Curvature of source motion

For a source passing through transit, the relationship between elevation and azimuth is quadratic :

              elevation = c₀ + c₁*az + c₂*az²

with coefficients that depend upon the source declination. The inclinometer readings are linearly related to elevation, so what is plotted as a result of transit inclinometry is, conventionally

              tilt = c₀ + c₁*(az-180) + c₂*(az-180)²

where tilt is in arcseconds and az is in degrees.

Since transit inclinometry is currently performed using a pseudo-source with a declination of -60^o10', I generated 'ephemerides' with 1-second-of-time resolution for fictitious sources with declinations between -59 and -61 degrees, for the periods 30 minutes either side of transit. These plots -- of elevation (in arcsec) versus azimuth (in degrees - 180) -- were each fitted with quadratic curves. The coefficient of the first order term was essentially zero, of course, and I recorded each second order coefficient. These vary roughly linearly with declination as shown below :

and, for this limited range of declinations around -60^o10', the relationship between the second coefficient and declination is

              c₂ = 58.4674 + 1.9477 * declination

with a s.d of about 0.01" from the fitting process.

For declinations around +75^o30' we find
              c₂ = 487.0383 - 7.75 * declination
[ These signs need reversing for the inclinometry data because the inclinometry readings get more +ve when the antenna tilts further down towards the horizon, but it is these formulae that are used in the reduction software mentioned in the other document].

The quadratic curve is subtracted from the data in order to estimate the size of blemishes on this ideal such as the transit step.

Puzzlingly, the data show the need for the first-order term, c₁, to be approximately 5 arcsec : i.e. :

              tilt = c₀ + 5*(az-180) + c₂*(az-180)²

This is incorporated into the plots of the 'quadratic-subtracted' data shown in the 'transit step archive'