# Difference between revisions of "Modulated scan high cadence LAT"

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=Geometrical considerations= | =Geometrical considerations= | ||

− | Let us assume that the observatory is | + | 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) | ||

+ | </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} | ||

+ | = \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