Astronomy & Celestial Mechanics
the sky, computed
Concept

Granulation — where it appears

The top of a convection zone seen from outside: cells about a thousand kilometres across, each a rising column ringed by cooler descending lanes. It biases every stellar radial velocity, because the rising material is brighter than the sinking material and so is over-weighted in the average.

Named by 7 essays across 3 fields — each of them below, with the objects they name alongside it.

Fringe visibility for a 47 mas disc at 575 nm. Fringe visibility against the separation of the two apertures, for a disc 47 milliarcseconds across seen at 575 nm. The solid curve is a uniform disc, |2J₁(x)/x| with x = πθB/λ; it is exactly one at zero baseline, where both apertures see the same wavefront, and falls to zero at 3.08 m — read off the drawn samples, and equal to 1.21967 λ/θ to better than one part in a million. That is the measurement: not a brightness, a baseline. The dashed curve is the same disc with linear limb darkening u = 0.4, whose null is 4.9% further out at 3.23 m — so the same observed null implies 47 mas as a uniform disc and 49.3 mas limb-darkened, and a diameter quoted without its model is a number without a unit. At the 2.54 m aperture of the telescope this was done on, the visibility is still 0.17: one mirror cannot reach the null, which is the same statement as saying it cannot resolve the star.

An angle of five hundredths of an arcsecond

No telescope has ever resolved a star other than the Sun, and stellar diameters are measured anyway — by finding the separation of two apertures at which the star's interference fringes vanish. What that returns is an angle; the radius arrives only when a distance is brought in, and the distance is the worse-known half.

starlight · Angular diameter A 28.7 km/s correction, and a 12.5 m/s planet underneath it. Two years of radial velocities of a star at ecliptic latitude 12°, orbited by a companion whose reflex semi-amplitude is 12.5 m/s — the Sun's own, from Jupiter. The upper panel is what the spectrograph measures: the Earth's motion about the barycentre of the solar system, amplitude 28.72 km/s, which is V⊕ cos β to a fraction of a per cent. The planet is in that curve and is 2,297 times smaller than it, which is a line thinner than the stroke it is drawn with. The lower panel is the same data after the correction, and the correction is not a fit: it is computed from an ephemeris, the observatory's position on a rotating deformable Earth, and the star's own coordinates and proper motion. To leave a centimetre a second it has to be right to one part in 2.9·10⁶ — the light-travel time across the Earth's orbit, the relativistic terms, and the fact that the star moves are all inside that budget. What remains is the planet, at 4333 days, and a scatter of 1.2 m/s that is the star rather than the instrument.

The metre per second that is not the star

A line shift is a speedometer, and the speed it reads is mostly the observer's. Getting to a metre a second means removing thirty kilometres of the Earth's own motion to a part in three million, and then confronting a floor that is the star's own surface rather than the instrument.

starlight · The Doppler effect Which gradient is steeper, and where. The temperature gradient radiation would need in order to carry the whole luminosity, against the gradient a rising blob of gas follows, through an n = 3 polytrope with a Kramers opacity and an energy generation going as ρT^4. Wherever the first exceeds the second the layer is unstable and convects. This model puts the outer boundary at 89 per cent of the radius; helioseismology measures the base of the Sun's convection zone at 71.3 per cent, and the gap between the two numbers is the price of a polytrope with one opacity law rather than a solar model. The curve also rises above the adiabatic value inside 4 per cent of the radius, which is a convective core — it appears when the energy generation is concentrated enough, and it is the structure a star burning by the CNO cycle has.

The part of a star that boils

Energy leaves a star's interior either by radiation or by bulk motion, and which one happens is settled by comparing two temperature gradients. The comparison decides that the Sun's outer third churns and its core does not, and that a massive star does exactly the opposite.

stars · Energy transport 14 radial orders of the Sun, at 135.1 μHz apart. The p-mode spectrum of the Sun — 1 solar mass in 1 solar radius — from the asymptotic relation with its second-order term, drawn as 14 radial orders of ℓ = 0, 1 and 2 under a Gaussian envelope centred on ν_max = 3,090 μHz. Two numbers are marked and they do very different work. The large separation, 135.1 μHz, is the spacing between consecutive ℓ = 0 modes and fixes the mean density. The small separation, 9.00 μHz, is 2.7 pixels on this axis — it fixes the age, and it is why the échelle diagram exists rather than being a convenience. The vertical axis is the measurement: each mode moves the surface by about 20.0 cm s⁻¹ at the peak, and brightens it by a few parts per million, which is why this was impossible before a decade-long velocity series. Each mode is one line: its true width is set by its lifetime and is far below a pixel here.

The interior read from a comb of frequencies

A star's surface moves by about twenty centimetres a second, in thousands of overlapping sound modes at once. Two numbers off that spectrum give a mass and a radius with almost no stellar model in the chain, and a third gives an age.

stars · Asteroseismology 4000 lines averaged into one profile, and 2.3 m/s out of it. A cross-correlation function: the average absorption profile obtained by shifting a mask of 4000 line positions across a spectrum and summing what falls under it. The faint curves behind are individual lines, each with its own depth, its own width and its own small offset; the heavy curve is what averaging them produces. The velocity is the position of the peak, and its precision is the width divided by the contrast, the signal-to-noise and the square root of the number of lines — 2.3 metres a second here. Nothing about this construction is a measurement of any one line. It is a measurement of where a weighted average of thousands of them sits, and the weights are a choice: a mask built for one spectral type applied to another weights the disagreement between the lines differently, and moves the peak.

A velocity that is an average of lines that disagree

A radial velocity measured to a metre a second is not measured from a line. It is the position of the peak of a cross-correlation against a mask of thousands of lines, and those lines do not agree with each other by hundreds of metres a second — because each one forms at a different depth in an atmosphere that is boiling.

starlight · The Doppler effect An exposure time chosen by the star. The fraction of a star's p-mode velocity signal that survives averaging over an exposure of the length on the horizontal axis, for an envelope of modes centred on νₘₐₓ with the stated width. The dashed curve is a single oscillation at νₘₐₓ = 3.1 mHz, a period of 5.4 minutes: it averages to exactly zero after one period and at every multiple. The solid curve is the real case, a band of modes 1.2 mHz wide, which cannot all be nulled at once: its first minimum is at 5.8 minutes, where 15% of the rms survives, and beyond it the residual falls only as one over the exposure time. Short exposures on bright stars sample the oscillations rather than averaging them, which is why a bright star observed quickly can be noisier than a faint one observed slowly.

An exposure time chosen by the star

A Sun-like star's surface rings at five minutes and boils on timescales from ten minutes to a day, and each of those motions shifts its spectral lines by more than the nine centimetres a second an Earth would. The ringing can be averaged away by an exposure tuned to the star; the slowest boiling cannot be averaged away within a night at all, and it sets the number of nights an Earth twin costs.

exoplanets · Reflex velocity Where the photons stop mattering and the star starts. Noise in a 13-hour window — the length of an Earth-like planet's transit across a Sun-like star — against the star's brightness, for a space telescope that reaches 17 parts per million in 6.5 hours on a twelfth-magnitude star. Photon noise falls by a factor of ten for every five magnitudes of brightness; the star's granulation, averaged over the window, stays at 5.1 parts per million whatever the brightness. The two are equal at magnitude 10.1; brighter than that, the star is the floor. The dashed line is an Earth's transit depth across the Sun, 84 parts per million: a single transit on a twelfth-magnitude star is detected at 6.4 standard deviations, and on the brightest stars at no better than 17, set by the star alone — and supergranulation, spots and faculae add more on these timescales.

A floor the star sets for itself

Above the atmosphere, with comparison stars and a big enough telescope, every noise in a measurement of a star's brightness can be beaten down except one: the star's own surface, a mosaic of convective cells each brighter or fainter than the rest. That flicker does not average away against anything, and it is larger for stars that are bigger and puffier than the Sun. For the brightest stars, an Earth's transit is limited by the star, not the photons — and the stars Kepler watched turned out twice as noisy as the Sun.

starlight · Photon noise

Named alongside it

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

ConvectionRadial-velocityStellar jitterConvective blueshiftCross-correlationHelioseismologyLine bisectorStellar radiusSurface gravitySystematic errorAdiabatic gradientAngular diameter

All concepts