The sensitivity calculator is a tool for estimating capabilities of Millimetron for doing astronomical observations. The sensitivity is calculated for different instruments in the Single Dish mode as well as in the Space VLBI mode. To use the calculator, one must select the instrument first, then adjust instrument parameters, such as the telescope main mirror temperature, sky background, instrument bandwidth, integration time, or leave them with default values, and press 'Calculate' button to obtain results in the format of a table.

In the single dish mode, Millimetron will operate with the following instruments:

- Shortwave Array Camera Spectrometer (SACS) camera, operating at 40(80?)–450 microns,
- Longwave Array Camera Spectrometer, or Fourier transfer spectrometer, 300–3000 microns,
- Grating spectrometer, which is a part of SACS,
- Heterodyne spectrometer.

The SACS camera is an array of detectors for wideband photometric observations in four bands: 43–80, 80–140, 140–230 and 230–450 microns. This division in four bands is very preliminary and can be changed in accordance with the request of astronomical community and technical limitations, so the user of the calculator can change the number of bands as well as their boundaries by entering a list of space-separated band boundaries in the corresponding field.

It is assumed that the receivers are high-sensitivity bolometers with NEP \(\sim10^{-19}\) W/Hz\(^{1/2}\), what makes the instrument sensitivity limited by the sky background. Due to this high sensitivity of bolometers, coolness of Millimetron mirrors, and large collecting area, the sensitivity of photometric observations is also very high, orders of magnitude better than it was for Herschel PACS camera. Thus, optimal integration times are of an order of 1 second. Even with such short integration, the RMS fluctuations are below the confusion limit created by distant galaxies. So, confusion must be taken into account for the most observations using SACS camera.

The output of the sensitivity calculator for this instrument includes the following columns for each band:

- short wave boundary of the band, microns,
- long wave boundary of the band, microns,
- sensitivity for a point source, in Jy, 1 RMS,
- sensitivity for an extended source, Jy/ster, 1 RMS,
- confusion limit, Jy,
- angular resolution, computed as diffraction limited above 100 microns, and 2 arcsec below.

The number of pixels in each band for this instrument is not yet defined. The field of view is limited by about 6 arcminutes.

This instrument is aimed at detecting weak spectral and spatial fluctuations of the intensity of the CMB, caused, e.g. by the Sunyaev-Zeldovich effect. The default values in the calculator are close to the instrument under development at Sapienza Univ. Roma.

The FTS has four bands: 100–200, 200–353, 353–667 and 667–1000 GHz. The number of bands and frequencies can be changed in the ’Bands boundaries’ field. Each band has its own set of detectors (pixels): for 100–200 GHz band there are 24 pixels covering 6 independent beams, for the second band 36 pixels, 9 beams, for the third one 48 pixels, 16 beams, and for the fourth band 100 pixels and 25 beams. The number of beams can be changed in the field ‘Beams per band’.

The spectral resolution of the FTS is defined by the field `Channel width', which is constant in the frequency domain. The instrument design allows to select channel width \(\gtrsim 1.5\) GHz.

Another important characteristic of a FTS is the throughput, i.e. the product of detector feedhorn area, \(A_\text{det}\), and the solid angle that the telescope’s entrance pupil subtends as seen from the detector, \(\Omega_\text{det}\). In the default configuration, the feedhorn diameter is chosen as \(1.22 \lambda_\text{max} / D\), where \(\lambda_\text{max}\) is the longest wavelength of each band. This is done to ensure the best angular resolution for observing compact astrophysical sources with FTS.

Note, that FTS can be optimized for measuring CMB spectral distortion by choosing appropriate bands and throughputs. E.g. one can try single band 100–2000 GHz with feedhorn of 7.5 arcmin diameter and throughput of 1200 \(\text{mm}^2\) sr.

The output of the sensitivity calculator for this instrument includes the following columns for frequency channel:

- low frequency channel boundary, GHz,
- sensitivity for an extended source, Jy/ster, 1 RMS,
- sensitivity, divided by the square root of number of independent beams in the current band,
- angular resolution, computed as diffraction limited above 100 microns, and 2 arcsec below.

The grating spectrometer is an instrument with moderate spectral resolution \(R \equiv \Delta \lambda / \lambda \sim 1000\) but with high sensitivity, instantaneous spectral coverage and productivity of observations. It is ideal for the broadband spectroscopy. One can consider BLISS (initially proposed for SPICA mission) as an example of design and capabilities of such an instruments.

Currently, there is little known on the possible design of the instrument for Millimetron. So the calculator user have to select him/herself the spectral resolution and observing wavelength. In reality spectral resolution will be a function of wavelength. The instrument may be optimized either for a single pixel observations, or for simultaneous observing a patch on the sky, like it was on Herschel PACS.

The sensitivity of the spectrometer also depends on the NEP of detectors used in the instrument. By default a value of \(5\times 10^{-20}\) \(\text{W/Hz}^{1/2}\) is used, which is current technology limit, but user can provide other values of NEP.

The results are represented in two blocks: first is the RMS sensitivity in \(\text{W/m}^2\) at the wavelength of interest. Second is a table, suitable for making a plot of sensitivity curve, on a predefined range of wavelengths, from 40 to 450 microns.

Heterodyne spectrometer can be used to study spectral lines in much more detail than a grating spectrometer or an FTS. It can have spectral resolution more than \(R>10^5\), which can be adjusted by software to meet the observer's needs. This comes at a price of lower sensitivity, limited by the quantum noise, and much lower instantaneous bandwidth of only few GHz.

The initial design for the ’Millimetron Heterodyne Instrument for the Far Infra red’, MHIFI, assumes that the instrument will not cover the whole FIR frequency range. It will have 3–7 bands centered on the most interesting spectral lines. Since the bands are not yet fixed, one can enter any wavelength in the current version of the Calculator. The other parameters are the spectral resolution and integration time. The MHIFI instrument also may have from 3 to 7 pixels on the sky in each band. The computation of sensitivity is yet very preliminary: it assumes the instrument sensitivity is at 10 quantum limits for all frequencies. In reality it will depend on band, since different bands will use different technologies.

The results include two blocks: first is the RMS sensitivity in \(\text{W/m}^2\) at the wavelength of interest. Second is a table, suitable for making a plot of sensitivity curve, on a predefined range of wavelengths, from 50 to 600 microns.

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 \(\text{W/Hz}^{1/2}\). The \(1\sigma\) point source flux
density for wideband cameras and for a grating spectrometer:
\[
S=\frac{\text{NEP}}{\Delta\nu \sqrt{\tau} A},
\]
\(\Delta\nu\) – width of frequency band, in Hz,

\(\tau\) – 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, \(\sigma_{s}=10\mu\text{m}\)),

\(c\) – speed of light.

For the Fourier transfer spectrometer (FTS) sensitivity for observing an extended source is computed differently (de Bernardis et al. 2012):

\[S_\text{FTS}=\frac{\text{NEP}}{\Delta\nu \sqrt{\tau}A_\text{det}\Omega_\text{det}},\]

where
\(S_\text{FTS}\) — flux density in Jy/ster,

\(A_\text{det}\Omega_\text{det}\) — throughput of the system,

\(A_\text{det}\) — detector area,

\(\Omega_\text{det}\) — solid angle that the telescope’s entrance pupil subtends as seen from the detector,

\(\delta \nu\) — width of a single spectral channel,

\(\Omega\) -- effective beam solid angle,

\[\Omega = 1.26 \left( \frac{\lambda}{D} \right)^2,\]

(this is computed for an ideal difraction-limited optics)

The NEP is computed as

\[\text{NEP}=\sqrt{\text{NEP}_{d}^{2}+\text{NEP}_{bg}^{2}},\]

\(\text{NEP}_{d}\) – intrinsic NEP of the detector,

\(\text{NEP}_{bg}\) – NEP created by the backgrounds (sky, mirror thermal emission). For each component it is (Lamarre 1986; Benford et al. 1998):

\[\text{NEP}_{bg}^{2}=4\int_{\nu-\Delta\nu/2}^{\nu+\Delta\nu/2} \frac{A_\text{det}\Omega_\text{det}}{\lambda^2} \epsilon \eta \alpha h^2 \nu^{2} n(\nu) \left[ 1+ \epsilon \eta \alpha n(\nu) \right]\text{d}\nu,\]

\[n(\nu) = \frac{1}{exp\left(\frac{h\nu}{kT}\right)-1},\]

\(n(\nu)\) — photon concentration in phase space (for a black body),

\(\epsilon\) — emissivity, \(\epsilon=0.001\) for the Millimetron primary mirror, \(\epsilon=1\) for CMB, \(\epsilon=3\cdot10^{-8}\) for Zodiacal light.

\(\Delta \nu\) — bandwidth of the FTS,

\(\alpha\) — optical efficiency,

\(\eta\) — main beam efficiency.

For a modified black body with slope \(\beta\):

\[n(\nu) = \frac{1}{exp\left(\frac{h\nu}{kT}\right)-1} \left(\frac{h\nu}{kT}\right)^\beta,\]

For the heterodyne receivers the \(\text{NEP}_{d}\) is computed as for 10 quantum limits:

\[\text{NEP}_\text{het}=20h\nu\sqrt{\Delta\nu}.\]

For the other detectors \(\text{NEP}_{d}\) is known.

For the computation of point source sensitivity we consider the case of a large detector, when its diameter is larger than the diffraction pattern size, \(1.22 \lambda / D\). In this regime

\[A_\text{det}\Omega_\text{det} = A\Omega,\] where \(\Omega\) — effective beam solid angle,

\[\Omega = 1.26 \left( \frac{\lambda}{D} \right)^2,\]

We use the following list of extended sources:

- CMB: \(T=2.73\) K, \(\epsilon=1.0\).
- Galactic thermal dust: \(T\), \(\epsilon\) and \(\beta\) depend on sky position.
- Zodiacal light: \(T=290\) K, \(\epsilon=3\times10^{-8}\).
- Cosmic Infrared Backgound created by distant galaxies: it varies on the sky, but we adopt \(T=18.8\) K, \(\epsilon = 4\times10^{-6}\), \(\beta=0.86\).

Besides this, we take into account mirror emission in the same manner, with \(T=4.5\) K, \(\epsilon=0.001\).

We use a Planck dust model map to extract dust parameters from a given location. The maps of \(T\), \(\epsilon\) and \(\beta\) included with the Calculator have resolution of 1 degree in Cartesian projection. The parameters are obtained by the Nearest grid point interpolation from the user input when the ‘Get background’ button is pressed. The input coordinates are given in degrees, longitude from 0 to 360 and latitude from -90 to 90.

The wideband photometric observations with a 10m aperture telescope will suffer from the confusion problem. The confusion is created by distant dusty galaxies. Due to a limited angular resolution in the FIR range, the images of these galaxies overlap, producing a background, which is strongly fluctuating from one place to another. Sources can be extracted from this background only if they are significantly brighter than the minimal value, called ‘confusion limit’, or if they have some well distinguishable peculiarities, e.g. different SED, variability, spectral fetures, polarization. For the wideband photometry (SACS camera instrument) the sensitivity calculator gives the values of confusion limit estimated in (Ermash et al. 2020, in prep.)

**The input parameters for each station**:

*Ant eff*– antenna usefull area ( 0.0 < Ant eff < 1.0 )*Ant area*– antenna geometrical area*T rcw*– reciever noise temperature*T amb*– antenna ambient temperature*T sky*– Sky temperature*n eff*– antenna scattering factor

**The common input parameters**:

*Frequency*– observation frequency*Bandwidth*– video bandwidth at correlator input (\(\Delta\nu\))*Coherence time*– coherent integration time (\(\tau_{c}\))*Solutoin interval*– correlator partial integration time*N channel*– number of channels in the correlator FFT*Bits per sample*– number of data quantization bits

**Output parameters**:

\(T_{sys}=(1+g)(T_{rcw}+n_{eff}T_{Sky}+(1-n_{eff})T_{amb}+T_{planck})\)

- \(T_{planck}\approx5\) K

\(SEFD=2761.32\,T_{sys}/A_{eff}\) Jy

\(A_{eff}=\)

*Ant area*\(\times\)*Ant eff*

**Interferometer sensitivity**:

\[S=\frac{1}{\eta\sqrt{2\Delta\nu\tau_{c}}}\sqrt{SEFD_{1}\,SEFD_{2}}\]

**Correlator search window**:

*Delay*– delay window size for the correlator fringe search*Fringe rate*– fringe rate window size for the fringe search*Range*– permitted error in the distance measurement towards the satellite*Velocity*– permitted error in the satellite velocity*Velocity range*– the channel width expressed in km/s.

- Benford D.J., Hunter T.R., Phillips T.G. 1998 IJIMW 19, 931
- de Bernardis P., Colafrancesco S., D'Alessandro G., et al. 2012 A&A 538, A86
- Lamarre J.M. 1986 Applied Optics 25, 870