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

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<math> | <math> | ||

(11) \quad\quad\quad \alpha\prime\prime_\mathrm{max} = \omega_0^2\frac{\cos \alpha_0}{\sin^3 \alpha_0} | (11) \quad\quad\quad \alpha\prime\prime_\mathrm{max} = \omega_0^2\frac{\cos \alpha_0}{\sin^3 \alpha_0} | ||

+ | </math> | ||

+ | |||

+ | ==Sky fraction== | ||

+ | |||

+ | Assume that the boresight sweeps between <math>\alpha_0</math> and <math>\alpha_1</math>. The range of declinations covered can be inferred from Eq.(1). Fractional sky area covered by these declinations is | ||

+ | |||

+ | <math> | ||

+ | (11) \quad\quad\quad f_\mathrm{sky} = \frac{1}{2}\cos\theta_0\cos\beta\left( | ||

+ | \cos\alpha_0 - \cos\alpha_1 | ||

+ | \right), | ||

+ | </math> | ||

+ | |||

+ | where <math>\theta_0</math> is the observatory latitude and <math>\beta</math> is the observing elevation. If the sweep is symmetric, <math>\alpha_1 = \pi - \alpha_0</math>, then | ||

+ | |||

+ | <math> | ||

+ | (12) \quad\quad\quad f_\mathrm{sky} = \cos\theta_0\cos\beta\cos\alpha_0. | ||

</math> | </math> | ||

## Latest revision as of 12:19, 5 November 2020

*October 25, 2019 - Reijo Keskitalo*

## Contents

# 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 in terms of the azimuthal rate

where the final form follows from application of . 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

where we have dropped the constant factor

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

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

## Equation of motion

From Eq.(5), assuming a constant observing elevation, we have

Solving for :

where we have chosen and set this as the beginning of the sweep.

At the azimuth is and we can solve the time it takes to sweep between and :

If the sweep is symmetric, , then

Differentiating Eq.(7) at gives us the maximum scan rate and acceleration:

## Sky fraction

Assume that the boresight sweeps between and . The range of declinations covered can be inferred from Eq.(1). Fractional sky area covered by these declinations is

where is the observatory latitude and is the observing elevation. If the sweep is symmetric, , then

## 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:

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:

## 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:

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.

## Conclusions

We found a simple mathematical form ( ) 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.

# Update including elevation nods

*September 10, 2020 - Mike Niemack*

Sigurd Naess proposed adding elevation nods to improve cross-linking in the large area LAT surveys. This was studied by Haruki Ebina (Cornell) working with Mike Niemack, Jason Stevens, Thuong Hoang, and Steve Choi and is being considered for SO and CCAT-prime observations. Here is the latest report on this analysis: File:Update on Modulated High cadence LAT survey strategies from Chile 20200910.pdf

Example position versus time file associated with Fig. 16 in the *20200910.pdf file above File:Input sinc data 20200910.txt

(a previous version of the report is available here: File:Update on Modulated High cadence LAT survey strategies from Chile 20200831.pdf)

In brief, executing ~1 deg amplitude elevation nods combined with wide varying azimuthal velocity scans appears to be a promising approach to achieve a high-cadence survey strategy with substantially better cross-linking than we can achieve with the current ACT strategy.

Thinking about the LAT and LATR optical performance, based on previous discussions I suggest that we keep the LATR at a fixed angle while the telescope elevation nods. This will result in small pointing offsets on the order of 4’ for the outer optics tubes that will need to be taken into account in the pointing model. The changes in beam shape will be tiny though, since 4’ is a small fraction of the angular size of an optics tube.