On Atmospheric Extinction and Atmospheric Refraction

Practical Experience:

You might have had similar life experiences: Is the morning sun less intense than the noon sun? Does the moon keep getting “brighter” as it rises from the east to the zenith? These phenomena are closely related to the “extinction” phenomenon of the Earth’s atmosphere!

As is well known, the Earth has a thick layer of atmosphere, which is both the source of our breath and a protective umbrella for our lives. It provides us with the ozone layer, blue skies, wind, frost, rain, dew, and even magnificent rainbows. However, from an astronomical perspective, the atmosphere is an “obstacle”; the dense atmosphere is unfavorable for clear observations of the universe. Therefore, astronomers have always hoped to build observatories at higher altitudes because the atmosphere there is thinner. To satisfy the thirst for higher clarity, people have even placed telescopes outside the Earth.

The influence of the Earth’s atmosphere on observation is mainly manifested in two aspects: one is atmospheric extinction, and the other is atmospheric refraction (these also seem to be favorite topics in astronomical Olympiads). The former refers to the weakening effect of the atmosphere on light, which is why the morning sun is not as bright; the latter refers to the bending effect of the atmosphere on light. For example, at the last moment we see the setting sun on the horizon, the sun has actually already fallen below the horizon.

Atmospheric Extinction:

“Extinction” refers to the process where celestial radiation is weakened by a series of influences before being received by the observer. During the propagation of light, it is mainly affected by interstellar extinction and atmospheric extinction (the Hubble Space Telescope does not have atmospheric extinction problems). In astrometry, any kind of extinction effect cannot be ignored.

For friends observing on the Earth’s surface, celestial radiation passing through the atmosphere to reach the ground will produce a decrease in intensity and a change in color, which is Atmospheric Extinction. We see stars “twinkling” because of atmospheric turbulence, which also belongs to atmospheric extinction. In daily life, we can clearly experience the significant effect of atmospheric extinction: on a sunny day at noon, the brightness of the sun is -26.7 magnitudes; in the morning or at dusk, the sun here has a brightness equivalent to only -15.8 magnitudes. From an astronomical perspective, atmospheric extinction has weakened the sun’s brightness by 23,000 times!

The impact of atmospheric extinction on celestial observation is also very obvious. Celestial photographs taken on Earth are often blurrier than those taken by space telescopes. You can discover this by comparing the “portraits” of Jupiter taken by the Hubble Space Telescope and the Nordic Optical Telescope (ground-based). For celestial data obtained in observations, we must first correct for atmospheric extinction and refraction before they can be used as data for the next step.

Influencing Factors:

There are many factors causing atmospheric extinction, which can be summarized into three categories: First, various molecules and atoms in the atmosphere (ozone and water vapor) absorb radiation, converting radiation energy into other forms of energy. This is a relatively small effect, weakening the brightness of celestial bodies by about 0.02 magnitudes. Another relatively large effect is the “Rayleigh scattering” phenomenon of atmospheric molecules, which reduces the brightness of celestial bodies by about 0.14 magnitudes. Finally, there is the scattering effect of suspended particles in the atmosphere (dust, water droplets, and high levels of atmospheric pollutants), which reduces the brightness of celestial bodies by 0.12 magnitudes. Under standard conditions (0^\circC and standard atmospheric pressure), the combined influence of these three factors on an observation point at sea level—that is, the average weakened magnitude—is 0.28.

Atmospheric extinction is related to the composition of the atmosphere, the wavelength of the radiation, and the thickness of the atmosphere the radiation passes through. Usually, blue light suffers more severe extinction than red light. As the thickness of the atmospheric layer that radiation passes through increases, the extinction effect also intensifies. Therefore, the main factor is, of course, the zenith angle (z, referring to the angle of the celestial body from the zenith).

The schematic diagram easily explains why the zenith angle has such a significant impact on atmospheric extinction: starlight coming from the zenith only needs to pass through the atmospheric thickness of AC; while non-zenith light must pass through the atmosphere of BC. Obviously BC > AC, so the weakening effect of the atmosphere is strengthened.

If the atmosphere is considered an ideal plane, then it is easy to have (z as the zenith angle): X \approx \frac{BC}{AC} = \frac{1}{\cos z} However, reality is always less than ideal; the atmosphere is annular, so we need to add a correction factor: X = \frac{BC}{AC} = \frac{1}{\cos z + 0.025e^{-11\cos z}}

This is the ratio of the atmospheric optical thickness in the direction where the zenith distance equals z to the atmospheric optical thickness in the zenith direction, which we call the Atmospheric Optical Mass. Here e is the base of the natural logarithm, with a value of 2.71828\dots. If we substitute z=45^\circ, we get BC/AC=1.414\dots, which means that at a zenith angle of 45^\circ, starlight needs to pass through 41.4\% more atmosphere.

The figure below shows the curve of atmospheric extinction changing with the zenith angle. It can be seen that initially, the zenith angle has little effect on atmospheric extinction, but it rises sharply after exceeding 75^\circ.

(The vertical axis is in magnitudes)

The relationship between the observed magnitude m' and the actual apparent magnitude m is: m' = m + kX

Here k is the extinction coefficient, which is a function of wavelength and also depends on the physical conditions in the atmosphere, so it varies by time and place. Approximately, we have:

Altitude also correspondingly affects atmospheric extinction. We know that the higher the altitude, the thinner the atmosphere, and the better the seeing (that is, the clearer the view). At altitudes of 0.5 km, 1.0 km, and 2.0 km, the extinction magnitudes at the zenith are 0.24, 0.21, and 0.16, respectively. Seasons also affect observations, leading to different degrees of extinction. The seeing of the winter sky is better than in summer; correspondingly, the worst seeing of the day is in the afternoon. The end of the article provides extinction magnitudes for different seasons, altitudes, and zenith angles for reference.

Atmospheric “Lens”:

From junior high school physics, we know that when light enters from a vacuum or a medium of one density into a medium of another density, the direction of light propagation changes. Similarly, the cosmic space outside the atmosphere can be approximately regarded as a vacuum, and when starlight passes through the atmosphere, the atmosphere acts like a lens, causing a similar deflection of light. This is atmospheric refraction.

The atmosphere refracts blue light more than red light, which is an important reason why the sky appears blue on sunny days. Like atmospheric extinction, the degree of atmospheric refraction depends on the “air mass” that the starlight needs to penetrate. There are many formulas and computer programs used to correct for atmospheric refraction. Duffett-Smith provided several formulas that can accurately calculate the atmospheric refraction rate. For zenith angles not exceeding 60^\circ, there is a commonly used simple estimation formula: r \approx k \tan z

Where:

• r is the increased altitude angle we observe due to atmospheric refraction.

• z is the zenith angle (90 degrees minus the altitude angle).

• k depends on the wavelength of light and the observer’s altitude.

It should be noted that the actual altitude angle equals (observed altitude angle - r). From the figure below, it can be seen that when the zenith angle is not large, the impact of atmospheric refraction is relatively small.

(The vertical axis is the refraction angle in arcseconds)

Under standard atmospheric pressure and sea-level conditions, the average value of k is 60.3'', while at an altitude of 2 km, the value of k is about 48.8'', which is about 19\% less than at sea level. In other words, the atmospheric refraction rate is also affected by altitude. For light with wavelengths of 400 (blue), 500, 600, to 700 (red) nanometers, the corresponding k values under standard atmospheric pressure are 60.4'', 57.8'', 57.4'', and 57.2''. So when the zenith angle increases, a narrow spectrum of infrared and ultraviolet light that was originally invisible can be seen in an excellent telescope. For example, at an altitude of 30^\circ, the spectral range can reach 5.5 arcseconds.

We are very familiar with the “deformation” of the sun and moon when they are rising or about to set. The circular sun becomes a flattened ellipse. One reason is the deflection of light by the atmosphere, which also includes interference from turbulence such as dust and water vapor in the atmosphere. To explain this elliptical shape, we must understand how fast the atmospheric refraction angle increases near the horizon: At altitude angles of 2^\circ, 1^\circ, and 0^\circ, the values of r are 18.4', 24.75', and 35.35', respectively. Since the sun or moon is a circular disk of about half a degree, the deflection angles of each point on the disk are different, so we see a flattened “sun” or “moon”. When the sun appears on the horizon, it is actually already below the horizon.

Final Words:

Although the atmosphere causes indelible interference to astronomical observations, its great role still makes us never hate it. It is no exaggeration to say that without the atmosphere, there would be no life. As intelligent beings, to eliminate the influence of atmospheric extinction, what we need to do is not to “get rid of” the atmosphere, but to step out of its envelope. Thus, we have various space telescopes, such as Hubble and the next generation, Webb!

Appendix 1:

Atmospheric Extinction Table for different altitudes (km) (Average values)

Atmospheric Extinction Table for different altitudes (km) (Winter)

Atmospheric Extinction Table for different altitudes (km) (Summer)

Appendix 2:

Note: There is a question in the Astronomy Olympiad asking whether the apparent magnitude of a celestial body changes between a sunny day and a cloudy day. The answer is no. The apparent magnitude here is defined by m - M = \log D - 5. This “apparent magnitude” depends on the luminosity of the celestial body and its distance from us, and is not affected by other external factors. As for the magnitude we see after it becomes dimmer or brighter, it is not “apparent magnitude” in the strict sense.

Appendix 3:

Regarding the calculation formula for “Atmospheric Optical Mass,” the formula listed in the article is the one proposed by Rozenberg in 1966. Many versions have since been derived with varying degrees of accuracy. For specific details, please refer to the link on this site: https://kexue.fm/archives/396

Article Sources:

• http://www.asterism.org/tutorials/tut28-1.htm

• http://www.cfa.harvard.edu/icq/ICQExtinct.html

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