Refraction

As light from a star goes through the atmosphere, it is bent due to the change 
in refractive index.  The first procedure to quantize the deviation in ray 
direction was to calculate a final change in angle.  This angle would be from 
the apparent look direction (tangent to the altitude) to the instrument to the 
point at which the ray leaves the atmosphere (outer limit of atmosphere chosen 
to be 120 km where refractive index is equal to one to the twelfth decimal).  
For worst possible conditions (lowest altitude of 10 km and tangent look 
direction), the deviation in angle was around 0.6°.  By this method, however, it 
would be difficult to relate this change to other parameters of interest or 
calculations (e.g. field of view).  

The second method is similar, whereby at each shell interface a new direction 
for the ray is calculated.  As well, calculations begin at the instrument and 
the ray is ‘back-tracked’ through the atmosphere.  The final deviation from the 
apparent line of sight is the summation of change in angle at each interface 
(i.e. the difference between the angle at which the ray strikes the shell 
interface and the angle at which it continues through the next shell is added 
together each time to give a total deviation angle).  In this way, it is 
possible to start at any altitude with any apparent look direction.  However, 
the worst case scenario is still the lowest tangent altitude (now at ground 
level).  Using this method with 5 km shells, and data for pressure and 
temperature from CIRA (January 60° N latitude), the deviation from the initial 
look direction to where the starlight actually originated, was calculated to be 0.3°.

This method was verified and correlated with members of the tomographic team.  
Further analysis performed with continuous refraction (instead of having shells 
and interfaces) lead to an estimate 0.5° deviation from apparent look direction 
(at ground level and tangent altitude). In order for consistency in tomography, 
this team could include two-dimensional ray tracing (with the suggested 
Runge-Kutta method) as well as the algorithms that have been developed.  From 
similar estimates of the effects of refraction by satellite studies, the 
estimated deviation of true look-direction may translate into only a few 
centimeters of deviation.  With the projected positional accuracy of our instrument, this effect would then be minor. 

One point that should be noted is that the ray can not be traced from space to a 
final tangent altitude.  The light source is estimated at infinity which leads 
to parallel rays of light incident on the atmosphere.  Thus, the stars within 
the field of view would appear to remain stationary if the instrument did not 
spin or swing.  Only those stars on the extreme outer edge may drop out of 
sight.  One can not track a ray through the descent in order to find specific changes in angle because the view is of a continuum of parallel beams. 

References

Allen, C.W., Astrophysical Quantities, The Athlone Press: London, 1973, p. 124.

Chamberlain, J.W.,  The Theory of Planetary Atmospheres, Academic Press: New York, 1978.