Sensitivity

Table 5 shows the VLA sensitivities expected for natural weighting of the visibility data. The values listed are the expected rms fluctuations due to thermal noise on an image, calculated using the standard formulae with the system temperatures and efficiencies listed. A maximum number of 25 antennas is used in these calculations (except for U band); at any given time, it is assumed that 3 of the possible 28 antennas are missing, one each for EVLA mechanical retrofitting, EVLA electronics outfitting, and EVLA commissioning. Hardware limitations prevent us from inputting more than 22 EVLA antennas to the old VLA correlator, and no more than 25 antennas will be available by the end of 2009. The tabulated sensitivity values are realized in practice except at low frequencies and in smaller configurations where the sensitivities are limited by confusing sidelobes from objects outside the image. The rms limit due to confusing sources for the VLA in D configuration is estimated in Table 5. Another case where the thermal rms noise will not be achieved is in imaging very bright objects where the residual image noise is due to baseline dependent errors (`closure errors' - see 3.11).

Table 5: VLA Sensitivity in Mid-2009

Frequency	Band Name		System	Antenna	Number	RMS (10 min)
(GHz)	Approximate	Letter	Temperature$^{(1)}$	Efficiency$^{(2)}$	Antennas	Sensitivity
	Wavelength	Code	(K)	(%)	(VLA+EVLA)	(mJy)
0.073 - 0.0745	400 cm	4	1000-10000	15	5+20	160$^{(3)}$
0.3 - 0.34	90 cm	P	150-180	40	$\leq 10$	$>4^{(3)}$
1.24 - 1.70	20 cm	L	35	55	5+20	0.061
4.5 - 5.0	6 cm	C	45	69	5+20	0.058
8.1 - 8.8	3.6 cm	X	35	63	5+20	0.049
14.6 - 15.3	2 cm	U	120	58	5+0	1.0
22.0 - 24.0	1.3 cm	K	50 - 80	40	5+20	0.11$^{(4)}$
40.0 - 50.0	0.7 cm	Q	80	35	5+20	0.27$^{(5)}$
Frequency	Wavelength	RMS Point-Source	Beam-averaged	Antenna	Peak/Total	RMS
		Sensitivity	Brightness	Primary	Confusing	Confusion
		(12 hours)	Sensitivity$^{(6)}$	Beam Size	Source	Level
			(D-config)	(FWHP)	in Beam	(D-config)
(GHz)		(mJy)	(mKelvins)	$\theta_{\rm PB}$	(Jy)	($\mu$Jy/beam)
0.073 - 0.0745	400 cm	17$^{(3)}$	2000	700$'$	20/350	lots
0.3 - 0.34	90 cm	0.18$^{(3)}$	21.3	150$'$	1.8/15	4400
1.24 - 1.70	20 cm	0.0071	0.8	30$'$	0.11/0.35	86
4.5 - 5.0	6 cm	0.0069	0.8	9$'$	0.002	3.6
8.1 - 8.8	3.6 cm	0.0057	0.6	5.4$'$	0.001	0.89
14.6 - 15.3	2 cm	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$
22.0 - 24.0	1.3 cm	0.013$^{(4)}$	1.5	2$'$	0.00001	--
40.0 - 50.0	0.7 cm	0.032$^{(5)}$	3.8	1$'$	--	--

All sensitivity calculations assume 43 MHz bandwidth per IF, (except for P-band and 4-band, where 3.125 MHz and 0.78 MHz are used), two IF pairs (four IFs), natural weighting, and an elevation of 45 degrees. Five VLA and 20 EVLA antennas are assumed, as will be the case in the period near the middle of 2009; since there will be only a few VLA antennas remaining with 2 cm installed, most information regarding this band has been removed. EVLA antennas are assumed to have the same performance as VLA antennas, although they will be significantly better at some bands (particularly at 6 cm = C band); since we mostly have ``transition'' EVLA receivers at present, the final system temperatures are not yet available. Performance will degrade for large zenith angles at high frequencies and for sources close to the galactic plane at low frequencies. The confusion limits (see Condon 2002, ASP Conf. 278, p. 155) for C configuration are approximately a factor of 10 less than those tabulated above for D configuration.

Footnotes:

1. Temperature ranges listed at P, K, and Q bands include sky temperature variations due to the galactic plane (P-band) or Earth's atmosphere (K band). System temperatures for VLA antennas at L band are increased substantially at elevations below $45^\circ$ due to ground pickup; this effect is considerably reduced (though not completely eliminated) in the EVLA antennas because of their larger L band feeds.
2. This is the system efficiency without the correlator factor (about 0.78). Efficiencies at U, K, and Q bands at low elevations ($< 30$ degrees) are considerably decreased due to gravitational distortions of the antenna figure.
3. Value listed for 74 MHz assumes observations near the galactic poles, and `3-D' imaging. Snapshot observations will not usually reach this level, as the confusion problem is insoluble with only snapshot u,v coverage. Full-beam A configuration deconvolution at 74 MHz will be calibration-limited due to the non-isoplanatic ionosphere. At 327 MHz the imaging will be very poor due to the limited number of receivers available with this sensitivity.
4. Listed sensitivity is for El = $45^\circ$ and very dry atmosphere at 22 GHz. A wet atmosphere can increase zenith opacity from 5 to 15 percent, and increase the sky temperature from 10 to 40 K.
5. Listed sensitivity is for El = $45^\circ$ and dry atmosphere at 43 GHz. A wet atmosphere can increase zenith opacity from 6 to 8 percent, and increase the zenith sky temperature from 15 to 24 K. Atmospheric attenuation and temperature increase dramatically with increasing frequency - sky temperatures exceeding 55 K and zenith opacity of 25 percent are expected at 49 GHz.
6. Values listed assume a uniform source that fills the synthesized beam in D configuration. Since natural weighting is assumed, the beam sizes are $\sim$50% larger in each dimension than the values given in Table 4.

For a comparison, Table 6 gives the sensitivity goals of the EVLA, including the expectations based on achieved values for those receivers that already have been implemented.

Table 6: EVLA Sensitivity Goals

Band$^{(1)}$	Band	${S_E}^{(2)}$	${S_E}^{(2)}$	Cont. Sens.	Line Sens.
	Code	req.	actual	$1\sigma$, 9 hr	$1\sigma$, 1 km s$^{-1}$, 9 hr
(GHz)		(Jy)	(Jy)	($\mu$Jy)	(mJy)
1 - 2	L	325	335	1.6	0.5
2 - 4	S	235	TBD	TBD	TBD
4 - 8	C	245	250	0.5	0.2
8 - 12	X	300	TBD	TBD	TBD
12 - 18	Ku	385	TBD	TBD	TBD
18 - 26.5	K	650	450	0.6	0.2
26.5 - 40	Ka	760	675	0.85	0.2
40 - 50	Q	1200	1400$^{(3)}$	1.8	0.4

Footnotes:

1. Performance parameters in the bands above 18 GHz are based on actual performance for the final receiver designs. Performance parameters for the 1-2 and 4-8 GHz bands are for interim systems; the final systems are expected to be slightly better. The remaining three bands are presently in the design and prototyping stages.

2. The parameter $S_E$ is the ``System Equivalent Flux Density'', a measure of the flux density of a natural radio source that would be required to double the system temperature. Thus, lower values of $S_E$ are better. $S_E$ is given by the equation $S_E = 5.62 T_{\rm sys}/\epsilon$, where $T_{\rm sys}$ is the total system temperature of the receiver and $\epsilon$ is the antenna aperture efficiency in the given band. Two values of $S_E$ are given, first the required value from the EVLA Project Book, followed by the actual achieved value.

3. The final performance of the 40-50 GHz systems is expected to be somewhat better than tabulated. Because the EVLA feed is at a different feed-circle location than the previous VLA feed, a new round of holography measurements and re-setting of antenna panels will still be necessary to optimize performance at the highest frequency band. There is some expectation for minor improvement in the 26.5-40 GHz band as well after panel re-setting.

In general, the expected point-source rms noise in mJy on an output image, for natural weighting, can be calculated with the following formula:

$$\Delta I_m = {K \over \sqrt{N(N-1)(N_{\rm IF} T_{\rm int} \Delta \nu_{\rm M}) }}\ {\rm mJy}$$ (1)

where $N$ is the number of antennas, $T_{\rm int}$ is the total on-source integration time in hours, $\Delta \nu_{\rm M}$ is the effective continuum bandwidth or spectral-line channel width in MHz, and $N_{\rm IF}$ is the number of IFs (from 1 to 4) or spectral line channels (from 1 to 512) which will be combined in the output image². $K$ is a system constant, equal to 1000, 50, 8.0, 7.8, 6.6, 27, 14, and 35³ for 4, P, L, C, X, U, K, and Q bands respectively. This constant $K$ also can be expressed in terms of system temperature and efficiency as:

$$K = {0.12 T_{\rm sys} \over \eta_a}$$ (2)

where $T_{\rm sys}$ is the system temperature, and $\eta_a$ is the antenna efficiency, as listed in Table 5. (A correlator efficiency of 0.78 has already been incorporated into this expression). For the more commonly used uniform weighting employing the robust weight scheme intermediate between pure natural and pure uniform weightings (available in the AIPS task IMAGR), the sensitivity will be a factor of about 1.2 worse than the listed values. To aid VLA proposers there is an exposure tool calculator on-line at http://www.vla.nrao.edu/astro/guides/exposure/ that provides a graphical user interface to these equations.

It is important to note that these listed sensitivities are calculated from data taken in optimum conditions. For many bands, the system temperature and gain are significant functions of elevation and weather conditions (see next section).

A useful alternate form of the point-source sensitivity equation is

$$\Delta I_m = {{42.4K}\over\sqrt{N_{\rm pts}N_{\rm IF}\Delta t_{\rm int}\Delta\nu_{\rm M}}}\ {\rm mJy}$$ (3)

where $N_{\rm pts}$ is the number of visibility points (which is listed in the AIPS header), and $\Delta t_{\rm int}$ is the integration time per visibility in seconds. $N_{\rm IF}$, $\Delta \nu_{\rm M}$ and $K$ are defined as above.

The beam-averaged brightness temperature measured by a given array depends on the synthesized beam, and is related to the flux density by

$$T_{\rm b} = \frac{S \lambda^2}{2k\Omega} = F \cdot S$$ (4)

where $T_{\rm b}$ is the brightness temperature (Kelvins) and $\Omega$ is the beam solid angle. For natural weighting (where the angular size of the approximately Gaussian beam is $\sim 1.5 \lambda/B_{\rm max}$), and $S$ in mJy per beam, the constant $F$ depends only upon array configuration and has the approximate value $F$ = 190, 19, 1.6, 0.15 for A, B, C, and D configurations, respectively. The brightness temperature sensitivity can be obtained by substituting the rms noise, $\Delta I_m$, for $S$. Note that Equation 4 is a beam-averaged surface brightness; if a source size can be measured the source size and integrated flux density should be used in Equation 4, and the appropriate value of $F$ calculated. A more detailed description of the relation between flux density and surface brightness is given in Chapter 7 of Reference 1, listed in Section 6.

Figure 1: System Temperatures vs. Frequency at K and Q bands. Above are plots of the system temperature (solid red line) versus observing frequency at K and Q bands. The data were taken during night time in May 2003, under very good weather conditions. This plot shows that the system temperature is stable across the band. In blue is the number of antennas that successfully locked at the given frequency.

Table 7: Sensitivity Ranges of VLA Bands

Band	0.9 x Nominal	0.5 x Nominal	Extreme Range
90 cm	305 - 337 MHz	303 - 342 MHz	298 - 345 MHz
20 cm	1240 - 1700 MHz	1170 - 1740 MHz	1150 - 1750 MHz
6 cm	4500 - 5000 MHz	4250 - 5100 MHz	4200 - 5100 MHz
3.6 cm	8080 - 8750 MHz	7550 - 9050 MHz	6800 - 9600 MHz
2 cm	14650 - 15325 MHz	14250 - 15700 MHz	13500 - 16300 MHz
1.3 cm	21200 - 25200 MHz	20600 - 25200 MHz	20400 - 25500 MHz
0.7 cm	40500 - 44500 MHz	39000 - 47500 MHz	38000 - 51000 MHz

For observers interested in HI in galaxies, a number of interest is the sensitivity of the observation to the HI mass. This is given by van Gorkom et al. (1986; AJ, 91, 791):

$$M_{\rm HI} = 2.36 \times 10^5 D^2 \sum S \Delta V ~{\rm M}_\odot$$ (5)

where $D$ is the distance to the galaxy in Mpc, and $S\Delta V$ is the HI line area in units of Jy km/s.

The sensitivity varies across each observing band. Table 7 gives the frequency ranges for each band at which the sensitivity degrades by 10% and by a factor of two for the VLA receivers. Also included are the maximum ranges over which the VLA receivers remain operative. At these extreme ends, the system sensitivity is typically 10 to 100 times worse than at band center. Furthermore, not all antennas will operate at these frequencies. For similar information for EVLA antennas, see Section 3.3. In Figure 1, we show the system temperature and number of operable antennas plotted as a function for frequency for K and Q bands.⁴ For all bands, consider consulting a VLA staff scientist if you wish to observe near the band edges.

In view of the importance of observations at the lower edge of the 20-cm band for studies of red-shifted HI, some special words are appropriate to describe the performance of the VLA receivers at frequencies below 1250 MHz. The roll-off of this band at the low frequency edge is very gentle, and useful observations at 1155 MHz and lower have been made. However, not all frequencies can be tuned, as tests have shown there are four 10 MHz wide `notches', centered at 1212, 1182, 1162 and 1150 MHz, within which the array can take no useful data. These notches exist in both RR and LL correlations but vary in shape and central frequency from antenna to antenna. A plot of the relative sensitivity between 1150 and 1250 MHz for the VLA receivers is shown in Figure 2.

Figure 2: VLA Sensitivity Between 1150 and 1250 MHz. This plot shows the relative sensitivity of the VLA at the lower end of the 20 cm band. The dashed line represents the sensitivity at the center of the band, so it can be seen that serious degradation in sensitivity is not notable until below 1190 MHz. The four `notches' are instrumental in origin.

The sensitivity at the low frequency bands (90 cm and 400 cm) is difficult to parameterize. There are two important effects which limit the sensitivity.

1. The diffuse galactic background contributes an important fraction of the total system temperature - especially at 400 cm, where it is the only important contributor. This means that the sensitivity will vary by nearly a factor of 10 between locations on the galactic plane and locations near the galactic poles.
2. The primary beam at both bands is very broad, and the sidelobe levels comparatively high, resulting in significant sensitivity to objects far from the target object. Because of the difficulty in imaging very large fields of view (due to the non-coplanar baseline effect, see Section 3.5.4, and the non-isoplanicity over large angles), the sidelobes of undeconvolved objects in non-imaged areas (essentially the entire 2$\pi$ steradians visible to the antennas!) will appear in the map of the target source. Use of the AIPS program IMAGR will permit removal of the major background objects, and should result in a sensitivity not worse than a factor of two higher than that expected on the basis of the system temperature.