Equations for the Millimetron sensitivity calculator

1 Introduction

The sensitivity calculator is a tool for making estimates of capabilities of Millimetron for doing observations. Currently it includes the calculation of the sensitivity for different instruments in the Single Dish mode as well as in the Space VLBI mode. It also calculates the confusion noise created by the distant submillimeter galaxies.

1.1 Single dish mode

The main input parameters which user can change are the central wavelength or frequency of the observation, the spectral resolution R=\lambda/\Delta\lambda (it can be expressed also in km/s) and the duration of the observation. User also can change the intrinsic sensitivity of the detectors (the NEP, see below), but normally it is selected automatically after the choice of the instrument.
Real instruments will have fixed values of some parameters, e.g. cameras will have a set of fixed wavelengths and bandwidths. Long slit and Fourier spectrometers will have choice of several possible values of spectral resolution and it also will be a function of the wavelength. This will be taken into account in future versions of the calculator.
The list of instruments also does not reflect their real assembly, e.g. the long wavength camera and the fourier spectrometer are actually the same instrument operating in different modes.

1.2 Space VLBI mode

In the interferometer mode the calculation of sensitivity requires the knowledge of the parameters of the ground-based telescopes. Several examples are available in the calculator, but the parameters can be also input by hand. The description of the parameters is given in the section Interferometry↓.

2 Single dish mode

2.1 Sensitivity

The sensitivity is computed from the Noise Equivalent Power (NEP), which characterizes the noise created by the detectors and the background radiation. NEP is measured in W/Hz^{1/2}. The 1\sigma flux density is, by definition:

S=\frac{\sqrt{2}\cdot NEP\cdot R}{\nu\cdot A\cdot\sqrt{\tau}},
\nu — frequency band,
R — spectral resolution,
\tau — integration time,
A — effective area of the telescope, which is given by Ruze equation:

A=0.8\frac{\pi D^{2}}{4}\exp\left(-\left(\frac{4\pi\sigma_{s}\nu}{c}\right)^{2}\right),
D — telescope primary mirror diameter (D=10 m),
\sigma_{s} — RMS accuracy of the mirror (currently, \sigma_{s}=5\mum),
c — speed of light.

2.2 NEP

The NEP is computed as

NEP=\sqrt{NEP_{d}^{2}+NEP_{bg}^{2}},
NEP_{d} — intrinsic NEP of the detector,
NEP_{bg} — NEP created by the backgrounds (sky, mirror thermal emission). For each component it is:

NEP_{bg}^{2}=2\epsilon\frac{h^{2}}{\nu^{2}}\int_{\nu-\Delta\nu/2}^{\nu+\Delta\nu/2}\frac{\nu^{4}}{\left(exp\left(\frac{h\nu}{kT}\right)-1\right)^{2}}\mathrm{d}\nu,
\epsilon — emissivity, \epsilon=0.05 for the primary mirror, \epsilon=1 for CMB, \epsilon=2.4\cdot10^{-6} for the Galactic Cirrus, \epsilon=3\cdot10^{-8} for Zodiacal light.
T=4.5K for the mirror, T=2.73K for CMB, T=30K for Galaxy, T=290 for Zodiakal light.
For the heterodyne receivers the NEP_{d} is computed as for 10 quantum limits:

NEP_{\mathrm{het}}=20h\nu\sqrt{\Delta\nu}.
For the other detectors NEP_{d} is known.

2.3 Saturation

The maximal flux density is limited by the saturation of the detectors. It is computed as:

S_{max}=\frac{\sqrt{2}\cdot NEP_{d}\cdot R}{\nu\cdot A\cdot\sqrt{\tau_{r}}}\cdot10^{\frac{dRange}{10}},
\tau_{r} — response time of the detectors or of the system. It is assumed to be 0.01 s.
dRange — dynamical range of detectors in dB.
One can find that at the wavelengths about 1 mm for the cameras the saturation by the CMB is reached. This means that only the differential measurements can be used at the wavelengths near the maximal flux density of the CMB. This justifies the choice of the differential Fourier spectrometer as the instrument for this band.

2.4 Angular resolution, field of view

The angular resolution is determined by the surface accuracy and assumed to be difraction-limited at \lambda>0.08 mm and 6 arcseconds otherwise:

\theta(\lambda)=\left\{ \begin{array}{lr}
2'' & :\lambda<0.08mm\\
1.22\frac{\lambda}{D} & :\lambda>0.08mm
\end{array}\right.
The minimal field of view (FoV) is 6 arcminutes at wavelength of 0.3 mm. It depends on the wavelength and soon we will provide a calculation for this.

2.5 Pixels and spectroscopy

For the cameras this field of view should be filled with pixels. If the pixel size corresponds to the angular resolution, there should be up to 100x100=10000 pixels which is not a problem for the current technology.
The number of pixels for the Fourier spectrometer also depends on the wavelength. It is now assumed to be 36 for \nu>700 GHz, 25 for 700>\nu>350 GHz, 9 for 350>\nu>130 GHz, 6 for \nu<130 GHz. This spectrometer will be a differential one, i.e. it will measure the difference of intensity in two fields. This allows to avoid saturation caused by CMB. The number of pixels above is for one field, and the total number of pixels is twice the numbers shown above.
The Long slit grating spectrometer will probably have two pixels on the sky, each observing a certain part of the spectrum.
The difference between grating and Fourier spectrometers is that the former will take the spectrum in the whole range 0.05 — 0.5 mm at once, while the Fourier spectrometer is a scanning one, i.e. it is measuring only a part of the spectrum with \Delta\nu=\nu/R simultaneously.
The number of pixels of the Heterodyne spectrometer is not defined yet. In principle, it can have several pixels. Its spectral resolution is adjustable and is determined by the onboard digital processing of the signal.

2.6 Confusion limit

The confusion created by distant galaxies is computed using Bethermin & Lagache 2011 model of galaxy counts. For the definition of the confusion limit we adopt the flux density limit at which the probability of not being able to distinguish two sources from one is P=0.1 (see also Dole et al. 2003 for the full description). This corresponds to the density of approximately 1/17 sources per beam. (For P=0.03 it is 1/58, for P=0.2 it is 1/8 and for P=0.5 it is 1/2.5.)
The calculation of the limit takes into account the angular resolution formula \theta(\lambda) shown above, i.e. the mirror is not assumed to be difraction quality at short wavelengths. The actual calculation involves numerical integration of the gridded data of Bethermin & Lagache, and in the calculator instead a set of numerical fits for different values of P is used. The fit quality is within 30%.
The model results for P=0.1 together with the fit are shown in the Figure:
figure calc_fig1.png
For the comparison on this Figure is shown the confusion limit for a diffraction quality 3.5m mirror.

3 Interferometry

The input parameters for each station:
The common input parameters:
Output parameters:
Interferometer sensitivity:
S=\frac{1}{\eta\sqrt{2\Delta\nu\tau_{c}}}\sqrt{SEFD_{1}\, SEFD_{2}}
Correlator search window: