Caution: The numbers quoted here are projected for the full array and assume ideal perfomance of the instrument

For more practical information on the actual sensitivity of the current configuration readers should consult the corresponding "Interferometric Modes", "Beam Formed Modes" and "Direct Storage Mode"  pages.

1 System Equivalent Flux Density

 

The System Equivalent Flux Density (or system sensitivity) is defined as

where k denotes Boltzmann's constant, η denotes the system efficiency factor (~ 1.0) , T_sys denotes the system noise temperature, A_eff denotes the total collecting area.

The system noise temperature consists of a sky brightness component and an instrumental component

For all LOFAR frequencies the sky brightness temperature is dominated by the Galactic radiation which depends strongly on the wavelength 

  and T_s0= 60 ± 20 K, for Galactic latitudes between 10 and 90 degrees. The instrumental noise temperature follows from measurements or simulations.

 

In Table 1 a lower limit for the effective area of the different LBA array selections is given. The upper limit is given by the number of dipoles times λ²/ 3 and is given in the last column (for 48 dipoles). Note that the inner array consists of only 46 dipoles. The effective area of each dipole in the array is determined by its distance to the nearest dipole ( d) within the full array:

This value is a lower limit of the actual effective area. 

Freq (MHz)	λ (m)	Aeff inner (46)	Aeff outer (48)	Aeff sparse (48)	48*Aeff,dipole
15	20.0	419.77	1973.4	1148.6	6400.0 *
30	10.0	419.77	1343.5	869.14	1600.0
45	6.67	415.37	693.61	558.64	711.11
60	5.00	347.37	398.18	371.19	400.00
75	4.00	239.67	256.00	247.43	256.00

Table 1 Effective area (m²) for the different LBA array selections as function of frequency (MHz) or wavelength (m). (*) an 81.34 m station has an area of 5196.3 m².

For an HBA dipole the effective area is limited by the available area in a tile. There are 16 dipole antennas within one 5 m by 5 m tile, hence, the dipole effective area is given by

The resulting HBA station effective area is given in Table 2. 

Freq (MHz)	λ (m)	Core(*)	Remote	European
120	2.50	600	1200	2400
150	2.00	512	1024	2048
180	1.67	356	711	1422
200	1.50	288	576	1152
210	1.43	261	522	1045
240	1.25	200	400	800

Table 2 Effective area (m²) for the HBA array as function of frequency (MHz) or wavelength (m). At 120 MHz the effective area is limited by the tile size. (*) There is two HBA fields per core station.

 

In Table 3 the resulting SEFD's (in single polarization) for the different LOFAR stations are given, where the above results for the sky temperature, the system temperature, and the effective area have been used. At 15 MHz and 30 MHz the outer array is considered, whereas at 45 MHz, 60 MHz, and 75 MHz the inner array is used. For the LBA the system sensitivities are based on an average between the minimum and maximum effective area as given in Table 1. The values for the European LBA station are based on a 65 m station. From the Table 3 it can be seen that in the LBA LOFAR is most sensitive at 60 MHz and in the HBA LOFAR is most sensitive at 150 MHz.

Freq. (MHz)	NL-Core (kJy)	NL-Remote (kJy)	EU-Remote (kJy)
15	483	483	519
30	89	89	41
45	48	48	19
60	32	32	15
75	51	51	25
120	3.6	1.8	0.89
150	2.8	1.4	0.71
180	3.2	1.6	0.81
210	3.7	1.8	0.92
240	4.1	2.0	1.0

Table 3 System Equivalent Flux Densities for the different LOFAR stations in single polarization.

 

2 Sensitivities

 

Caution: The numbers quoted here are projected for the full array and assume ideal perfomance of the instrument

For more practical information on the actual sensitivity of the current configuration readers should consult the "Examples of deliverable LOFAR products" page.

The sensitivity ΔS (in Jy) of a single dipole (or half an ``antenna'') is defined as follows : 

where δν denotes the bandwidth (in Hz) and δ t (in s) denotes the total integration time. An antenna that consists of two (equal) dipoles placed perpendicular to each other has a sensitivity of

For a station the overlap in effective area from different dipoles has to be taken into account. Using the SEFD of a station, given in the previous section, for a single polarization, the sensitivity of a station (in single polarization) is given by

  In an image signals from different baselines and two polarizations are combined. For LOFAR this includes signals from stations with different effective areas. The noise level in a LOFAR image is given by

where W denotes a factor for increase of noise due to the weighting scheme (which depends on the type of weighting: natural, uniform, robust, ...), N_c and N_r denote the number of core and remote stations respectively, S_core and S_remote denote the SEFDs for the core and the remote stations respectively. This expression can easily be extended to include European stations.

In case of an array having N equal stations (or dishes) the following familiar result is retrieved

In Table 4 the expected noise levels in LOFAR images are given. The images are based on 1 hour integration time, 2 polarizations, and 4 MHz bandwidth. A weighting factor W of 1.3 is applied to incorporate the effect of weighting. SEFD's of Table 3 have been used.

Table 4 Theoretical LOFAR sensitivity for 1 hour integration time, an effective BW of 3.57 MHz, and dual polarization. A weighting factor of 1.3 is applied. The columns correspond to the 6 stations in the Superterp, the 23 core stations, the 40(planned) stations in the Netherlands and the 48 stations (in the Netherlands and the rest of Europe).

 

3 Bandpass

There are several contributions to the frequency dependent sensitivity of LOFAR to incoming radiation (the bandpass). At the correlator, a digital correction is applied within each 0.2 MHz subband to remove the frequency-dependent effects of the conversion to the frequency domain. The station beam is also strongly frequency dependent, except at the beam pointing centre. Finally, the physical structure of the individual receiving elements causes a strongly peaked contribution to the bandpass near the resonance frequency of the dipole. In the case of the LBA dipoles, the nominal resonance frequency is at 52 MHz. However, as can already be seen in Figure 1, the actual peak of the dipole response is closer to 60 MHz.

Determining the combined bandpass (referred to as the “global bandpass”), which includes all frequency dependent effects in the system that are not yet corrected post-correlation, can be achieved during calibration. To illustrate this, the bright quasar 3C196 has been observed using the core and remote stations (see Figure 1).

Figure 1 The bright quasar 3C196 has been observed using the core and remote stations. Observations of 3C196 between 15–78 MHz were obtained in two observing sessions (15–30 MHz in one session, and 30–78 MHz in the other). BBS was used to calibrate the data. For each subband, a system gain was determined, and the gain amplitude was taken as the value of the global bandpass at the frequency of the particular subband. Left: The sensitivity of various LOFAR stations as a function of frequency between 10 and 78 MHz. The different relative sensitivities of the stations are due to incomplete station calibration across the array. Right: The resulting global bandpass of LOFAR between 10 and 78 MHz, based on the median bandpass for the individual stations.