3. ALTITUDE CALCULATIONS

3.1 A SIMPLE APPROACH 

Figure 3.1.1: A simplified two-dimensional diagram of the Earth's attitude on the 6'th of December 2002.

We first start out with a simple approach to calculate the altitudes for our event. A sketch of the elevation of the Earth and the solar zenith angle for the 6'th December 2002 is shown in Figure 3.1.1. Initially we assume no refraction of the solar rays as they penetrate the atmosphere. The location of the Auroral Stations is  78o 12 ' 10'' North and 15o 49 ' 30'' East. At 10:00 UT the solar depression angle was 10.91o. The azimuth angle to the sun was 168.79o, which gives us an azimuth angle of observation close to 21o for the MSP. The altitude h that it is necessary to climb in order to see the sun at Longyearbyen is then calculated by simple triangular functions. Using the above parameters as input, an altitude of h = 117 km is obtained. The distance D-A of Fig. 3.1.1 is close to 607 km with an altitude h' = 29 km. The angles are obtained using the Solar Calculator provided by the Surface Radiation Research Branch of NOAA (http://www.srrb.noaa.gov/highlights/sunrise/azel.html).

The above calculations indicate that if we look at the horizon from Longyearbyen on the 6'th of December 2002 (10:00 UT), then everything we see illuminated by the sun above point D must be higher than 29 km. Note that we then assume the target to be vertical in extent at that point. The horizontal extent of the emitting target must be found as well.

3.2 A MORE DETAILED DESCRIPTION
The below calculations are based on work conducted on twilight observations of airglow [cf.Chamberlain, 1961]. We will adopt the resulting formulas to obtain an altitude estimate of the emitting layer, under the assumption that the layer is a result of the direct and immediate action of sun light. If we first assume the configuration without the presence of an absorbing atmosphere, the intersection between the observer's line of sight and solid-Earth shadow may be visualized as in Fig. 3.2.1. Note that the below figures are re-productions from Chamberlain [1961].

Figure 3.2.1: 3D-diagram for height determination without atmosphere. 

In Fig. 3.2.1 the point Ps is the intersection of the solid-Earth shadow and the point O, the observer's line of sight. z is the zenith distance. Df is the azimuth from the sun. Here we assume that the great circles on which a and b are measured are perpendicular to the terminator. They intersect at the sub solar point as seen in Fig. 3.2.2.

Figure 3.2.2: The geocentric celestial sphere. The point P's is the projection of Ps onto the celestial sphere. g is the length of the arc.

THE SHADOW OF THE SOLID EARTH
The above Figures 3.2.1 and 3.2.2 show the situation when atmospheric screening is neglected. If we allow for refraction and atmospheric screening the situation becomes a little bit more complicated. A 2D sketch is shown in Fig. 3.2.3.

Figure 3.2.3: Two - dimensional geometry of twilight scattering.

Note that in the above Fig. 3.2.3 the point of the observer O has been projected onto the plane containing the great circle b. The angles b and g are not in general in the same plane. z0 is the height where the incident ray passes just above the screening height h0 and intersect with the observers line of sight. Note that the effect of refraction lowers the rays.

The Apparent Height zs, the height of the shadow from the Solid Earth with no refraction, is as found from the triangle CPsQs, where C is the center of the Earth;

  zs = R ( sec b -1 ) .   (3.2.1)

R is the radius of the Earth. Furthermore, from the triangle COPs, using the law of sines, we obtain the van Rhijn formula

                           sin q s = ( R / (R+zs ) ) sin z  = cos b sin z ,   (3.2.2)

where

          g = z - q s .   (3.2.3)

The final required relation is given by the spherical triangle in Fig. 3.2.2

sin b = cos g sin a - sin g cos a cos Df .   (3.2.4)

In our case, the known variables are a, Df, R and z.  The easiest way to solve these equations is to adapt an iterative solution. The method was first reported by Chamberlain [1958]. We start the iteration by assuming a height zs(0). From equation (3.2.2) we obtain q s(1). The next step is to compute g(1) from equation (3.2.3) and b(1) from (3.2.4), and finally zs(1) from (3.2.1). The process is then repeated, and as stated by Chamberlain,  it will happily converge. Table 3.2.1 shows the results for our event. 

qmsp [o [o [o [o Ls [km]  qs [o  N  zs [km] 
90 0 10.91 0.00     0.00 0.00 2 117.01
100 10 10.74 0.18   19.63 9.82 3 113.44
110 20 10.58 0.35   39.29 19.65 4 109.92
120 30 10.40 0.54   60.21 29.46 4 106.23
130 40 10.20 0.76   83.67 49.24 5 102.15
140 50   9.96 1.02 112.97 48.98 6   97.23
150 60   9.63 1.37 151.95 58.63 7   90.84
160 70   9.13 1.91 211.66 68.09 8   81.50
170 80   8.19 2.90 322.22 77.10 14   65.55
180 90   5.64 5.63 624.79 84.37 75   30.92

Table 3.2.1. Calculated apparent heights. Location is the Auroral Station in Adventdalen (N 78o 12 ' 10'', E 15o 49 ' 30''). Date is 06.12.2002 and time is 10:00 UT. The solar depression angle a = 10.91o and the azimuth angle of observation Df = 20.79o. The Greek letters are according to the above equations. qmsp is the scan angle of the Meridian Scanning Photometers [MSP] (South is 180o).

The results in Table 3.2.1 corresponds well to our first attempt to get the height. See section 3.1. The number of iterations are set according to a height resolution of 0.1 km. The parameter Ls = g R is the distance from the observer to Ps', which corresponds to the distance A-C in Fig. 3.1.1.

HEIGHT MEASUREMENTS WITH ATMOSPHERIC SCREENING AND REFRACTION

Figure 3.2.4: 3D geometry of observation with atmospheric screening.

In order to continue, we must include the effect that the atmosphere itself casts a shadow. Fig. 3.2.4 explains the situation. ho is called the atmosphere's Screening height [Vegard, 1940]. The aim is the to get the Actual shadow height z0 of intersection with the line of sight when we also include refraction. The height of intersection of the un-refracted ray is called z1. See also Fig. 3.2.3. First of all, if the observations are done in zenith, then

z1(1) - zs = ho sec b.   (3.2.5)

Note that the subscript (1) refers to zenith observations. Furthermore, the actual height will be lowered due to refraction corresponding to    

dz (b) = z1(1) - z0(1) = 0.020 [N(ho)/N(0)] ( R + ho ) tan b sec b,   (3.2.6)

where N is the density of the atmosphere. The factor 0.020 is from the fact that a star that rises or sets is observed to refract 0.01 radian. A tangential ray passing completely through the lower atmosphere will suffer a refraction of 0.020 [N(ho)/N(0)] radian. The total change in shadow height due to the lower atmosphere can now be written has

z0(1) - zs = ho sec b - dz.   (3.2.7)

From Fig. 3.2.4 we get the following triangular relations

q = s cos c ,   (3.2.8)

z0(1) - z0 = q tan b,   (3.2.9)

z0 - zs = s cos qs,   (3.2.10)

and

cos c = sin qs cos Df.   (3.2.11)

Eliminating q, s, and cos c from Equations (3.2.8) to (3.2.11) yields

(z0(1) - z0) / (z0 - zs ) = tan qs tan b cos Df .   (3.12)

Finally, combining Equation (3.2.12) with (3.2.7), we find

z0 - zs = ( ho sec b - dz ) / ( 1 + tan qs tan b cos Df ).   (3.2.13)

The procedure is then as follows: The parameters obtained in Table 3.2.1 can now be used to calculate the refraction term dz from Equation (3.2.6). The actual height z0 including screening and refraction then follows from equation (3.2.13). Table 3.2.2 shows the results for our event.

qmsp [o z [o b [o g [o  Ls [km]  qs [o  N  zs [km]   dz [km]   Lo [km]   zo [km] 
90 0 10.91 0.00     0.00 0.00 2 117.01 4.50     0.00 124.73
100 10 10.74 0.18   19.63 9.82 3 113.44 4.43   20.93 120.99
110 20 10.58 0.35   39.29 19.65 4 109.92 4.36   41.89 117.31
120 30 10.40 0.54   60.21 29.46 4 106.23 4.28   64.19 113.45
130 40 10.20 0.76   83.67 49.24 5 102.15 4.20   89.50 109.18
140 50   9.96 1.02 112.97 48.98 6   97.23 4.10 120.67 104.04
150 60   9.63 1.37 151.95 58.63 7   90.84 3.95 162.51  97.37
160 70   9.13 1.91 211.66 68.09 8   81.50 3.73 226.67  87.63
170 80   8.19 2.90 322.22 77.10 14   65.55 3.34 346.11  71.10
180 90   5.64 5.63 624.79 84.37 75   30.92 2.28 675.75  35.97

Table 3.2.2. Calculated Apparent heights and Actual heights including atmospheric Screening and refraction. Location is the Auroral Station in Adventdalen (N 78o 12 ' 10'', E 15o 49 ' 30''). Date is 06.12.2002 and time is 10:00 UT. The solar depression angle a = 10.91o and the azimuth angle of observation Df = 20.79o. The screening height ho is set to 12 km.

Let the point P'  be the projection of P onto the celestial sphere. Then Lo is the distance from O to P'. It is found by first using the cosine law

M 2 = R 2 + ( R+zs ) 2 - 2 R ( R + zs ) cos g,   (3.2.14)

where M is the distance OPs. The total field of view distance (OP) is then simply

M T = M + s.   (3.2.15)

Secondly, the law of sines gives us

sin g' / MT = sin ( p - z ) / (R + zo ),   (3.2.16)

and finally

Lo = R g'.   (3.2.17)

We now have a mathematical tool to understand our measurements.