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(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 electro-magnetic 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:
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.
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