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
(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 groundbased 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 (
), which characterizes the noise created by the detectors and the background radiation.
is measured in W/Hz
. The 1
flux density is, by definition:
— frequency band,
— spectral resolution,
— integration time,
— effective area of the telescope, which is given by Ruze equation:
— telescope primary mirror diameter (
m),
— RMS accuracy of the mirror (currently,
m),
— speed of light.
The
is computed as
— intrinsic NEP of the detector,
— NEP created by the backgrounds (sky, mirror thermal emission). For each component it is:
— emissivity,
for the primary mirror,
for CMB,
for the Galactic Cirrus,
for Zodiacal light.
for the mirror,
for CMB,
for Galaxy,
for Zodiakal light.
For the heterodyne receivers the
is computed as for 10 quantum limits:
For the other detectors
is known.
2.3 Saturation
The maximal flux density is limited by the saturation of the detectors. It is computed as:
— response time of the detectors or of the system. It is assumed to be 0.01 s.
— 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 difractionlimited at
mm and 6 arcseconds otherwise:
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
GHz, 25 for
GHz, 9 for
GHz, 6 for
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
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
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:
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:

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 ()

Coherence time — coherent integration time ()

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:
Interferometer sensitivity:
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.