How tides turn a cluster into a stellar stream

Compare two gravitational accelerations

The cluster centre and a star near its edge both feel the Galaxy. Their relative fate depends on the difference between those accelerations, not the total Galactic attraction. In the frame rotating with a circular cluster orbit, the centrifugal contribution also enters.

Let the circular radius be $R$ , the angular frequency $\Omega$ , the small radial-oscillation frequency $\kappa$ and the cluster mass $m_c$ . A local expansion requires a cluster much smaller than its orbital radius.

Displace a star radially by a small $x$ . The host acceleration changes by $-\Phi_h^{\prime\prime}(R)x$ , while the centrifugal term changes by $+\Omega^2x$ . Thus the outward coefficient is $\Omega^2-\Phi_h^{\prime\prime}=4\Omega^2-\kappa^2$ . Balance that linear term against the cluster attraction $Gm_c/x^2$ to obtain the tidal scale below.

r_t^3=\frac{Gm_c}{4\Omega^2-\kappa^2}

This local tidal scale assumes a circular orbit and a positive denominator. Eccentric clusters encounter a tidal field that changes along their path.

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

Worked example: a ten-thousand-solar-mass cluster

For a flat circular-speed curve, increasing radius decreases angular frequency. At the same radius the radial epicyclic frequency is higher than the orbital angular frequency. Differentiating gives their relation and the tidal denominator.

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

\Omega=\frac{v_c}{R},\qquad\kappa^2=R\frac{d\Omega^2}{dR}+4\Omega^2=2\Omega^2

Flat means constant circular speed, not constant angular frequency.

r_t=\left(\frac{Gm_cR^2}{2v_c^2}\right)^{1/3}\simeq46.4\,\mathrm{pc}

Use

m_c=10^4M_\odot

,

R=15\,\mathrm{kpc}

and

v_c=220\,\mathrm{km\,s^{-1}}

. This analytic example describes a hypothetical cluster; the encounter experiment begins with an already prepared stellar arc.

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

Where does the radial frequency come from?

For a star moving in a fixed spherical potential, conservation of angular momentum lets us write its radial motion as motion in an effective potential. The extra centrifugal term is a way of retaining the tangential motion while solving for radius. It is not an additional physical mass. For this radial derivation choose the first two axes in this star’s orbital plane and the third along its angular momentum. The normal component used below then equals the angular-momentum magnitude. Radius here is distance from the centre within that orbital plane. An inclined star’s component along the Lab reference axis cannot replace this orbital-plane normal component.

At a circular guiding radius $R_g$ , the first radial derivative vanishes. Displace the radius slightly while keeping the same angular momentum. Expanding the radial force to first order gives a harmonic oscillator whose squared frequency is the curvature of that effective potential. A positive curvature gives a restoring force.

This explains why two clocks enter the tidal calculation. One describes going around the centre; the other describes the small inward–outward oscillation. They coincide in some special potentials, but a flat circular-speed host has different clocks.

\Phi_{\rm eff}(R)=\Phi_h(R)+\frac{L_z^2}{2R^2},\qquad\kappa^2=\Phi_h^{\prime\prime}(R_g)+3\Omega^2

The circular condition is the balance between the host radial force and the centrifugal contribution. Expanding about that condition produces the frequency used above.

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

The same location can conceal different futures

Two neighbouring stars with slightly different velocities will not remain neighbours. We therefore need positions and their conjugate momenta together: phase space. In our inertial Cartesian specific coordinates, momentum equals velocity.

Hamiltonian motion stretches and folds a small phase-space element while preserving its phase-space volume. A one-dimensional sky count can become sparse without contradicting that statement, because the projection hides distance, velocity and other directions.

The left-hand ellipse is a harmonic oscillator in position–velocity coordinates. It is not the spatial outline of a stellar orbit.

Harmonic phase-space ellipse and diverging orbital phases — Left: position and velocity trace a harmonic phase-space ellipse. Right: phases accumulate at different rates for two near-circular reference orbits. Both panels are analytic illustrations.

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

An action measures amplitude; an angle tracks progress

For periodic motion, the area enclosed by the phase-space path provides a useful amplitude coordinate: an action. Its partner angle describes progress around a cycle. That angle need not be the visible spatial azimuth.

J=\frac1{2\pi}\oint p\,dq,\qquad J_{\rm harmonic}=\frac E\omega

For the harmonic oscillator the action is the ellipse area divided by a full turn. At fixed frequency, larger energy means larger area.

\theta(t)=\theta(0)+\Omega(J)t

In the appropriate action–angle coordinates of an integrable fixed potential, actions stay fixed while angles advance at their frequencies. Real GD-1 action inference would additionally require its phase-space information and an adopted Galactic potential.

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

Worked example: one percent on either side of the reference

A nearly circular orbit in a flat-speed host has $L_z=R_gv_c$ . Set the speed and length units to one, and compare $L_z=0.99$ with $L_z=1.01$ . After eight time units:

\Omega=\frac{v_c^2}{L_z},\qquad\theta_{1.01}-\theta_{0.99}=8\left(\frac1{1.01}-\frac1{0.99}\right)\simeq-0.160\,\mathrm{rad}

The larger-angular-momentum orbit trails by about 9.2 degrees. This example uses the near-circular guiding-centre relation; eccentric orbits require their own actions and frequencies.

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

Connect the picture to the experiment

The experiment assigns 192 distinct initial states and integrates each trajectory. It omits escape from the cluster, so its initial snapshot cannot validate the tidal-radius calculation. It preserves the next question: how an already distributed set of orbits responds to an external encounter.

The independent reader also measures a small radial-oscillation period between successive radial maxima. A maximum followed by a minimum is usually only half a radial period; that distinction matters when testing the analytic frequency.

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

How tides turn a cluster into a stellar stream

Compare two gravitational accelerations

Worked example: a ten-thousand-solar-mass cluster

Where does the radial frequency come from?

The same location can conceal different futures

An action measures amplitude; an angle tracks progress

Worked example: one percent on either side of the reference

Connect the picture to the experiment

Exercises and explanations

Try the prediction

Follow neighbouring orbits

Sources and calculation scope