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Stellar streams, encounters and observed projections

Stellar streams · L2

From a flyby force to a much later stream gap

Sandbox Physics ·

A flyby does not deposit a ready-made hole. It leaves a position-dependent change in velocity. To understand the gap, first calculate that change, then let the stars continue moving in the host potential.

Fix the geometry and the units

Use a right-handed inertial frame centred on the Galaxy. At nominal encounter the reference star is at (R0,0,0)(R_0,0,0) and moves along positive second axis. Take R0=15 kpcR_0=15\,\mathrm{kpc} and v0=220 km s−1v_0=220\,\mathrm{km\,s^{-1}}, giving a time unit of about 66.7 million years.

In those units prescribe rp(t)=(1+Wt,t,b)\mathbf r_p(t)=(1+Wt,t,b). The perturber shares the reference second-axis speed; WW is their first-axis relative speed, and bb is the vertical offset. This path does not respond to the stars.

The Plummer scale aa sets the spatial smoothing and keeps the central potential finite; it is not a hard surface radius. MM is the imposed perturber mass. Potential and specific energy are measured in the velocity unit squared.

t0=R0v0≃66.7 Myr,Tcircle=2πt0≃419 Myrt_0=\frac{R_0}{v_0}\simeq66.7\,\mathrm{Myr},\qquad T_{\rm circle}=2\pi t_0\simeq419\,\mathrm{Myr}
The time unit is length divided by speed. A full reference circular orbit in the unperturbed host takes an additional full-turn factor; the unit is not the orbital period.
Φ(r,t)=ln⁡∣r∣−μ∣r−rp(t)∣2+a2,μ=GMR0v02\Phi(\mathbf r,t)=\ln|\mathbf r|-\frac{\mu}{\sqrt{|\mathbf r-\mathbf r_p(t)|^2+a^2}},\qquad\mu=\frac{GM}{R_0v_0^2}
The host is spherical logarithmic; the imposed perturbation is Plummer-softened. Positions and softening length are dimensionless here.

Reading: Erkal & Belokurov (2015), Forensics of Subhalo-Stream Encounters, arXiv:1412.6035v2 · Sandbox Physics standalone stellar-stream scientific reader v1

First solve a short-encounter limit

During the encounter only, approximate the stream as a straight line translating along the second axis. A star has along-stream offset yy. If the encounter is short and the kick small, integrate the force on that line. The odd first-axis contribution cancels.

The impulse integral below switches back to dimensional quantities: tt is now physical encounter time. Lengths are in kiloparsecs and relative speed in kilometres per second, with consistent length–time conversion in their product.

∫−∞∞dt(W2t2+C2)3/2=2WC2,C2=y2+b2+a2\int_{-\infty}^{\infty}\frac{dt}{(W^2t^2+C^2)^{3/2}}=\frac{2}{WC^2},\qquad C^2=y^2+b^2+a^2
Substitute relative path length divided by the transverse scale, or evaluate the standard antiderivative at both infinities. Multiplying by the appropriate force numerator gives the two kick components below.
Δvy=−2GMyW(y2+b2+a2),Δvz=2GMbW(y2+b2+a2)\Delta v_y=-\frac{2GMy}{W(y^2+b^2+a^2)},\qquad\Delta v_z=\frac{2GMb}{W(y^2+b^2+a^2)}
This equation uses dimensional lengths, mass and speed. A positive along-stream offset is pulled back; a positive vertical impact pulls upward. Changing the encounter geometry requires projecting the force into the new directions.

Reading: Erkal & Belokurov (2015), Forensics of Subhalo-Stream Encounters, arXiv:1412.6035v2 · Sandbox Physics standalone stellar-stream scientific reader v1

Worked example: a kick with two signs

Use thirty million solar masses, a=0.25 kpca=0.25\,\mathrm{kpc}, b=0.10 kpcb=0.10\,\mathrm{kpc}, W=200 km s−1W=200\,\mathrm{km\,s^{-1}} and y=0.30 kpcy=0.30\,\mathrm{kpc}. The squared-length sum in the denominator is 0.1625 square kiloparsecs.

Use the gravitational constant G=4.30091×10−6 kpc (km s−1)2M⊙−1G=4.30091\times10^{-6}\,\mathrm{kpc}\,(\mathrm{km\,s^{-1}})^2M_\odot^{-1} so kiloparsecs, kilometres per second and solar masses can be inserted consistently.

Δvy≃−2.38 km s−1,Δvz≃+0.794 km s−1\Delta v_y\simeq-2.38\,\mathrm{km\,s^{-1}},\qquad\Delta v_z\simeq+0.794\,\mathrm{km\,s^{-1}}
The along-stream magnitude is three times the vertical magnitude because the offset is three times the vertical impact. Both signs follow attraction toward the perturber.

Reading: Erkal & Belokurov (2015), Forensics of Subhalo-Stream Encounters, arXiv:1412.6035v2 · Sandbox Physics standalone stellar-stream scientific reader v1

Why the slowed stars can eventually lead

After the encounter, a small tangential slowdown reduces angular momentum and guiding radius in a flat-speed host. The smaller mean orbit has a higher angular frequency. A star initially pulled back can therefore complete subsequent circuits sooner. A forward-kicked star can do the reverse.

The cold circular-stream analysis distinguishes initial compression, expansion and late folds. After a fold, multiple initial positions can reach one final position. Their density contributions must all be added; a single-valued Jacobian is no longer sufficient.

δΩΩ≃−δvTvc,δθ≃δΩ t\frac{\delta\Omega}{\Omega}\simeq-\frac{\delta v_T}{v_c},\qquad\delta\theta\simeq\delta\Omega\,t
This keeps the secular guiding-centre term and omits bounded epicyclic motion. Applying the example slowdown for 500 million years gives a secular near-circular estimate of about positive 4.65 degrees; the full trajectory also retains radial oscillations.

Reading: Erkal & Belokurov (2015), Forensics of Subhalo-Stream Encounters, arXiv:1412.6035v2 · Sandbox Physics standalone stellar-stream scientific reader v1

Turn the changed tracks into a count profile

In an ideal one-dimensional cold stream, let ψ0\psi_0 label initial azimuth and ψ(ψ0,t)\psi(\psi_0,t) be its later corotating azimuth. A short interval contains the same stars before and after the mapping, so its number equals density per radian times angular length.

Stretching an interval to twice its former length halves the density; compressing it to half its length doubles the density. The number has not changed. This is the simplest worked example of the mapping Jacobian.

Once the mapping folds, several initial intervals arrive at one final location. Add their positive count contributions. A zero derivative produces a cold caustic, formally divergent in this ideal description. A warm velocity distribution and finite bins spread that singularity; the experiment therefore reports finite tracer counts rather than plotting an infinite density.

λ(ψ,t)=∑ψ0↦ψλ0(ψ0)∣∂ψ/∂ψ0∣\lambda(\psi,t)=\sum_{\psi_0\mapsto\psi}\frac{\lambda_0(\psi_0)}{\left|\partial\psi/\partial\psi_0\right|}
The sum is over all initial labels reaching the plotted angle. It is an ideal cold-stream relation, distinct from counting our finite warm ensemble.

Reading: Erkal & Belokurov (2015), Forensics of Subhalo-Stream Encounters, arXiv:1412.6035v2 · Sandbox Physics standalone stellar-stream scientific reader v1

Why integrate the actual encounter?

The experiment applies the moving Plummer force throughout the flyby rather than assigning an instantaneous kick. Stars move while being perturbed and continue to feel the host. This lets us ask how much the short-encounter approximation misses.

The four original comparisons use saved states from about 16.7 million years after encounter onward. Their maximum impulse-versus-full position difference, divided by the peak full-versus-host position perturbation, is about 9–14 percent; the analogous velocity ratio is about 10–16 percent. The denominator is the perturbation signal, not the orbital radius or total speed. This measures a physical model difference, not integrator error or observational noise.

The public reader checks a separate high-order integration, half-step refinement, direct impulse quadrature and phase-space volume. These numerical comparisons cover the 25 registered points individually; accuracy between them requires further validation.

ϵr=max⁡t≥0.25,j∥rjimpulse−rjfull∥max⁡t≥0.25,j∥rjfull−rjhost∥\epsilon_r=\frac{\max_{t\ge0.25,j}\left\|\mathbf r_j^{\rm impulse}-\mathbf r_j^{\rm full}\right\|}{\max_{t\ge0.25,j}\left\|\mathbf r_j^{\rm full}-\mathbf r_j^{\rm host}\right\|}
Time here is in the experiment units; maxima run over both late saved times and star identities. Replace positions with velocities for the velocity ratio. These four comparisons come from the earlier offline calculation; the new package defaults recompute 25 numerical cases and the analytic impulse limit.

Reading: Erkal & Belokurov (2015), Forensics of Subhalo-Stream Encounters, arXiv:1412.6035v2 · Sandbox Physics standalone stellar-stream scientific reader v1

External work replaces closed-system stellar energy conservation

A prescribed moving potential depends explicitly on time. The stars alone are not a closed system. Their energy can change even in a correct integration.

In the same dimensionless units, H=12∣v∣2+ΦH=\tfrac12|\mathbf v|^2+\Phi is specific total energy: kinetic energy plus both host and perturber potential terms.

The saved work and torque make the appropriate balance test possible. A self-consistent perturber orbit, recoil or tidal mass loss would require additional equations.

dHdt=∂Φp∂t=Fp⋅vp,dLdt=r×Fp\frac{dH}{dt}=\frac{\partial\Phi_p}{\partial t}=\mathbf F_p\cdot\mathbf v_p,\qquad\frac{d\mathbf L}{dt}=\mathbf r\times\mathbf F_p
Force, energy and angular momentum are specific quantities. Evaluate the force and imposed-path velocity at the same state; no compensating recoil energy is computed.

Reading: Erkal & Belokurov (2015), Forensics of Subhalo-Stream Encounters, arXiv:1412.6035v2 · Sandbox Physics standalone stellar-stream scientific reader v1

Exercises and explanations

  1. Double both mass and relative speed at fixed geometry. Is the impulse unchanged?

    Read the explanation

    Yes in this short-encounter formula, since their ratio is unchanged. A finite-duration calculation need not be identical because the duration and host response also change.

  2. Does increased stellar energy prove an integration error?

    Read the explanation

    First subtract the explicit-time work of the moving potential. A residual beyond the registered tolerance is a balance failure; the energy increase alone is not. The prescribed body and stars are not evolved as a closed self-consistent system.

Try the prediction

Test the short-encounter picture

  1. Compare compact and fast flybys, predicting the impulse trend before looking at the same late saved time.
  2. Drag the impact marker and connect the vertical response to the same prescribed trajectory; distinguish physical height from display enlargement.
  3. Export work, angular momentum and particle states and compare the balance residuals.

Sources and calculation scope

  1. Erkal & Belokurov (2015), Forensics of Subhalo-Stream Encounters, arXiv:1412.6035v2

    All 14 pages actually read. Sections 2–3 and equations 1–23 derive the short-encounter circular-stream model; sections 4–6 compare simulations and physical degeneracies. Printed pages 3/9 actually viewed. The imposed warm ensemble is not the paper’s tidal-release calculation.

  2. Sandbox Physics standalone stellar-stream scientific reader v1

    Own Python integrates 25 finite cases anew, checks an independent high-order reference, half steps, analytic limits and saved-record counts. Default zero network. Optional exact-source galpy exercise retains poor fit/nonconvergence/no-coverage failures. No App/Engine, completed tracks, originals or observed-derived figures in the ZIP.

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