Airmass & Atmospheric Extinction

What airmass actually measures

Every photon that reaches a ground-based telescope has just finished a swim through the atmosphere. Airmass is the bookkeeping for how long that swim was. We define it as the column of air the light traversed relative to the shortest possible column — the one straight up. Look at the zenith (directly overhead) and the light crossed exactly one airmass; that is the definition, X = 1. Tip your telescope toward the horizon and the same atmosphere is now a long slanted slab, so the light crosses many airmasses.

For a flat, uniform, plane-parallel atmosphere the geometry is pure trigonometry. If z is the zenith angle (0° overhead, 90° at the horizon), the path length is the vertical thickness divided by cos(z):

X = sec(z) = 1 / cos(z)

That single formula carries the whole intuition: at the zenith cos(0°) = 1 so X = 1; at 60° from the zenith cos(60°) = 0.5 so X = 2; at 70°, X ≈ 2.92. But sec(z) diverges to infinity as z → 90°, which is physically wrong — the real atmosphere is a thin curved shell wrapped on a round planet, so the slant path stays finite. The widely used Kasten & Young (1989) fit replaces the bare secant with a curved-atmosphere correction and yields a maximum of about X ≈ 38 at the true horizon. In practice the divergence matters only below ~10° altitude; above 70° altitude the plain secant is accurate to better than a percent.

Extinction: where the photons go

Crossing more atmosphere means losing more light, and that loss is atmospheric extinction. It is the sum of several physical channels:

• Rayleigh scattering — light bounces off air molecules (mostly N₂ and O₂). The cross-section scales as 1/λ⁴, so it dominates in the blue and ultraviolet.
• Aerosol (Mie) scattering — dust, sea salt, smoke, and volcanic haze. The wavelength dependence is gentle (roughly 1/λ to 1/λ¹·³) and highly variable night to night.
• Molecular absorption — ozone removes a broad chunk of the green-yellow (the Chappuis band) and nearly all of the ultraviolet; water vapor and O₂ carve absorption bands in the red and near-infrared.

Because extinction acts multiplicatively on flux — each layer transmits a fixed fraction — it adds linearly in magnitudes, which are logarithmic. That gives astronomy's workhorse relation:

m_obs = m₀ + k · X

Here m_obs is the measured magnitude (fainter = larger number), m₀ is the magnitude you would measure with no atmosphere at all (the "above-the-atmosphere" value), and k is the extinction coefficient in magnitudes per airmass. The relation is a straight line in the plane of magnitude versus airmass; observe a star at several altitudes through the night, fit the line, and the slope is k while the intercept at X = 0 is the corrected magnitude. This is the heart of all-sky photometric calibration.

Reddening: why low stars turn orange

The 1/λ⁴ steepness of Rayleigh scattering means k is not one number — it depends on wavelength. Blue light is scattered out of the beam far more aggressively than red. At a good dark site the visual (V, ~550 nm) coefficient is around 0.15 mag/airmass, the blue (B, ~440 nm) is closer to 0.25–0.30, and the red (R, ~640 nm) drops to ~0.10. So as airmass grows, a star's blue light is depleted faster than its red light, and the transmitted color shifts redward — this is atmospheric reddening.

It is exactly the everyday physics of sunset. At the zenith the noon Sun crosses one airmass and looks white; at the horizon it crosses ~38 airmasses, the blue is almost entirely scattered into the surrounding sky (which is why the daytime sky is blue), and what remains is the familiar orange-red disk. The same effect makes bright low stars like Sirius "twinkle" through a rainbow of colors: turbulence steers the already-reddened, dispersed beam, and your eye samples a flickering spectrum.

Putting numbers on it

The table below shows airmass and the corresponding extinction in the V band for a representative coefficient k_V = 0.15 mag/airmass. The brightness ratio is 10^0.4·Δm relative to the zenith.

Altitude	Zenith angle z	Airmass X	Extra extinction (k·X, V)	Brightness vs. zenith
90° (zenith)	0°	1.00	0.15 mag	1.00×
60°	30°	1.15	0.17 mag	0.98×
45°	45°	1.41	0.21 mag	0.95×
30°	60°	2.00	0.30 mag	0.87×
20°	70°	2.90	0.44 mag	0.76×
10°	80°	5.6	0.84 mag	0.53×
~0° (horizon)	~90°	~38	~5.6 mag	~0.006× (×1/170)

The lesson observers internalize: keep targets high. Beyond an airmass of ~2 (altitude 30°) the extinction, the differential color term, and the atmospheric turbulence all rise steeply, and the limiting magnitude of a survey collapses. This is why telescope queues prioritize objects near transit and why "airmass" is a column in every observing log.

How extinction varies with site, band, and night

The extinction coefficient is not a universal constant — it is a property of the air above a particular telescope on a particular night. The table contrasts typical values.

Condition / band	k (mag/airmass)	Dominant cause
U band (~365 nm), good site	0.45–0.60	Rayleigh + ozone, steep in UV
B band (~440 nm), good site	0.25–0.30	Rayleigh scattering
V band (~550 nm), good site	0.10–0.20	Rayleigh + aerosols + ozone
R band (~640 nm), good site	0.07–0.12	Aerosols
I band (~800 nm), good site	0.04–0.08	Aerosols + water bands
High-altitude site (Mauna Kea, 4200 m)	lower across all bands	less air, less aerosol/water above
After a major volcanic eruption	+0.1 to +0.3 in V	stratospheric aerosol layer

High, dry sites (Mauna Kea at 4200 m, the Atacama at 2400–5000 m) win twice: there is simply less air column overhead, and that column is poorer in the water vapor and aerosols that drive the most variable part of extinction. Stratospheric aerosols injected by eruptions such as Pinatubo (1991) raised measured V extinction at observatories worldwide for years.

The second-order color term

For precise photometry the simple m = m₀ + k·X is not quite enough, because extinction varies across a filter's bandpass. A blue star and a red star seen through the same V filter at the same airmass are not dimmed by exactly the same amount — the blue star's light is weighted toward the steeper part of the extinction curve. Astronomers add a second-order color term:

m_obs = m₀ + k′·X + k″·(color)·X

where k′ is the first-order (gray) coefficient and k″ couples airmass to the star's color index (e.g. B−V). The term is small — a few hundredths of a magnitude per airmass per magnitude of color — but it is the difference between 1% and 3% photometry, and it matters enormously for surveys like LSST that chase millimagnitude calibration across the whole sky.

Common misconceptions

• Airmass is a mass. No — it is a dimensionless path-length ratio, normalized to 1 at the zenith. The name is historical.
• sec(z) works everywhere. It is excellent above ~70° altitude but diverges at the horizon; use Kasten & Young near the ground.
• Extinction dims all colors equally. It is strongly wavelength-dependent (∝1/λ⁴ for Rayleigh) — that is the whole reason for reddening.
• The extinction coefficient is fixed. It changes with site, altitude, season, humidity, dust, and volcanic activity, and must be re-measured each night for precise work.
• Reddening here equals interstellar reddening. Both redden, but atmospheric reddening is removable by observing higher; interstellar reddening from dust between stars is intrinsic to the line of sight.
• A star at the horizon is only a little fainter. At the horizon it is dimmed by ~5.6 magnitudes in V — a factor of ~170 — which is why deep imaging at low altitude is futile.