Difference between revisions of "Modulated scan high cadence LAT"

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(Created page with "=Introduction= This post refines the work shown in High_cadence_LAT_from_Chile. We show how to modulate the scan rate to achieve maximally uniform integration depth acros...")
 
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=Geometrical considerations=
 
=Geometrical considerations=
  
Let us assume that the observatory is located at latitude <math>\theta_0</math>
+
Let us assume that,
 +
* <math>\theta_0</math> is the observatory latitude, measured in radians from the Equator
 +
* <math>\alpha</math> is the observing azimuth, measured in radians from North.  East is at <math>\pi/2</math>
 +
* <math>\beta</math> is the observing elevation, measured in radians from the horizon.
 +
* <math>\theta</math> is the declination on the celestial sphere.
 +
 
 +
Then we can write the declination as a function of the observatory latitude and observing azimuth and elevation.
 +
 
 +
<math>
 +
\theta = \sin^{-1}\left(\cos \theta_0\cos\beta\cos\alpha + \sin\theta_0\sin\beta\right)
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</math>
 +
 
 +
Moreover, we can write for the rate at which declination changes whilst scanning at constant azimuthal rate
 +
 
 +
<math>
 +
\frac{\mathrm d\theta}{\mathrm d\alpha}
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= \frac{\cos\theta_0\cos\beta\sin\alpha}{\sqrt{1-\left(\cos\theta_0\cos\beta\cos\alpha + \sin\theta_0\sin\beta\right)^2}}
 +
= \frac{\cos\theta_0\cos\beta\sin\alpha}{\cos\theta}
 +
</math>
 +
 
 +
The time, <math>T</math>, spent observing at a specific declination is inversely proportional to the rate at which the declination changes
 +
 
 +
<math>
 +
T(\theta) \propto \left(\frac{\mathrm d\theta}{\mathrm d\alpha}\right)^{-1} = \frac{\cos\theta}{\cos\theta_0\cos\beta\sin\alpha}
 +
</math>
 +
 
 +
If we consider the fact that the sky area at each latitude is proportional to <math>\cos\theta</math>, the integration time per unit sky area, <math>t</math>, is
 +
 
 +
<math>
 +
t(\theta_0, \beta, \alpha) = \frac{1}{\cos\theta_0\cos\beta\sin\alpha}
 +
</math>

Revision as of 08:22, 25 October 2019

Introduction

This post refines the work shown in High_cadence_LAT_from_Chile. We show how to modulate the scan rate to achieve maximally uniform integration depth across a maximal sky area.

Geometrical considerations

Let us assume that,

  • is the observatory latitude, measured in radians from the Equator
  • is the observing azimuth, measured in radians from North. East is at
  • is the observing elevation, measured in radians from the horizon.
  • is the declination on the celestial sphere.

Then we can write the declination as a function of the observatory latitude and observing azimuth and elevation.

Moreover, we can write for the rate at which declination changes whilst scanning at constant azimuthal rate

The time, , spent observing at a specific declination is inversely proportional to the rate at which the declination changes

If we consider the fact that the sky area at each latitude is proportional to , the integration time per unit sky area, , is