Calculating the GBT time required to detect a galaxy cluster with MUSTANG-2

Authors: Charles Romero and Emily Moravec

Last updated: 2024-12-30.

Usage

You can use this notebook as a template to calculate the required GBT time to detect a cluster of a given mass and at a given redshift with MUSTANG-2. You will need to install M2_ProposalTools (see https://m2-tj.readthedocs.io/en/latest/index.html for details). This notebook is based on the “Example case of estimating observing time for assumed A10 clusters” example on M2_ProposalTools (https://m2-tj.readthedocs.io/en/latest/SinglePointing_M2.html which can be downloaded at https://github.com/CharlesERomero/M2_TJ/blob/master/docs/source/SinglePointing_M2.ipynb).

About

Hello potential MUSTANG-2 observer that wants to know how much telescope time is required to detect a given a cluster of a specific mass at a particular redshift at a particular signal-to-noise ratio. This notebook walks you through creating an expected compton-y signal map for your given parameters (\(z\) and \(M_{500}\)), running your map through the MUSTANG-2 transfer functions to take into account the filtering of MUSTANG-2, and convolving this model map with the MUSTANG-2 beam. It then calculates the required observing time for your cluster based on your SNR requirement and the peak of the simulated SZ signal from the physical parameters you give it (\(z\) and \(M_{500}\)).

Note

The time estimates calculated in this notebook are made assuming using the MIDAS pipeline for data processing.

Imports and constants

[1]:

import M2_ProposalTools.WorkHorse as WH
import numpy as np
import astropy.units as u
import M2_ProposalTools.FilterImages as FI
import scipy

[2]:

# Constants needed for calculations - it is recommended that you keep these default parameters.
h70            = 1                          # Normalization of Hubble parameter
fwhm           = 10.0                       # Roughly the resolution of MUSTANG-2
pixsize        = 2.0                        # Map pixel size (arcseconds) - making a map - package does beam convolution and filtering
s2f            = np.sqrt(8.0*np.log(2.0))   # Conversion between FWHM and sigma
pix_sigma      = fwhm/(pixsize*s2f)         # Gaussian sigma, in pixel size
y2k            = -3.3                       # Approximate conversion of compton-y to Kelvin. -3.3 is conservative whereas -3.4 is perhaps more accurate.

MUSTANG-2 Parameters

Change these parameters based on your science. Consult https://gbtdocs.readthedocs.io/en/latest/references/receivers/mustang2/mustang2_mapping.html for the corresponding row.

[3]:

scansize  = 3.5 # Scan size: 3' or 3.5' is recommended
mapping_speed_uJy = 57 # mapping speed in uJy/beam root hour
mapping_speed_uK = 74 # mapping speed in uK root hour
K_to_J = (mapping_speed_uJy/mapping_speed_uK) # conversion of Kelvin to Janskys

Function for calculating the observing time for a single cluster

We calculate the required observing time using the radiometer equation:

\(t \propto 1/\sigma^2\)

where \(t\) is observing time in hours and \(\sigma\) is the sensitivity/RMS/noise of the observation.

We can set up a proportional relationship with the radiometer equation such that

\(t_2/t_1 \propto (\sigma_1/\sigma_2)^2\)

where \(t_2\) is the required integration time that you are solving for, \(\sigma_1\) is the RMS corresponding to your desired map size (see https://gbtdocs.readthedocs.io/en/latest/references/receivers/mustang2/mustang2_mapping.html#mustang-2-mapping-information), \(t_1=1\) as the mapping speed are in within 1 hour, and \(\sigma_2\) is your desired sensitivity.

Note that using the fact that mapping speed = \(ms = \sigma_1 * (t_1)^{1/2}\) and the radiometer equation in a proportion like the above you can derive that \(t_2 = (ms / \sigma_2)^2\).

[4]:

def calculate_observing_time(M500,z,SNR_required):
    print("-"*5,"Cluster Properties","-"*5)
    print("M500 (1e14):",(M500/1e14).value)
    print("Redshift:",z)

    # Create expected signal map
    ymap = WH.make_A10Map(M500,z,pixsize=pixsize,Dist=True) # create compton-y map
    mymap = WH.smooth_by_M2_beam(ymap,pixsize=pixsize) # convolve signal map (y) with MUSTANG-2 beam

    # Filter signal map
    tab = WH.get_xfertab(scansize) # 2D array of values used to create filtered map
    filtered_model_ymap = FI.apply_xfer(mymap,tab,pixsize) # apply transfer function to compton-y map and create a filtered map
    snr_smoothing_kernel = pix_sigma*0.9 # when an SNR map is created there is smoothing applied - it is recommended to keep this smoothing at 90% of M2 beam (so keep 0.9)
    yxfer = scipy.ndimage.gaussian_filter(filtered_model_ymap, snr_smoothing_kernel)

    # Convert model map to observable values (uJy/beam or uK)
    uKmap      = yxfer*y2k*1e6 # convert compton-y map to uK (1e6 is to convert from K to uK)
    uJymap     = uKmap * K_to_J

    # Calculate peak value wrt the scale of interest (some values are negative)
    print("-"*5,"Signal Peak Stats","-"*5)
    SZpeak_yraw  = np.max(ymap) # peak of y-map
    print("Unfiltered y peak:", SZpeak_yraw)
    SZpeak_yconvolved  = np.max(mymap)
    print("Beam-convolved y peak:", SZpeak_yconvolved)
    SZpeak_yxfer  = np.max(yxfer)
    print("Filtered y peak:", SZpeak_yxfer)
    SZpeak_uK  = np.min(uKmap) # this is negative so calculate minimum
    print("Filtered peak, uK:", SZpeak_uK)
    SZpeak_uJy = np.min(uJymap) # this is negative so calculate minimum
    print("Filtered peak, uJy/beam:", SZpeak_uJy)

    # Calculate the observing time based on the Compton-y peak in uJy/beam
    print("-"*5,"Telescope Time Stats","-"*5)
    target_sensitivity = SZpeak_uJy/SNR_required # calculate the detection threshold based on the required SNR
    print("Your target sensitivity or 1-sigma level in uJy/beam is:",np.abs(target_sensitivity))
    # from the radiometer and a proportional relationship (see math in comments above) we can calculate the observing time required
    ObsTime = np.round((mapping_speed_uJy/target_sensitivity)**2,5) # from unit analysis this variable is in units of hours
    print("On-source time to",str(SNR_required),"sigma peak detection (hrs):",ObsTime)
    TotalTimeRequest = ObsTime * 2 # include the required GBT high frequency factor of 2 to account for overheads
    # GBT time gets scheduled in units of 0.25 hr so convert and
    TelTime = np.ceil(TotalTimeRequest*4)/4.0

    print("Total telescope time for this cluster (hrs):",TelTime)

    return TelTime

Calculate the telescope time for one cluster

[5]:

# Physical parameters of your cluster
M500 = 3*1e14*u.M_sun # Mass of cluster
z = 1.0 # The redshift of the cluster
obstime_single_cluster = calculate_observing_time(M500=M500,z=z,SNR_required=7)

----- Cluster Properties -----
M500 (1e14): 3.0
Redshift: 1.0
----- Signal Peak Stats -----
Unfiltered y peak: 9.619897496889924e-05
Beam-convolved y peak: 8.860120011199452e-05
Filtered y peak: 4.829389221185476e-05
Filtered peak, uK: -159.3698442991207
Filtered peak, uJy/beam: -122.7578530412146
----- Telescope Time Stats -----
Your target sensitivity or 1-sigma level in uJy/beam is: 17.536836148744943
On-source time to 7 sigma peak detection (hrs): 10.56446
Total telescope time for this cluster (hrs): 21.25

Why do we require the SNR=7? Based on experience the MUSTANG-2 instrument team highly recommends that you adopt a \(7\sigma\) threshold, but depending on your exact science objective, a different threshold may be adopted. Feel free to discuss this with the instrument team.

Calculate the telescope time for multiple clusters

[6]:

# Physical parameters of your cluster
M500s = np.array([1,2,3,4,5,6,7,8])*1e14*u.M_sun # Mass of clusters
zs = np.array([1.0,1.0,1.0,1.0,1.0,1.0,1.0,1]) # Matching redshifts for the M500s list

[7]:

TotalTime = 0 # Initiate total time counter
for M500,z in zip(M500s,zs):
    print("="*50)
    obstime_cluster = calculate_observing_time(M500=M500,z=z,SNR_required=7)
    TotalTime += obstime_cluster
print("="*50)
print("Total Telescope Time (hrs): ", TotalTime)

==================================================
----- Cluster Properties -----
M500 (1e14): 1.0
Redshift: 1.0
----- Signal Peak Stats -----
Unfiltered y peak: 2.7103546504401935e-05
Beam-convolved y peak: 2.4135944165655142e-05
Filtered y peak: 1.4965757817167562e-05
Filtered peak, uK: -49.387000796652956
Filtered peak, uJy/beam: -38.04133845147592
----- Telescope Time Stats -----
Your target sensitivity or 1-sigma level in uJy/beam is: 5.434476921639417
On-source time to 7 sigma peak detection (hrs): 110.01052
Total telescope time for this cluster (hrs): 220.25
==================================================
----- Cluster Properties -----
M500 (1e14): 2.0
Redshift: 1.0
----- Signal Peak Stats -----
Unfiltered y peak: 6.0365334122178264e-05
Beam-convolved y peak: 5.530305199658348e-05
Filtered y peak: 3.1817993020714536e-05
Filtered peak, uK: -104.99937696835796
Filtered peak, uJy/beam: -80.87789847562708
----- Telescope Time Stats -----
Your target sensitivity or 1-sigma level in uJy/beam is: 11.553985496518154
On-source time to 7 sigma peak detection (hrs): 24.33807
Total telescope time for this cluster (hrs): 48.75
==================================================
----- Cluster Properties -----
M500 (1e14): 3.0
Redshift: 1.0
----- Signal Peak Stats -----
Unfiltered y peak: 9.619897496889924e-05
Beam-convolved y peak: 8.860120011199452e-05
Filtered y peak: 4.829389221185476e-05
Filtered peak, uK: -159.3698442991207
Filtered peak, uJy/beam: -122.7578530412146
----- Telescope Time Stats -----
Your target sensitivity or 1-sigma level in uJy/beam is: 17.536836148744943
On-source time to 7 sigma peak detection (hrs): 10.56446
Total telescope time for this cluster (hrs): 21.25
==================================================
----- Cluster Properties -----
M500 (1e14): 4.0
Redshift: 1.0
----- Signal Peak Stats -----
Unfiltered y peak: 0.00013376973214360876
Beam-convolved y peak: 0.00012461706064739533
Filtered y peak: 6.52603290828719e-05
Filtered peak, uK: -215.35908597347722
Filtered peak, uJy/beam: -165.88470135794867
----- Telescope Time Stats -----
Your target sensitivity or 1-sigma level in uJy/beam is: 23.697814479706953
On-source time to 7 sigma peak detection (hrs): 5.7854
Total telescope time for this cluster (hrs): 11.75
==================================================
----- Cluster Properties -----
M500 (1e14): 5.0
Redshift: 1.0
----- Signal Peak Stats -----
Unfiltered y peak: 0.00017266723341949052
Beam-convolved y peak: 0.00016159098229221895
Filtered y peak: 8.1695551395762e-05
Filtered peak, uK: -269.5953196060146
Filtered peak, uJy/beam: -207.66125969652475
----- Telescope Time Stats -----
Your target sensitivity or 1-sigma level in uJy/beam is: 29.66589424236068
On-source time to 7 sigma peak detection (hrs): 3.69177
Total telescope time for this cluster (hrs): 7.5
==================================================
----- Cluster Properties -----
M500 (1e14): 6.0
Redshift: 1.0
----- Signal Peak Stats -----
Unfiltered y peak: 0.00021264462426804625
Beam-convolved y peak: 0.00019890428730061485
Filtered y peak: 9.761530156346558e-05
Filtered peak, uK: -322.1304951594364
Filtered peak, uJy/beam: -248.12754356875507
----- Telescope Time Stats -----
Your target sensitivity or 1-sigma level in uJy/beam is: 35.44679193839358
On-source time to 7 sigma peak detection (hrs): 2.58581
Total telescope time for this cluster (hrs): 5.25
==================================================
----- Cluster Properties -----
M500 (1e14): 7.0
Redshift: 1.0
----- Signal Peak Stats -----
Unfiltered y peak: 0.0002535355941774716
Beam-convolved y peak: 0.00023785709856947226
Filtered y peak: 0.0001135652820943107
Filtered peak, uK: -374.7654309112253
Filtered peak, uJy/beam: -288.6706697559438
----- Telescope Time Stats -----
Your target sensitivity or 1-sigma level in uJy/beam is: 41.238667107991965
On-source time to 7 sigma peak detection (hrs): 1.91047
Total telescope time for this cluster (hrs): 4.0
==================================================
----- Cluster Properties -----
M500 (1e14): 8.0
Redshift: 1.0
----- Signal Peak Stats -----
Unfiltered y peak: 0.00029521992026628467
Beam-convolved y peak: 0.0002786569572687501
Filtered y peak: 0.0001299492606572224
Filtered peak, uK: -428.8325601688339
Filtered peak, uJy/beam: -330.3169720219396
----- Telescope Time Stats -----
Your target sensitivity or 1-sigma level in uJy/beam is: 47.188138860277085
On-source time to 7 sigma peak detection (hrs): 1.4591
Total telescope time for this cluster (hrs): 3.0
==================================================
Total Telescope Time (hrs):  321.75