Sections · Airmass

Vol. Science · Astronomy · The air

Airmass
and the horizon

Straight up, a star is seen through one thickness of air. Near the horizon the path is longer, so the star is dimmer, and the air lifts it. Relative air mass on this desk is the Kasten and Young 1989 approximation, m = [sin γ + 0.50572 (γ + 6.07995)^−1.6364]⁻¹, with the apparent elevation γ in degrees. Their Table II, integrated through the ISO Standard Atmosphere (1972), lists 1.0000 at 90°, 1.9939 at 30°, 5.5841 at 10°, and 38.0868 at the horizon. The plane-parallel shortcut is sec(z) = 1/sin γ. It is 2 at 30° of altitude and has no finite value at 0°. Extinction in magnitudes is k times that air mass. The slopes are published medians and nights from Krisciunas and colleagues: Mauna Kea’s summit, the 2800 m level, and two sea-level nights at 75 m. Refraction is Bennett’s 1982 formula from the apparent altitude, R = cot(h + 7.31/(h + 4.4)) arcminutes. Sæmundsson’s 1986 formula goes the other way, from true altitude: R = 1.02 cot(h + 10.3/(h + 5.11)). Meeus multiplies both by (P/1010) × (283/(273+T)), which is 1 at 1010 hPa and 10°C. Sunrise on the twilight desk is the standard −0.833°: about 34′ of refraction plus about 16′ of solar semi-diameter. Catalog brightness, with this air left out, is the magnitude desk. What the horizon puts overhead is the local sky. What an aperture collects of the light that gets through is the telescope desk.

Educational calculator. A classroom line of sight. Not a forecast, not an ephemeris, and not a promise of what a given night will do. Within a few degrees of the horizon, real refraction depends on the temperature profile, which a single ground temperature does not fix. The extinction slopes were fit at ordinary airmasses. Carrying k × X out to the horizon is that straight line, not a second measurement. Catalog magnitudes here are visual. A B, J, H, or K slope is not added to a V magnitude. The shell in the drawing is thicker than the real air so the ray can be seen.

Interactive · Airmass, extinction, refraction

Line of sight through a schematic shell · thickness exaggerated

Line of sight
Zenith

Opens at 30° of apparent altitude. sec(z) is 2. Kasten and Young are 1.9943. Table II lists 1.9939.

Altitude · apparent elevation

Object · catalog V, extinction left out

Site · published extinction slopes

Band · magnitudes per airmass

Limiting magnitude · a smaller number is brighter

Altitude deg

Limit mag

Pressure hPa

Temp °C

Kasten & Young · 30° · Vega · summit V

1.9943

Relative air mass · sec(z) = 2.0000

Airmass1.9943
sec(z)2.0000
k0.11 mag / airmass
Dimming0.219 mag
Catalog V+0.03
Through the air+0.25
Versus limitBrighter than 6.0
Refraction1.72′
True altitude29.971°

At 30° the approximation is 1.9943 and sec(z) is 2.0000. Table II lists 1.9939 here. At the horizon sec(z) is unbounded and the approximation is 37.92. Table II lists 38.0868.

Summit V is the 1987 median, 0.11 magnitude per airmass. Vega’s catalog +0.03 becomes +0.25. That is still brighter than the limit 6.0. A smaller magnitude is brighter.

Bennett at 30°, 1010 hPa, and 10°C is 1.72′. The horizon value of the same formula is 34.5′, beside NOAA’s about 34′. The setting-Sun button puts the center one semi-diameter, 16′, above the horizon.

The sliders need JavaScript. The relations are still Kasten and Young, m = [sin γ + 0.50572 (γ + 6.07995)^(−1.6364)]^(−1). sec(z) = 1/sin γ. Table II is 1.0000 at 90°, 1.9939 at 30°, 5.5841 at 10°, and 38.0868 at 0°. Extinction in magnitudes is k × m. Summit medians from Krisciunas et al. 1987: B 0.19, V 0.11, J 0.10, H 0.05, K 0.07. The hazy sea-level V on 19 December 1989 is 0.341 ± 0.020. Bennett at apparent altitude 0°, 1010 hPa, and 10°C is 34.5′. NOAA’s sunrise standard is −0.833°, about 34′ plus about 16′. Not a forecast.

What the three numbers do

A longer path, a dimmer star, a lifted limb.

Airmass. One airmass is the vertical path. The plane-parallel formula treats the air as a flat slab, so the path is sec(z). That is an excellent match to Kasten and Young through the middle of the sky: 2 against 1.9943 at 30°. By 10° the slab says 5.76 and the approximation says 5.586. On the horizon the slab has nowhere to stop, and the curved, thinning atmosphere in their Table II is 38.0868. This page evaluates the approximation, which returns 37.92 at 0°. The paper prints a relative error of 0.432% on that row. The elevation in the formula is the apparent one, the altitude you see, because refraction has already bent the ray.

Extinction. A Bouguer plot is magnitude against airmass. The slope is k, in magnitudes per airmass, so the dimming is k × X. Krisciunas et al. 1987 give summit medians B 0.19, V 0.11, J 0.10, H 0.05, and K 0.07. The 1990 paper’s 2800 m median V is 0.172, and its sea-level Table 2 at Keaau, 75 m, is V 0.341 ± 0.020 on the hazy night of 19 December 1989 and V 0.187 on the clearer night of 20 December. Blue dims more than visual. The infrared slopes are smaller. Those k values are the fits from the nights they measured. They are not a new horizon photometry. Five magnitudes of dimming is a factor of 100 in flux, the same scale as the magnitude desk.

Refraction. The air lifts the apparent place of a star. Bennett’s formula takes the apparent altitude and returns the lift in arcminutes. At 0° and the standard 1010 hPa and 10°C it is 34.5′. NOAA’s sunrise and sunset use about 34′ of that allowance, plus about 16′ of solar semi-diameter, as the geometric altitude −0.833°. The lower limb of the setting Sun is lifted more than the upper limb, because the lift grows quickly in the last degree. The vertical diameter comes out shorter than the horizontal one. The horizontal size on this page is about 32′, twice the about-16′ semi-diameter. The vertical size is Bennett’s formula applied to the two limbs. It is not a photograph. Sæmundsson’s formula starts from the true altitude instead. Fed the true altitude that Bennett implies, the two stay within a tenth of an arcminute on the checks this page runs.

Presets · cited altitudes and cited magnitudes

What each button sets, and the source of that number
Button	Set on the controls	Why that number
Zenith · 90°	Apparent altitude 90°. The object, site, and band stay as they are.	Table II lists relative air mass 1.0000. sec(z) is 1. Bennett’s residual at the zenith is under 0.01′. Refraction at the zenith is zero. The formula is not forced to print zero.
30°	Apparent altitude 30°.	sec(z) is exactly 2. Table II lists 1.9939. The approximation on this page is 1.9943. This is where the flat-slab picture still holds.
10°	Apparent altitude 10°.	Table II lists 5.5841. sec(z) is 5.76. The gap is already visible, and it is not yet the horizon.
Horizon · 0°	Apparent altitude 0°.	The approximation is 37.92. Table II lists 38.0868, with a printed relative error of 0.432%. sec(z) is unbounded. Bennett is 34.5′ at 1010 hPa and 10°C.
Setting Sun	The Sun, with the center at 16′ of apparent altitude. That is one semi-diameter up, so the lower limb is on the apparent horizon before differential refraction is applied.	NOAA’s about 16′ is the semi-diameter inside the −0.833° sunrise standard. The about 34′ is the refraction allowance at the horizon. The button shows both, and the flattened vertical diameter from Bennett.

Literacy · the words on the readout

Apparent altitude: The elevation you see, measured up from the horizon. Kasten and Young and Bennett both take this angle. The true altitude is lower by the refraction. At the horizon the difference is the whole point of the page.
sec(z): One over the sine of the apparent altitude. The path through a flat slab. It matches the curved atmosphere until the last 10° or so, and then it runs away. The drawing is curved on purpose. The painted length is not the airmass. The number is.
k × X: Magnitudes of dimming. k is the slope those papers measured, in magnitudes per airmass. X is the Kasten and Young air mass at the slider. The summit V button is 0.11. The hazy 75 m V button is 0.341 ± 0.020. The same star can be an easy naked-eye object at the zenith and gone at the horizon.
Limiting magnitude: The faintest magnitude you are willing to count. 6.5 and 6.0 are the ideal range in NASA’s Imagine the Universe note. 4.0 is Sky & Telescope’s Bortle class 9 inner-city edge, a limiting magnitude of 4.0 or less. The slider runs from −2 to 15 so a bright planet and a bright telescopic limit both fit. Those ends are the control. A smaller magnitude is brighter. The verdict uses V only.
Why the Sun looks flat: Refraction is a vertical lift, and the lift is larger at a lower altitude. The bottom of the Sun is lifted more than the top. The left and right limbs sit at the same altitude, so the horizontal diameter stays about 32′, twice NOAA’s about 16′. Bennett’s formula supplies the vertical diameter. On the setting-Sun button the lower limb is at the apparent horizon, the lower limb is lifted by 34.5′, and the vertical diameter is shorter than 32′. Cold air and high pressure increase the factor (P/1010)×(283/(273+T)). They do not replace the temperature profile of the lowest air, which is what actually sets the horizon lift on a given evening. The atmosphere desk is that layered air. Skyglow, which is a different way to lose the faint end, is the light-pollution desk.

Sources · the formulas and the presets

Cited, not invented

Kasten, F., and Young, A. T., Applied Optics 28, 4735–4738 (1989) — Revised optical air mass tables and approximation formula. Relative air mass m(γ) = [sin γ + 0.50572 (γ + 6.07995)^−1.6364]⁻¹, γ the apparent elevation in degrees. Table II is the integral through the ISO Standard Atmosphere (1972): 38.0868 at 0° (relative error 0.432%), 5.5841 at 10°, 1.9939 at 30°, 1.1543 at 60°, and 1.0000 at 90°. This page evaluates the approximation. At 0° that evaluation is 37.92.
Young, A. T., Applied Optics 33, 1108–1110 (1994) — A later air-mass and refraction note, for true zenith angle. This desk stays with the 1989 apparent-elevation formula. It does not mix the two fits.
Krisciunas, K., et al., PASP 99, 887 (1987) — Atmospheric extinction at Mauna Kea. Summit medians, their 4200 m level: B 0.19, V 0.11, J 0.10, H 0.05, K 0.07 magnitudes per airmass. The 2800 m median V in that paper is 0.18. The button for 2800 m uses the later median below, and says so.
Krisciunas, K., PASP 102, 1052 (1990) — Extinction at 75 m, 2800 m, and 4205 m on the island of Hawaii. Median V at 2800 m is 0.172. The B mean at 2.8 km in the scale-height set is 0.322 ± 0.019. Table 2, Keaau, elevation 75 m: 19 December 1989, noticeable haze, V 0.341 ± 0.020 and B 0.498 ± 0.055. 20 December 1989, no volcanic haze, strong scintillation, V 0.187 ± 0.046 and B 0.334 ± 0.033. B is for a star of B−V = 0. The paper’s summit means (V 0.125 ± 0.007, B 0.206 ± 0.011) are a different summary from the 1987 medians. The summit button keeps the medians.
Bennett, G. G., Journal of Navigation 35, 255–259 (1982) — Astronomical refraction from apparent altitude. R = cot(h + 7.31/(h + 4.4)), R in arcminutes, h in degrees. Compared in that paper with Garfinkel’s algorithm, the standard the nautical almanac offices were using. This page does not reprint an accuracy bound the abstract does not state.
Sæmundsson, Þ., Sky & Telescope 72, 70 (1986) — Refraction from true altitude. R = 1.02 cot(h + 10.3/(h + 5.11)), R in arcminutes, h in degrees. This page feeds it the true altitude implied by Bennett and shows the difference.
Jean Meeus, Astronomical Algorithms — the pressure and temperature factor used with both formulas, (P/1010) × (283/(273+T)), P in millibars (hPa) and T in °C. 1010 hPa and 10°C give a factor of 1. 1010 hPa is 101.0 kPa. The NOAA solar-calculation details are the sunrise note this desk shares with the twilight page: 0.833° for refraction at the horizon plus the size of the solar disk. The NOAA solar equations use zenith 90.833°.
NASA NSSDCA — Sun Fact Sheet — visual magnitude V(1,0) −26.74. Moon Fact Sheet — apparent visual magnitude −12.74 at mean opposition. Venus Fact Sheet — maximum visual magnitude −4.8.
Ducati, J. 2002 (VizieR II/237), as listed by SIMBAD: Sirius V = −1.46, Vega V = 0.03, Polaris V = 2.02. Polaris is typed a classical Cepheid, so 2.02 is approximate for a given night.
NASA GSFC Imagine the Universe — under ideal conditions, stars as faint as magnitude 6.0 to 6.5. Sky & Telescope — How dark is your night sky? — Bortle class 9, limiting magnitude 4.0 or less. The idea of that bright sky is the light-pollution desk.

Classroom line of sight. Not a forecast. At 30° with the summit V median, Vega’s +0.03 becomes +0.25. At the horizon the same slope is about four magnitudes, and the hazy sea-level slope is about thirteen. Bennett at 0°, 1010 hPa, and 10°C is 34.5′. NOAA’s sunrise figure remains the about 34′ inside −0.833°. Earthrise preview: NASA / Bill Anders (public domain).

Keep going

The clock

Altitude	Apparent elevation, 0° to 90°. Zenith, 30°, 10°, and the horizon are rows in Kasten and Young’s Table II. The setting-Sun button sets 16′, NOAA’s about-16′ semi-diameter, and selects the Sun.
Object	Catalog V with extinction left out. Sun −26.74, full Moon −12.74, Venus −4.8, Sirius −1.46, Vega +0.03, Polaris +2.02. The Moon, Venus, and Polaris are marked approximate in the sources.
Site and band	Summit medians from 1987, the 2800 m median V and mean B from 1990, and the two Table 2 nights at 75 m. The before-and-after magnitude is computed only in V. Other bands show k × X and do not rewrite the V catalog value.
Limit	6.5, 6.0, and 4.0 are the cited naked-eye anchors. The slider runs from −2 to 15. Those ends are the control.
Pressure and temperature	The Meeus factor on Bennett and Sæmundsson. 1010 hPa and 10°C give 1. The slider ends, 900 to 1100 hPa and −20°C to 40°C, are the control, not a climate record.

Shell	A thick band so the ray is visible. The real air is a thin skin. The length of the painted line is not the airmass.
Zenith ray	Dashed, for comparison. Altitude 90° is that ray.
Sun	When the Sun is selected, a small disk shows the vertical squash. The disk is a shape, not the angular size on the sky. A horizon line is drawn when the disk is low enough to meet it.

Chart	Left and Right step the altitude by one degree while the canvas is focused. Up and Down step it by a tenth of a degree.
Buttons	Enter or Space selects. The pressed button is aria-pressed. Targets are at least 44 pixels on a side.
Phone	Below 900 pixels the long section row becomes a single Sections control. Open it for the same links. The site menu is the Menu button in the top bar.
This dialog	The question mark, or ?, opens it. Escape closes it. Tab stays inside until you close it. Click the dim page behind the card to close it.

Airmass
and the horizon

A longer path, a dimmer star, a lifted limb.

Cited, not invented

Twilight

Magnitude

Local sky

Telescope

Light pollution

Atmosphere

Airmass desk

Inputs

Drawing

Keyboard and touch

Airmassand the horizon

A longer path, a dimmer star, a lifted limb.

Cited, not invented

Twilight

Magnitude

Local sky

Telescope

Light pollution

Atmosphere

Airmass
and the horizon