation forces and physical light speed

Release from a circular parent orbit

Before release, let a large parent orbit at radius r under essentially unreduced stellar gravity. A fragment inherits its circular speed. Immediately afterward, radiation pressure reduces the fragment’s central attraction, but does not instantaneously reduce that inherited speed. Substituting the speed into the new specific orbital energy gives the escape test directly.

For a fragment with beta below one half, the new orbit is bound and the release point is its periapsis. At one half the two-body energy vanishes; above one half it is positive. This famous threshold assumes a circular parent and a negligible velocity kick. For an eccentric parent or a velocity kick, evaluate the fragment’s energy from its actual release position and velocity.

v02=GM⋆r,ϵβ=v022−GM⋆(1−β)r=GM⋆r(β−12)v_0^2=\frac{GM_\star}{r},\qquad \epsilon_\beta=\frac{v_0^2}{2}-\frac{GM_\star(1-\beta)}{r}=\frac{GM_\star}{r}\left(\beta-\frac12\right)
This is specific energy in the pressure-reduced two-body potential, before planetary work or PR drag is included.
aβ=r1−β1−2β,eβ=β1−β(0≤β<1/2)a_\beta=r\frac{1-\beta}{1-2\beta},\qquad e_\beta=\frac{\beta}{1-\beta}\quad(0\leq\beta<1/2)
Use the energy relation for semimajor axis and the inherited angular momentum for eccentricity.

Reading:

Worked orbit: same release, very different excursion

Release a grain at 20 astronomical units with beta 0.2 and no kick. The new semimajor axis is 26.7 astronomical units and its eccentricity is 0.25. Its periapsis remains 20 astronomical units; apoapsis reaches 33.3. An annulus of parent bodies can therefore feed a wider dust halo without any planet being present.

A grain with beta 0.6, released in the same way, has positive two-body energy. It does not disappear at the instant of release. The calculation continues to follow it until it crosses the stated 160 astronomical-unit outer domain. That distinction matters observationally: a departing grain can still be warm enough to contribute to the image while it is inside the field. The mass ledger calls this an outer-domain crossing; a separate energy diagnostic states whether the crossing grain was energetically unbound.

There is a useful warning hidden in this example. A broad halo can be a radiative outcome of an axisymmetric release. Its existence alone is not proof that a planet scattered the grains. Compare the no-planet preset before assigning every outer feature to gravitational stirring.

Reading:

Why a crushing cascade can approach a particular slope

Now temporarily replace the finite release with an idealized steady cascade. Suppose relative speed and catastrophic strength are size independent, the fragment law is self-similar, comparable-sized projectiles dominate destruction, and source and sink sizes are far apart. Let the differential number distribution be a power law. Number in a logarithmic interval then scales as radius to the power one minus q, and mass in that interval as radius to the power four minus q.

The comparable-size collision rate per target is number density in that interval times cross section and speed. Multiplying this rate by the interval’s mass gives the mass flux down the cascade. A steady, scale-independent flux selects the exponent seven halves. This scaling argument is conditional: size-dependent strength, radiation removal near the smallest sizes and nonlocal projectile populations can change it. It is not a proof that a finite numerical cloud has reached this state. Dohnanyi’s 1968 report distinguishes the simplified mass exponent eleven sixths from the more complete numerical values in Table III. Its example mass exponent 1.837 converts to a radius exponent 3.511. The seven-halves result here is the declared scaling limit, not an exact quotation of every original collision model.

dNds∝s−q,Nln⁡s∝s1−q,Mln⁡s∝s4−q\frac{dN}{ds}\propto s^{-q},\qquad N_{\ln s}\propto s^{1-q},\qquad M_{\ln s}\propto s^{4-q}
A logarithmic size interval contributes an extra factor of radius compared with a linear interval.
tcoll−1∝s3−q,M˙ln⁡s∝s7−2q,q=72t_{\rm coll}^{-1}\propto s^{3-q},\qquad \dot M_{\ln s}\propto s^{7-2q},\qquad q=\frac72
Constant mass flow follows only under the assumptions stated above.

Reading: Stark & Kuchner 2009 · collisional grooming, fixed arXiv 0909.2227v1 · Dohnanyi 1968 · NASA-CR-99280, sections III–IV and Table III

What the experiment actually initializes

The production experiment starts with eight resolved radii, approximately 0.173 to 379 micrometres, and a prescribed finite release spectrum with the same numerical exponent. That is an initial condition, not an emergent steady result. Each parcel carries a physical number of grains, and the initial normal optical depth fixes the total cross section. There is no continuous injection after the start.

Subsequent destructive encounters transfer mass between bins, while radiation transports grains to different positions and open boundaries remove them. The next chapter makes these transfers explicit. Reading the distinction carefully prevents a common mistake: choosing the expected steady slope at the beginning and then presenting the resulting histogram as a successful derivation of that slope.

Reading:

Saved calculation, readable without the Lab

Solar star, planet mass 0.003 solar masses, circular 13 AU orbit, initial normal optical depth 0.0003. Eight grain bins, 384 initial weighted parcels, RK4 at 128 steps per planet orbit, finite-cell catastrophic collisions and a 128-per-bin compression limit. The representative record is eight planet periods old.

QuantitySaved resultInterpretation
Elapsed time374.42 yr374.42\,\mathrm{yr}A real finite integration window
Weighted collision events590Statistical packet events, not individual grain count
Maximum normal optical depth0.0019440.001944Finite-cell column estimate
Largest raw kinetic compression jump3.16%3.16\%Before explicit velocity-moment correction
Conservative probe angle range4.6919 rad4.6919\,\mathrm{rad}Separate 150-period record, not the whole collisional cloud

The stochastic four-period refinement changes 850 µm flux by about 0.7–4.0% and 70 µm flux by 8.7–32%. Centroid and low-order density moments change enough that individual cloud clumps are not a converged planet inference. The independent conservative probe only establishes bounded angle behavior for its specified initial state and saved 150-period window.

Inspect actual refinement records · Inspect the all-parent finite stress counterexample

Predict a release halo

  1. Load the no-planet release and compare the small and large size populations. Identify which can leave the 160 AUdomain.
  2. Compare 70 and 850 micrometres while paused. Explain the difference using cross section and temperature.

Exercises with explanations

  1. A circular-parent fragment has beta 0.4. Predict its semimajor axis, eccentricity and apoapsis when released at 20 astronomical units.

    Read the explanation

    The semimajor axis is 60 astronomical units, eccentricity two thirds and apoapsis 100 astronomical units. The periapsis stays at the 20 astronomical-unit release point. This large excursion appears without planetary scattering.

  2. If the number of grains per logarithmic interval is constant, is their mass per interval constant?

    Read the explanation

    No. Constant logarithmic number means q equals one. Individual mass still grows with the cube of radius, so the mass per logarithmic interval grows with the cube of radius. A histogram must state whether it counts grains, area or mass.

Sources and calculation scope

  1. Stark & Kuchner 2009 · collisional grooming, fixed arXiv 0909.2227v1

    Primary full text: sections II.1–II.4, III and IV; local rates, grid/noise tests and the limits of fragment reassignment. This weighted initial-value operator is not a reproduction of their steady-stream solver.

  2. REBOUNDx 5.1.0 · radiation forces and physical light speed

    Official force documentation and example; independent records use actual REBOUND 5.2.2 IAS15 plus REBOUNDx 5.1.0, with physical-c PR and barycentric coordinates converted to the star frame.

  3. Dohnanyi 1968 · NASA-CR-99280, sections III–IV and Table III

    Public original report dated 22 July 1968; section IV and Table III distinguish the ideal eleven-sixths mass slope from numerical collision-model slopes. The later journal paper is JGR 74, 2531–2554, 15 May 1969, DOI 10.1029/JB074i010p02531. This course derives a restricted scale-free limit.

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