>(This article originally appeared in the Q2 2003 STAR Newsletter and was written by Jerry Watson. -Ian)
Astronomical Refraction- The Basics –
by Jerry Watson
At a Club meeting some time ago the subject of the computation of the time of sunset was discussed. We noted that when the visible Sun is just sitting on the horizon the ‘real’ Sun is already just below the horizon. The visible (apparent) Sun sets a few minutes later than the real Sun; likewise the apparent Sun rises a few minutes before the real Sun. Since we define sunrise and sunset to occur when the top limb of the visible Sun just touches the horizon, the length of the day will be slightly longer than if we determined when the real Sun is just below the horizon. This time discrepancy is a consequence of the bending (refraction) of light from astronomical objects by the Earth’s atmosphere. Let’s explore the phenomenon of astronomical refraction with help from one of my old undergraduate textbooks: “Introduction To Physical Meteorology” by Hans Neuberger, Penn State University, pp 270, 1957. (Boy, am I dating myself.) The figure and tables below are from this source.
Refraction of light on traversing the atmosphere occurs for the same reason light waves bend when passing through a lens. Light, which travels at the famous c = 186,000 miles per second in a vacuum, slows down when it passes through some medium. The property of a material that measures its ability to slow down electromagnetic radiation is called the index of refraction (n) defined by:
light speed in a vacuum c
n = ————————————- = ————–
light speed in medium cm
Since cm is less than c, then n is greater than 1.0. Some typical values are:
- glass = 1.6 water = 1.33 air (sea level) = 1.0003 .
Note that light is slowed by about 38% when passing through glass, but by only .03% when going through air. Nevertheless, the latter value is enough to displace the location of an object near the horizon upward from its true location by about 1/2 degree.
The index of refraction for white light through air is related to its density, and hence to air pressure (p) and temperature (T), namely,
(n-1)106 = 79 (p/T)
The multiplication of both sides by one million (106) is for convenience in displaying values of (n-1). In this equation p is in millibars (mb) and T is absolute temperature in Kelvin degrees (K). At sea level p = 1013 mb, and on taking T = 0°C = 273 K, the equation gives a value of 293 (that is, n = 1.000293). As pressure increases and/or temperature decreases, air density and the index of refraction both increase. From the point of view of an incoming light beam, air pressure steadily increases from p = 0, the vacuum of interplanetary space, to that at sea level. Temperature, on the other hand, increases or decreases downward depending on which layer of the atmosphere the beam is passing through. While the details are relevant to the precise value of n at any particular altitude, suffice it to say that p and T combine so that air density increases downward nearly logarithmically from zero at the ‘top of the atmosphere’ to about 1.25 kg/m3 at sea level. Likewise the index of refraction increases along the light path; this causes an ever-increasing downward curvature of the light beam until it arrives at the ground.
The table below shows the variation of (n-1)106 with temperature when the pressure is that at sea level. As expected, the values decrease as T increases. The table also shows that the index of refraction at any particular temperature depends on wavelength. [Note: 1 μm = 1 micrometer = 1 millionth of a meter, and is equivalent to 1 micron in older books.]
The value of 293 that we calculated in the previous paragraph (T = 0° C) is seen to be appropriate for a wavelength in the middle of the visible spectra where the eye is most sensitive (wave length = 0.557 μm). One sees that red light (.766 μm) is bent less on passing through the atmosphere than blue light (.398 μm). This dependence on wavelength is called dispersion and also occurs as light passes through a simple lens. The result is chromatic aberration; a focused star appears to be surrounded by colorful rings. Vigorously ‘twinkling’ (scintillating) stars on a night of poor seeing may also exhibit chromatic scintillation, flickering red and blue colors, especially for stars near the horizon.
Given the value of the index of refraction, the actual path a light beam takes to the ground is governed by Snell’s law and depends on the vertical air density profile and on the angle at which the light beam enters the atmosphere (angle of incidence, i). We can assume surfaces of constant density to be concentric with the Earth’s surface; any particular density surface is at a radial distance r from the Earth’s center. At sea level r = 1 while at a height of 40 kilometers (’top of the atmosphere’ for all practical purposes) r = 1.006. The equation for the curved light beam takes the form:
nr sin i = constant
The angle i is always measured with respect to the perpendicular to the surface of constant density, that is, to the direction of the radial line from the Earth’s center through the point where the light beam meets the density surface. For a light beam from a star directly over your head at the zenith, i = 0 degrees and the above equation states that i must remain zero all the way to the ground. This light path does not deviate from its original direction; just as a light beam entering the middle of a lens straight on is not refracted.
However, for any other star between the zenith and the horizon, the light beam encounters the atmosphere at an angle, and i has a value other that zero. In this case i will vary along the light path since the product nr varies. To see what all this has to do with the apparent displacement of astronomical objects from their true location in the sky, let’s refer to the following figure.
An observer O on the surface is at the distance r0 from the Earth’s center C. Surfaces of constant density and index of refraction n are shown concentric with the ground. A star whose light beam comes from direction S enters the atmosphere at point P, where it makes an angle of incidence i with the radial line rh. The dashed line continues the path of this beam as if there were no atmosphere; the beam would pass over the head of the observer. However, the observer actually sees object S at Sʹ, because the refracted ray is bent toward the ground and arrives at O from the apparent direction O-Sʹ. Z and Z0 are the zenith angles representing the direction of the original beam and the apparent beam, respectively. The difference between these angles, Z-Z0, is the astronomical refraction, θ. One can see that astronomical objects will appear higher in the sky by an amount given by the angle θ.
The angle θ, and also the angle Φ, can be obtained by integrating appropriate equations between the top of the atmosphere and the surface, providing the variation of n (therefore of pressure and temperature) is known along the light path. For a normal vertical variation of pressure and a temperature of 8.5C, one obtains the values in the following table. Incidentally, the author of this table, F. W. Bessel, is the astronomer and mathematician, who in 1838 was the first to determine the distance to a star (61 Cygni).
In the table the values of θ are in arcminutes (60 min. = 1 deg.). Note that θ remains less than 10 min. to within 5 deg. of the horizon (Z0 = 85 deg.). As the horizon is approached, θ increases rapidly. For an object on the horizon (Z0 = 90 deg.), θ is more than half a degree. The real object is almost 35 arcminutes below the horizon.
The pronounced differential refraction near the horizon causes the Sun to appear distinctly flattened, since the bottom of the Sun is refracted upward more than the top of the Sun.
Finally, returning to our original concern regarding the time of sunrise and sunset, note first that the Earth’s rotation corresponds to a rate of about 1 degree per 4 minutes of time. Given an astronomical refraction of about ½ degree at the horizon, the apparent Sun will rise about 2 minutes earlier than the real Sun, and set about 2 minutes later. The length of the day is thus increased a little more than 4 minutes as compared to what it would be if the planet had no atmosphere.
