Astronomy & Celestial Mechanics
the sky, computed
Concept

The Shapiro delay — where it appears

The extra light travel time a signal accumulates while passing through a mass's gravitational field. In a binary pulsar it recurs once per orbit with a shape giving both the companion's mass and the inclination, which makes it the cleanest way to weigh a neutron star.

Named by 4 essays across 3 fields — each of them below, with the objects they name alongside it.

Three ways for a timing model to be wrong, and three shapes that say which. Timing residuals over 6 years for the Crab pulsar, one curve per kind of error in the model, in microseconds. A position error of 1.2 mas leaves a sinusoid of period exactly one year — measured off the drawn curve as 1.000 — with amplitude (a/c)·δθ·cos β = 2.7 μs, because the error is being projected onto a baseline that is the Earth's own orbit and nothing about the pulsar. An unmodelled proper motion of 0.9 mas/yr leaves the same sinusoid with an envelope growing linearly: twice as large at 6 years as at 3. An error of one part in 10⁹ in Ṗ leaves a parabola, the second integral of a frequency drift, whose second derivative is constant to 4e-12 across the span — and that fractional error is deliberately minute, because anything larger produces a residual thousands of times the other two and draws them as flat lines. The shapes do not resemble each other, which is the whole reason a pulsar is an instrument rather than a clock: fitting them simultaneously delivers a position, a proper motion and — from the annual curvature term, not drawn here — a parallax, all from the arrival times of pulses and no image of anything. What is left when every known shape has been removed is the science: glitches, red noise, and the correlated residual between pairs of pulsars that a timing array exists to find.

A clock read against a model of everything in between

A pulsar delivers arrival times and nothing else. Everything else is a model, and what is measured is the difference between the model and the arrivals — a residual whose shape says which term is wrong, and whose annual sinusoid is a position measured from pulses rather than from an image.

stars · Pulsars A measured mass of 2.08 deletes an equation of state. Mass against radius for three neutron-star equations of state, each a polytrope P = Kρ² integrated through the Tolman–Oppenheimer–Volkoff equation from the centre outwards until the pressure reaches zero. Every sequence rises, turns over and falls; only the rising part is stable, because past the maximum adding mass makes the star smaller and the smaller star cannot hold itself up. The maxima here are 1.60, 2.10, 2.57 solar masses, in the order of increasing stiffness — the same nuclear matter with a slightly harder response to compression supports a heavier star, and nothing else in the calculation changes. The horizontal band is PSR J0740+6620, whose mass of 2.08 ± 0.07 solar masses comes from the Shapiro delay of its own pulses passing its companion, which is a timing measurement and involves no model of the star at all. It sits above the maximum of one of the three, and those are not disfavoured but excluded: an equation of state that cannot hold up two solar masses is wrong, whatever else recommends it. The two lines at the left are exact and no star may cross them — the Schwarzschild radius, and the bound above it inside which the speed of sound in the matter would exceed the speed of light. A caution about the curves themselves: a Γ = 2 polytrope is a stand-in for nuclear matter and runs a kilometre or two large in radius at fixed mass, so read the ordering and the maxima rather than the radii.

A radius that decides what matter can be

Nobody can make matter at four times the density of an atomic nucleus, and no calculation settles what it does there. What can be done is to weigh a neutron star — every candidate description of that matter predicts a heaviest star it could hold up, and a single measured mass above that value deletes the description permanently.

stars · Degeneracy Three passes, and the fourth is below the noise. How far the position is still wrong after each pass of the light-time iteration, for five targets, on a logarithmic scale of kilometres. The first pass uses the body's position now and is wrong by exactly the distance the body travels while its light is in transit — 6254 kilometres for Mars near opposition, which is about twenty-five arcseconds and larger than any residual in an ephemeris fit. Each further pass multiplies the error by the body's speed divided by the speed of light, so the lines are straight and their slopes are that ratio and nothing else. Three passes take every one of these below a metre, which is why the loop in an ephemeris code has a fixed trip count rather than a convergence test. The same argument run backwards is why the correction cannot be applied to the observation instead: the direction light arrived from is what the observation is, and the position it corresponds to is not known until the body's orbit is.

Where a planet is and where it is seen

An ephemeris is fitted to observations, and no observation is of a position. It is of a direction light arrived from, at a time that is not the time the light left, bent by a Sun that is nowhere near the line of sight. Three corrections stand between the two, and all three are larger than the residuals.

orbits · Ephemerides Four media, and only two of them care about the frequency. The four propagation delays between a tracking station and a spacecraft, in metres of round-trip range, against the angle between the target and the Sun. The troposphere contributes a few metres and depends only on the elevation; the ionosphere is smaller at X band and scales as the inverse square of the frequency; the solar plasma rises steeply towards conjunction and scales the same way; and the Sun's gravitational delay rises as a logarithm and has no frequency dependence at all. That last distinction is the whole of the calibration strategy: transmitting and receiving at two widely separated frequencies measures the two plasma terms and removes them, leaving the troposphere to a weather model and the relativistic term to theory. Near conjunction the plasma exceeds everything else by an order of magnitude, which is why a spacecraft passing behind the Sun is both untrackable and the best available laboratory for measuring the relativistic term.

Four media between the antenna and the spacecraft

A deep-space range measurement is a round-trip time, and the signal spends that time passing through a troposphere, an ionosphere, the solar wind and a curved region of spacetime. All four delay it, none of them is the orbit, and the whole of the navigation depends on removing them.

spaceflight · Radiometric navigation

Named alongside it

The objects these essays reach for when they reach for this one.

Solar conjunctionAberrationApparent placeAstrometric placeBarycentric coordinate timeBarycentric correctionCausality limitCompactnessDegeneracy pressureDispersionDispersion measureDual-frequency

All concepts