Modulated scan high cadence LAT

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October 25, 2019 - Reijo Keskitalo


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,

  • \theta_0 is the observatory latitude, measured in radians from the Equator
  • \alpha is the observing azimuth, measured in radians from North. East is at \pi/2
  • \beta is the observing elevation, measured in radians from the horizon.
  • \theta 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.


(1) \quad\quad\quad \theta = \sin^{-1}\left(\cos \theta_0\cos\beta\cos\alpha + \sin\theta_0\sin\beta\right)

Moreover, we can write for the rate at which declination changes in terms of the azimuthal rate


(2) \quad\quad\quad \frac{\mathrm d\theta}{\mathrm d t} = \frac{\mathrm d\theta}{\mathrm d\alpha}\frac{\mathrm d\alpha}{\mathrm d t}
 = \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{\mathrm d\alpha}{\mathrm d t}
 =  \frac{\cos\theta_0\cos\beta\sin\alpha}{\cos\theta}\frac{\mathrm d\alpha}{\mathrm d t}

where the final form follows from application of (1). The time, T, spent observing at a specific declination is inversely proportional to the rate at which the declination changes


(3) \quad\quad\quad T(\theta) \propto \left(\frac{\mathrm d\theta}{\mathrm d t}\right)^{-1} = \frac{\cos\theta}{\cos\theta_0\cos\beta\sin\alpha} \left(\frac{\mathrm d\alpha}{\mathrm d t}\right)^{-1}

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


(4) \quad\quad\quad T_a(\theta_0, \beta, \alpha) \propto \frac{1}{\cos\beta\sin\alpha} \left(\frac{\mathrm d\alpha}{\mathrm d t}\right)^{-1}

where we have dropped the constant factor \cos\theta_0

From (4) it is evident that for the observing depth to be uniform ( T_a to be constant), the azimuthal scanning rate must satisfy


(5) \quad\quad\quad \frac{\mathrm d\alpha}{\mathrm d t} \propto \frac{1}{\cos\beta\sin\alpha}

and explicitly, the azimuthal rate on the sky, \omega, will depend on some base azimuthal rate on the telescope, \omega_0, as


(6) \quad\quad\quad \omega = \frac{\omega_0}{\cos \beta\sin\alpha}


Hardware limits

Modulating the azimuthal scanning rate is limited by maximum azimuthal rate of the telescope and, to a lesser degree, the azimuthal acceleration of the telescope.

Here is a plot of the modulation factor as a function of the telescope azimuth:

Az vs modulation.png

It is obvious that sweeping close to 180 degrees is unfeasible as it would require the telescope to move very fast as it approaches the turnaround. A factor of 2 or 3 (throw of 160 or 170 degrees) is achievable, especially if the base scanning rate is low enough.

Pushing for extreme azimuth ranges is not even required for the sky area, as we quickly enter the domain of diminishing returns:

Fsky vs az.png


Example

Assuming a base scanning rate of 0.75 deg/s, low acceleration of 1 deg/s^2, throw between 20 and 160 degrees and observing elevation of 30 degrees:


Modulated scan.png

The top row shows one simulated back-and-forth scan, the bottom row shows 10 consecutive sweeps. The telescope never moves faster than 2.75 deg/s in azimuth.


Hit maps

We simulated full year hit maps with a reduced focal plane and sampling rate using constant and modulated scan rates. We considered three observing elevations: 30, 40 and 50 degrees and chose the azimuth ranges to keep the maximum modulation factor at 2.75:

  • 30 deg elevation : Az = [19, 161] deg
  • 40 deg elevation : Az = [21, 159] deg
  • 50 deg elevation : Az = [26, 154] deg

In the following plot, left column hit maps use constant scan rate, right column is from the modulated scan rate. The titles of each panel show the raw, signal and noise dominated fskies.

All high cadence hit maps.png


Conclusions

We found a simple mathematical form ( 1/\sin\mathrm{Az} ) of modulated scan rate that achieves uniform integration depth across all declinations. Allowing for a non-constant azimuthal scanning rate seems essential if we are to achieve high cadence (for transient science) and effective sky area (critical for Neff and other science targets). Once the hardware limits are known, it is straightforward to define the scan strategy that fits within those limits.