# Flux and bandpass calibration

MeerKAT makes use of J1939-6342 and J0408-6545 as the flux and bandpass calibrators. Due to MeerKAT’s wide field of view and sensitivity, there is structure introduced into the bandpass response from secondary sources in the field. It is necessary to use a multi-component model for the primary calibrators in order to derive a stable bandpass calibration and accurate flux scale. This is vital at the UHF band, as shown below, but it is also advisable to do so at L-band for accurate bandpass calibration.

The L-and UHF-band calibrator models used by the SDP pipeline, in wsclean format, can be found here.

# J1939-6342

J1939-6342 (Figure 1) has an additional ~20% flux contribution from other sources in the field at 900 MHz, and up to 40% at 600 MHz with its spectrum turning over at ~1 GHz. These sources cause time-variable ripples across the bandpass in both phase and amplitude (examples shown in Figure 2). While the standard CASA flux density models can be used for L-band data reduction, it is essential to use multi-component models for UHF observations.

**Figure 1:**The spectrum of J1939-6342 (Reynolds 1994).

**Figure 2:**Effects of off-axis sources surrounding J1939-6342, in the UHF band, on visibility amplitudes on a selection of baselines. Multiple scans are overplotted to illustrate the time-variability.

A full discussion of the observations and models are included in this commissioning report. The SDP calibrator pipeline is currently using 90 components. This is sufficient for most cases.

# J0408-6545

This source has somewhat lower interference from other sources in the field (1% contribution at L-band); there is a bright double-lobed source at the 1% (integrated flux) level 5.5 arcmin away from the central source.

## Using CASA setjy for non-standard flux models

There isn’t a standard flux model available for J0404-6545 in CASA. The code below shows how to set the model (using only the primary source in the field).

```
def casa_flux_model(lnunu0, iref, *args):
"""
Compute model:
iref * 10**lnunu0 ** (args[0] + args[1] * lnunu0 + args[1] * lnunu0 ** 2 + args[0] * lnunu0 ** 3)
"""
exponent = np.sum([arg * (lnunu0 ** (power ))
for power, arg in enumerate(args)], axis=0)
return iref * (10**lnunu0) **(exponent)
def fit_flux_model(nu, s, nu0, sigma, sref, order=5):
from scipy.optimize import curve_fit
from scipy.special import binom
"""
Fit a flux model of given order from :
S = fluxdensity *(freq/reffreq)**(spix[0]+spix[1]*log(freq/reffreq)+..)
Very rarely, the requested fit fails, in which case fall
back to a lower order, iterating until zeroth order. If all
else fails return the weighted mean of the components.
Finally convert the fitted parameters to a
katpoint FluxDensityModel:
log10(S) = a + b*log10(nu) + c*log10(nu)**2 + ...
Parameters
----------
nu : np.ndarray
Frequencies to fit in Hz
s : np.ndarray
Flux densities to fit in Jy
nu0 : float
Reference frequency in Hz
sigma : np.ndarray
Errors of s
sref : float
Initial guess for the value of s at nu0
order : int (optional)
The desired order of the fitted flux model (1: SI, 2: SI + Curvature ...)
"""
init = [sref, -0.7] + [0] * (order - 1)
lnunu0 = np.log10(nu/nu0)
for fitorder in range(order, -1, -1):
try:
popt, _ = curve_fit(casa_flux_model, lnunu0, s, p0=init[:fitorder + 1], sigma=sigma)
except RuntimeError:
log.warn("Fitting flux model of order %d to CC failed. Trying lower order fit." %
(fitorder,))
else:
coeffs = np.pad(popt, ((0, order - fitorder),), "constant")
return [nu0] + coeffs.tolist()
# Give up and return the weighted mean
coeffs = [np.average(s, weights=1./(sigma**2))] + [0] * order
return [nu0]+ coeffs.tolist()
def convert_flux_model(nu=np.linspace(0.9,2,200)*1e9 , a=1,b=0,c=0,d=0,Reffreq= 1.0e9) :
"""
Convert a flux model from the form:
log10(S) = a + b*log10(nu) + c*log10(nu)**2 + ...
to an ASA style flux model in the form:
S = fluxdensity *(freq/reffreq)**(spix[0]+spix[1]*log(freq/reffreq)+..)
Parameters
----------
nu : np.ndarray
Frequencies to fit in Hz
a,b,c,d : float
parameters of a log flux model.
Reffreq : float
Reference frequency in Hz
returns :
reffreq,fluxdensity,spix[0],spix[1],spix[2]
"""
MHz = 1e6
S = 10**(a + b*np.log10(nu/MHz) +c*np.log10(nu/MHz)**2 + d*np.log10(nu/MHz)**3)
return fit_flux_model(nu, S , Reffreq,np.ones_like(nu),sref=1 ,order=3)
#name=0408-65 epoch=2016 ra=04h08m20.4s dec=-65d45m09s a=-0.9790 b=3.3662 c=-1.1216
a=-0.9790
b=3.3662
c=-1.1216
d=0.0861
reffreq,fluxdensity,spix0,spix1,spix2 = convert_flux_model(np.linspace(0.9,2,200)*1e9,a,b,c,d)
f_cal_alt = 'J0408-6545'
setjy(vis=msfile,
field=f_cal_alt,
spix=[spix0, spix1, spix2, 0],
fluxdensity = fluxdensity,
reffreq='%f Hz'%(reffreq),
standard='manual')
```

# Applying a full sky model to a CASA measurement set

The calibrator models are in `wsclean`

format.

The `crystalball`

package can be used to populate the ‘`MODEL_DATA`

' column of a measurement set. This replaces CASA’s own `setjy`

task as `crystalball`

utilises all the components in the sky model. `setjy`

by default assumes a single point source model at the phase center of the flux calibrator field. A model image may also be provided for use with `setjy`

.

`crystalball`

can be used as follows:

`$ crystalball observation.ms -sm skymodel.txt -f 0`

for the measurement set “observation.ms”, sky model “skymodel.txt” and flux calibrator field ID “0”.

Calibration can proceed as normal after this step.