Astronomy & Celestial Mechanics
the sky, computed
Generator

The tidal-stream generator

Two doors, 1.14 tidal radii apart on either side, and neither at the same height
Two doors, 1.14 tidal radii apart on either side, and neither at the same height. The effective potential along the line joining a 10⁵ solar-mass cluster to the centre of its host galaxy, in the frame that turns with the cluster's orbit at 20.0 kiloparsecs. The deep well in the middle is the cluster; the two maxima either side of it are the saddles of the combined field, and they sit at 1.14 tidal radii, which is what the tidal radius is defined to approximate. A star that wanders above the level of a saddle can leave, and it can only leave through one of the two: inward, toward the galactic centre, or outward, away from it. That is the first fact about a stream and it is geometric rather than statistical. The second is the energy. A star released at the inner saddle is on a slightly smaller orbit than the cluster and one released at the outer saddle on a slightly larger one, differing by about 512 squared kilometres a second either way, and in a galaxy where the period grows with radius the inner star runs ahead while the outer one falls behind. Two arms, one leading and one trailing, from one door each.

The effective potential along the line joining a 10⁵ solar-mass cluster to the centre of its host galaxy, in the frame that turns with the cluster's orbit at 20.0 kiloparsecs. The deep well in the middle is the cluster; the two maxima either side of it are the saddles of the combined field, and they sit at 1.14 tidal radii, which is what the tidal radius is defined to approximate. A star that wanders above the level of a saddle can leave, and it can only leave through one of the two: inward, toward the galactic centre, or outward, away from it. That is the first fact about a stream and it is geometric rather than statistical. The second is the energy. A star released at the inner saddle is on a slightly smaller orbit than the cluster and one released at the outer saddle on a slightly larger one, differing by about 512 squared kilometres a second either way, and in a galaxy where the period grows with radius the inner star runs ahead while the outer one falls behind. Two arms, one leading and one trailing, from one door each.

2 essays call tidal-stream. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Every figure listed here is the same construction drawn at different numbers, so a correction to one is a correction to all of them.

A stream that is not the orbit it came from. 456 stars released in pairs from the two saddles of a 10⁵ solar-mass cluster over 4.0 billion years, integrated in a halo whose circular speed is 220 kilometres a second, drawn with the progenitor's own orbit. The cluster runs between 10 and 25 kiloparsecs and the orbit is the thin closed-looking curve; the stars are everything else. The point of the figure is the discrepancy. Stars leaving through the inner saddle are on slightly smaller orbits, turning round at a median of 24.77 kiloparsecs rather than the progenitor's 25.00, and therefore running ahead; stars leaving through the outer saddle reach 25.22 and fall behind. The whole spread is 1.8 per cent of the apocentre, which is the number worth carrying: an offset far too small to see in this drawing builds the entire stream, because it acts for four billion years. The two arms are therefore not merely displaced along the orbit, they are on different orbits, and the track a survey measures is a family of them rather than any single one. Fitting a Galactic potential by demanding that a stream lie along an orbit is wrong by exactly this much, and the size of the error grows with the mass of the progenitor, because the mass is what sets the distance between the two doors. Galaxies

A stream is not the orbit it came from

A cluster torn apart by a galaxy leaves a thin trail of stars across the sky, and the obvious thing to do with it is fit an orbit. That is wrong by a knowable amount, because stars leave through two doors with a small energy offset and end up on a family of orbits rather than on one.

 A gap's depth is one number: 2GmT ÷ v b². The central density of the gap against impact parameter, 4 billion years after the encounter, for subhaloes of 10⁶, 10⁷, 10⁸ solar masses. Points are read off the drawn density profiles; the curves are 1/(1 + 2GmT/v b²), the stretch the map applies at the encounter point, and the two agree to 3.0 per cent wherever the histogram has enough stars left in the gap to measure a depth at all. Everything about the encounter enters the depth through that one combination. The consequence is the horizontal reading: a gap of a given depth is produced by every point along a locus on which the mass rises as the square of the impact parameter — half depth at 0.42, 1.33, 4.19 kiloparsecs for the three masses drawn, an exponent of 0.500 against the half the algebra requires. What breaks that particular degeneracy is the gap's width, which scales as the impact parameter itself while the depth does not: rescale s by b and the map is identical, so the profile is one shape stretched. Depth and width together give the impact parameter and the product mT. They do not give the mass. Galaxies

A hole that says mass times time

A gap in a stellar stream is the strongest evidence available that dark subhaloes exist, and its depth depends on the perturber's mass, the impact parameter and the elapsed time only through one combination. Two of those three are unobservable, so a gap is a measurement of a product.

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