Atmospheric Extinction and Refraction

By: Mike Luciuk

Ah, our atmosphere. It protects us, nourishes us, and gives us life. However, it also acts as a distorting filter when we view the sky. Another tutorial addresses the atmosphere’s turbulence effects that reduce resolution of telescopes. We’ll now cover two more areas that affect our view of the heavens. It should be pointed out that major observatories have established unique algorithms and procedures to deal with extinction and refraction for their specific locations. The formulas below are more useful for amateur use.

EXTINCTION

Atmospheric extinction is the reduction in brightness of stellar objects as their photons pass through our atmosphere. The effects of extinction depend on transparency, elevation of the observer, and the zenith angle, the angle from the zenith to one’s line of sight. Therefore, looking vertically, the zenith angle is 0⁰, and is 90⁰ at the horizon.

It’s obvious that as the zenith angle increases, light from stellar objects must pass through more atmosphere, decreasing brightness. Therefore, a star viewed near the zenith appears much brighter than when it nears the horizon. We’ll examine a formula (reference 1.) that can quantify this effect.

If we assume that one’s air mass at the zenith is unity, the number of air masses X at zenith angle z is

Where e has the value of 2.71828...

For example, if z = 45⁰,

Therefore, at a zenith angle of 45⁰, stellar light passes through about 41.4% more air mass than at the zenith.

Note that the number of air masses at the horizon is 40. That is, stellar light passes through 39x more atmosphere than at the zenith.

There are three factors that can be quantitively considered to assess the effect of extinction. Molecular absorption, mainly due to atmospheric ozone and water, is a minor one, about 0.02 magnitudes per air mass. More importantly, Rayleigh scattering by air molecules accounts for up to 0.14 magnitude increases per air mass. Finally, aerosol scattering (dust, water and manmade pollutants) adds about 0.12 magnitudes per air mass. The average total effect at sea level is the sum of these factors, in the order of 0.28 magnitudes per air mass at Standard Temperature and Pressure, (STP = 760 mm Hg, 0⁰ C). Note that stellar objects are, therefore, 0.28 magnitudes brighter at the top of our atmosphere. At elevations of 0.5 km, 1.0 km, and 2.0 km, the extinction effects are about 0.24, 0.21, and 0.16 magnitudes per air mass, respectively. Therefore, mountain observatories have smaller extinction (and also refraction) effects. Extinction in winter is smaller than in summer due to less atmospheric water. Finally, Rayleigh scattering affects blue light more than red so as zenith angle increases there’s a corresponding reddening of a stellar object.

Extinction becomes significant when altitudes are lower than about 45⁰. At sea level, zenith extinction is 0.28 magnitudes, and at an altitude of 45⁰ it’s 0.40 magnitudes, an increase of only 0.12 magnitudes. However, at an altitude of 12.5⁰, extinction is 1.28 magnitudes, an increase of 1.00 magnitudes greater than the zenith’s. The effect becomes much more dramatic at even lower altitudes. At the horizon, the magnitude effect is 11.2!

The Sun’s magnitude at noon is about –26.7, but at the horizon, it would only be in the order of –15.8, a loss of brilliance almost 23,000x from that of noon. In addition, reddening would be very prominent.

REFRACTION

As stellar light passes through the atmosphere, it is refracted just as through a lens. Blue light is refracted more than red light. Like extinction, the amount of atmospheric refraction depends on the amount of air mass that light has to traverse.

There are many formulas and computer programs available to correct for refraction. Duffett-Smith has several formulas for accurate refraction predictions. A simple approximation (reference 2.) that is useful for zenith angles up to 60⁰ is

where
r is the increased perceived altitude due to refraction

z is the zenith angle (90⁰ – altitude)

k depends on the wavelength of the light and the elevation of the observer above sea level.

Note that the actual altitude of a refracted object is (perceived altitude – r). This refraction effect is relatively minor until the zenith angle approaches 90⁰.

At STP, k averages about 60.3". However, at an elevation of 2 km, k is about 48.8", a 19% reduction in refraction from sea level. For 400 (blue), 500, 600 and 700 (red) nm wavelengths, the corresponding STP k values are 60.4", 57.8", 57.4", and 57.2", respectively. So as z increases, a tiny spectrum becomes visible in good telescopic seeing, with blue at the top and red at the bottom. For example, at an altitude of 30⁰, the size of the spectrum would be about 5.5 arc seconds.

We’re familiar with the distortion of a setting Sun or Moon. Recall that these bodies average about 30 arc minutes in size. The following image illustrates the flattening effect. It also includes the effects of atmospheric turbulence.

To explain this oval shape, we must examine how rapidly atmospheric refraction increases as we near the horizon. At altitudes of 2⁰, 1⁰ and 0⁰ the respective angles of refraction, r,are 18.4’, 24.75’, and 35.35’. The differential refraction of an extended body like the Sun or Moon, where the angle of refraction is greater at the bottom of the body than at its top results in its flattened shape. When the Sun appears to be at the horizon, it has actually set.

REFERENCES

Mike Luciuk