In article ,
says...
On Sun, 30 Sep 2012 12:05:18 -0700, "W. eWatson"
wrote:
It seems what I'm after is misinterpreted. I was talking with a friend
about some variable star work he does at his domed observatory. Somehow
I asked him at what is the lowest elevation he surveys. He said 30
degrees, about two atmospheres of thickness. He then said," maybe near
the horizon, it's 47 thickness." That's what I'm looking for.
How many thickness of atmosphere will I see from zero to ninety degrees
above the horizon.
As an idealized approximation, consider the Earth as a sphere sitting
at the origin of the usual x-y-z Cartesian coordinates. Without loss
of generality, we can limit ourselves to the z=0 plane. We end up
with the circle
x^2 + y^2 = R^2
where R is on the order of 4000 miles but you can choose whatever
units and accuracy you like.
If the atmosphere has a depth of r miles, its boundary forms a
concentric circle outside the first with the formula
x^2 + y^2 = (R+r)^2
Also without loss of generality, we can put the observer at (0,R), the
top of the circle (corresponding to the north pole).
Obviously, looking up vertically (90 deg) means looking through r
miles of atmosphere. And looking out horizontally (0 deg) means
looking through sqrt(2Rr+r) miles.
When looking up at an angle e, you look through the atmosphere to the
point where the line
y = sin e * x + R
meets the above circle. Replacing y in equation of the circle
produces an equation which is easy enough to solve for x and then
compute y.
What this doesn't account for is atmospheric density. If we assume
the atmosphere is 20 miles thick, then looking horizontally passes
through 400+ miles of atmosphere. While this is only 20 times as much
as looking up, most of it is much thicker and probably has greater
visual impact. So while it is only 20 times as thick, it might well
be 47 times the impact.
Since the density is a continuous function of altitude, the formula
for computing impact is probably an obnoxious integral. It would take
into account that as you move out horizontally, the altitude increases
slowly and the rate of increase itself increases as the ground level
Your approximation of the atmosphere is called the "homogeneous
atmosphere" which is the simplest possible atmospheric model: one then
assumes that the air density is constant everywhere up to some "upper
surface" where the atmosphere would end (somewhat like the oceans of the
Earth). If the atmosphere was a liquid rather than a mix of gases, this
approximation would work well. However, the "thickness of the
atmosphere" would not be as large as 20 miles, but instead about 8
kilometers (= about 5 miles). That's what you get if you take the ground
air pressure (= the weight per unit area of the atmosphere) and divide it
by the ground air density and then also by the Earth's acceleration of
gravity. The "homogeneous atmosphere" approximation would yield some 40
airmasses in the horizontal direction compared to the vertical direction.
The second simplest approximation to the Earth's atmosphere is the
"isothermal atmosphere": here one assumes that the atmosphere is an ideal
gas which has the same temperature everywhere, and one also neglects the
(small) changes in the Earth's acceleration of gravity at different
altitudes. The result is that both the air pressure and the air density
decreases exponentially with elevation. There's a quantity here called
the "scale height" of the atmosphere, which is the difference in
elevation over which the pressure and temperature decreases by a factor
of e (= the base of the natural logarithms). Interestingly, the scale
height in the isothermal atmosphere approximation is exactly the same as
the "thickness of the atmosphere" in the homogeneous atmosphere
approximation: about 8 kilometers. And the air masses at different
altitudes above the horizon is indeed described with a ghasty integral --
the solution to that integral is called the Chapman function. This
integral lacks an analytic solution, but there are useful approximations
available.
To proceed further, we need some quantities:
T = absolute temperature
k = Boltzmann's constant
m = average molecular weight of the air
g = the Earth's acceleration of gravity
H = scale height = k*T / (m*g)
z = zenith distance
r = radius of curvature of the surface of the Earth
X = r / H
y = sqrt( 0.5 * X ) * abs(cos(z))
With these quantities, a useful approximation to the Chapman function
ch(X,z) is:
ch(X,z) = sqrt(pi/2 * X) * exp(y*y) * erfc(y)
where erfc(y) is the complementary error function: erfc(y) = 1 - erf(y),
which can be approximated as:
0 = y = 8:
a = 1.0606963
b = 0.55643831
c = 1.0619896
d = 1.7245609
erfc(y) = exp(-y*y) * (a + b * y) / (c + d * y + y*y)
ch(X,y) = sqrt(pi/2 * X) * (a + b * y) / (c + d * y + y*y)
8 = y = 100:
f = 0.56498823
g = 0.06651874
erfc(y) = exp(-y*y) * f / (g + y)
ch(X,y) = sqrt(pi/2 * X) * f / (g + y)
There are also series expansions of the erfc function:
http://en.wikipedia.org/wiki/Erfc
A series expansion which is useful in this case is:
erfc(y)*exp(y*y) = (1/(y*sqrt(pi))) * ( 1 + SUM(n=0,1,2,...)((-1)^n * ( 1
* 3 * 5 * ... * (2*n-1) ) / ( (2*y*y)^n ) ) )
which can be written as:
erfc(y)*exp(y*y) = (1/(y*sqrt(pi))) * ( 1 + A )
where
A = SUM(n=0,1,2,...)((-1)^n * ( 1 * 3 * 5 * ... * (2*n-1) )/((2*y*y)^n))
This series diverges for every y, but if y is large enough (i.e. larger
than approx. 4 to 5), only the first few terms need to be included.
If the zenith angle is exactly 90 degrees, then y = 0 and erfc(y) = 1 and
the Chapman function formula can be simplified to:
ch(X,90_deg) = sqrt(pi/2 * X)
which, for common atmospheric conditions, yields:
T = absolute temperature = ( 273 + 15 ) = 288 K
k = Boltzmann's constant = 1.3806488E-23 J/K
m = average molecular weight of the air = 28.97 * 1.66053886E-27 kg
g = the Earth's acceleration of gravity = 9.80665 m/s^2
H = scale height = k*T / (m*g) = 8428 m = 8.428 km
X = 6378 / 8.428
ch(X,90_deg) = sqrt(pi/2 * 6378/8.428) = 34.5
i.e. at the horizon one sees 34.5 air masses if the temperature is +15
deg C.
For zenith angles larger than 90 degrees (i.e. if you look below the
horizon), you get:
ch(X,z) = 2*exp(X*(1-sin(z))) * ch(X*sin(z),90_deg) - ch(X,180_deg-z)
ch(X,z) = 2*exp(X*(1-sin(z))) * sqrt(pi/2 * X * sin(z)) - ch(X,180_deg-z)
These formulae may be useful when you're observing from a high
mountaintop.
The paper describing these calculations was originally published in 1972
by F.L. Smith and Cody Smoth. The paper, titled "Numerical evaluation of
Chapman's grazing incidence integral ch(X, Chi)" is available from the
link below. Unfortunately, it's not for free, you must pay to get a copy:
http://www.agu.org/pubs/crossref/197...19p03592.shtml
Willian Swider extended this to cases where the scale height varies with
altitude. His paper "The determination of optical depth at large solar
zenith angles" can be obtained from this link. Again, this paper is not
for free, you must pay to get it:
http://www.sciencedirect.com/science...3206336490056X
None of these formulae accounts for the refraction in the Earth's
atmosphere. They were intended to be used for studies of the Earth's
ionosphere, where atmospheric refraction of light is insignificantly
small.