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# SIGNAL Help

SIGNAL calculates the number of photons per Å detected from a source of given apparent magnitude (per arcsec2 if extended), as:
```
10.^(-mag/2.5)
* photons/sec/A/m^2 giving mag = 0 at the top of the atmosphere
* transmission of atmosphere at given airmass
* exposure time in sec
* unobstructed area of main mirror in m^2
* measured throughput of telescope/instrument
* quantum efficiency of detector in the given band
```
Accuracy is typically +-20%. Counts from the sky are calculated in a similar way. For spectroscopic observations, signal-to-noise is per pixel step in wavelength for point sources, and per pixel for extended ones. For imaging, the counts obtained above are multiplied by the effective bandwidth of the filters in Angstrom, and the signal-to-noise is calculated within a 2-FWHM-diameter aperture for a point source and per pixel for extended sources.

The program can be used before observing to estimate the exposure time needed for a particular experiment, and at the telescope to check that the expected number of photons (counts × gain) is detected by the CCD.

The calculation made by SIGNAL can be summarised as follows:
```(1) DEFINE/CALCULATE RELEVANT INPUT PARAMETERS

Nobj = photons/A (per arcsec^2 if extended) from object, calculated as above
Nsky = photons/A/arcsec^2 from sky, calculated as above

BAND = equivalent width of filter in A (integral T(l)dl
where T(l) is transmission, l is wavelength)
FWHM = object fwhm (intrinsic and due to seeing) in arcsec
P    = number of pixels over which integration carried out
SLIT = slit width, or diameter of fibre or lenslet (arcsec)

(2) CALCULATE SIGNAL-TO NOISE

(2a) IMAGING AND SLIT SPECTROSCOPY:

Imaging, point source:
N is number of (detected) object photons
S is number of (detected) sky photons per pixel

N = Nobj * BAND
S = Nsky * BAND * (arcsec/pixel)^2
P = pi * (FWHM/(arcsec/pixel))^2
(This assumption radius = FWHM is slightly pessimistic.  Optimum S:N
ratio is achieved for radius = 2/3 * FWHM, according to
Naylor 1998, MNRAS 296 339.)

Imaging, extended source:
N is number of object photons/arcsec^2
S is number of sky photons per pixel

N = Nobj * BAND * (arcsec/pixel)^2
S = Nsky * BAND * (arcsec/pixel)^2
P = 1

Spectroscopy, point source:
N is number of object photons per pixel step in wavelength
S is number of sky photons per pixel

N = Nobj * A/pixel
S = Nsky * SLIT (arcsec) * arcsec/pixel * A/pixel
P = 2 * FWHM (arcsec) / arcsec/pixel

Spectroscopy, extended source:
N is number of object photons per pixel
S is number of sky photons per pixel

N = Nobj * SLIT (arcsec) * arcsec/pixel * A/pixel
S = Nsky * SLIT (arcsec) * arcsec/pixel * A/pixel
P = 1

For point sources, the sky counts are those from a
2-FWHM-diameter circular aperture for imaging, and from
a 2-FWHM * slit-width rectangle for spectroscopy.
For extended sources, the calculations are per pixel.

(2b) INTEGRAL-FIELD OR FIBRE SPECTROSCOPY:

Spectroscopy, point source:
N is number of object photons per pixel step in wavelength
S is number of sky photons per pixel step in wavelength
(not per pixel)

N = Nobj * A/pixel
S = Nsky * 3.14159/4. * SLIT(arcsec)^2 * A/pixel
P ~ 2 (depends on instrument)

Spectroscopy, extended source:
Same calculation as for point source, except that object mag
is integrated over circular aperture of diameter SLIT.
```
The predicted counts are based on actual throughput measurements for all currently-offered instruments.

The AF2 throughputs are for reflection mode. For echelle mode, the throughputs for orders 3, 4, 5, 6, 7 are factors 1.0, 1.3, 1.3, 2.0, 3.2 lower respectively.

The program does not take into account losses due to colour and ND filters or to polarisation optics in spectrograph; to the lower grating efficiency at large angles of incidence; or to vignetting at large field radius.

The original web interface to the program was written August 1998 by Ashley James of UCL (ING summer student). It was rewritten August 2003, in PHP, by Robert Greimel.

### Instruments and gratings

'N/A' in the menu indicates no longer available as a common-user instrument (for coding reasons, this label also temporarily appears against non-ING instruments).

For the latest news on available instruments, see the ING astronomy page under 'Overview'.

In reflection mode, WYFFOS is normally used with ISIS diffraction gratings. It can also be used with IDS gratings, but SIGNAL doesn't cater for this option.

Imaging calculations ignore the grating selected.
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### Detectors

The current default detector assignments are shown on the instruments overview page .

For more information on each detector, see the ING detectors page .

Selecting INGRID or NAOMI/INGRID forces selection of the Rockwell detector.
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### Band and effective bandwidth

For imaging, the U, B, V, R, I and Z bandwidths used correspond to the 50-mm CuSO4/UG1 or liq-CuSO4 U filters, the Harris BVRI set, and the RGO glass Z filter (although for actual broad-band imaging, e.g. with ACAM, most observers will nowadays use the Sloan u, g, r, i and z filters). The J, H and Ks bandwidths are those of the old WHIRCAM J, H and Kshort filters.

Note that the 125-mm glass U filter has a factor 3 lower throughput than the 50-mm U. Note also that the throughputs of the Harris B filters are up to 25% less than that of the KPNO B filters.

The effective bandwidth of a filter is taken to be the integral over T(l)dl, where T(l) is the transmission of the filter and l is the wavelength, i.e. the area under the filter transmission curve, measured in A, see ING filter effective bandwidths. SIGNAL assumes sensible defaults for broad-band filters.

The approximate wavelengths used for spectroscopy are given in Å in the menu.

For FOS, U,B are for second order, V,R,I are first order.
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### Object type

The calculations can be carried out for point sources, with specified FWHM and apparent magnitude; or for extended sources, with specified apparent mag per square arcsec.

For fibre and integral-field spectroscopy, point sources are assumed, i.e. SIGNAL assumes that the given mag is for one fibre or lenslet. For observations of an extended object, just give the mag per fibre or lenslet.
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### Apparent magnitude

By default, SIGNAL expects apparent magnitudes to be expressed in one of the Johnson/Bessell U B V R I Z J H or K bands ('Vega' magnitudes).

Information about converting SDSS u g r i z magnitudes to U B V R I can be found on the SDSS web pages.

If you want to specify to SIGNAL an Oke AB apparent magnitude, rather than a Vega magnitude, add 100 and enter the total. E.g. if mag(AB) = 24, enter 124.

If you want to specify the intensity in Jy (10-26 W/Hz/m2), multiply by -1 and enter the result. E.g. if S(Jy) = 0.01, enter -0.01.

For fibre or integral-field spectroscopy, the mag is assumed to be per lenslet or per fibre.

SIGNAL converts Vega magnitudes to SI units using the calibrations given by Bessell (1979, PASP, 91, 589) and Bessell and Brett (1988, PASP, 100, 1134), which are similar to those given by Johnson (1966, Ann Rev Astr Astrophys, 4, 193). These can be expressed as intensities in Jy for mag = 0 in each band:

``` U     B     V     R     I     Z     J     H     K
1810  4260  3640  3080  2550  2200  1570  1020   640
```
(These are the values used in version 14.3 of SIGNAL, and they are unlikely to change much.)

Emission-line objects - imaging
To estimate for an imaging observation the signal-to-noise with which a point emission-line object will be detected, divide the emission-line intensity S (in W/m2) by the effective bandwidth (see above) of the filter in Hz (e.g. a 50-A Hα filter with a transmission of 100% would have an effective bandwidth of ~ 3.5x1012 Hz) to calculate the mean intensity across the filter bandpass, in W/m2/Hz, then divide by 10-26 to convert to Jy. The intensity in Jy can then be entered as noted above.

For an extended object with known emission-line surface brightness S, in W/m2/arcsec2, a similar calculation can be carried out, to obtain surface brightness in Jy/arcsec2, which can then be entered in SIGNAL to estimate S:N per arcsec2.

Example: an object has a surface brightness of 10-15 erg/s/cm2/arcsec2 in the Hα emission line. This corresponds to 10-18 W/m2/arcsec2. Imaged through an Hα filter with a 50-A bandpass (i.e. bandpass 3.5 x 1012 Hz), this corresponds to 2.9 x 10-31 W/m2/Hz/arcsec2, or 2.9 x 10-5 Jy (r mag ~ 20) per arcsec2.

To predict the S:N for an 1800-sec ACAM imaging exposure of such an object through an Hα filter in bright-of-moon, enter the appropriate parameters in SIGNAL, i.e. 'apparent mag' = -0.00003, time = 1800 sec, band = 'R', bandwidth = 50 A, object = 'extended' and sky = 'b' (for bright). SIGNAL should predict ~ 2100 photons/pixel from the object, ~ 10300 photons/pixel from sky and S:N ~ 19 (i.e. sky photon noise dominates).

[NB as a sanity check of the number of detected photons, and given that the energy of an Hα photon is E = hc/λ = 2.9 x 10-19 J, one can convert the surface brightness 10-18 W/m2/arcsec2 into a photon arrival rate at the top of the atmosphere ~ 3.4 photons/s/m2/arcsec2. For exposure time 1800 sec, WHT collecting area 12.5 m2, assumed atmosphere/WHT/ACAM/CCD efficiency ~ 0.42, and pixel size 0.25 arcsec, this indeed corresponds to ~ 2000 photons/pixel, as calculated by SIGNAL.]

Emission-line objects - spectroscopy
For an estimate of the signal-to-noise with which an emission line (intensity S W/m2) is detected in a spectrum of a point object, it's probably simplest to first predict the number of detected photons N:

```N = S * telescope area * throughput * exp time / Enu
```
where the telescope area and throughput of the system (atmosphere, telescope, instrument, detector) take the values reported by SIGNAL, and where Enu is the energy of one photon (e.g. 3.0 x 10-19J for an Halpha photon, wavelength 6563 A). The predicted N can then be combined with SIGNAL's estimate of the counts from the sky, and the known readout noise, to yield an estimate of the signal-to-noise.
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### Exposure time

Any value of exposure time can be entered.

In practice, the minimum useful exposure time is usually determined by the time taken for the shutter to open and close (~ 0.1 sec).

Most of the observing time is expended on exposures shorter than half an hour. Longer exposures (1) will suffer a large number of cosmic-ray hits, and (2) incur a larger risk of lost time in the event of a technical failure.
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### Object FWHM

The image FWHM (seeing convolved with object size) is used to determine both vignetting by the slit and the number of pixels over which to integrate for point-source observations:
```  pi * (FWHM / arcsec/pixel)**2 for imaging
2  *  FWHM / arcsec/pixel     for spectroscopy
```

For NAOMI/INGRID, SIGNAL uses the appropriate throughput and pixel scale, but doesn't attempt to model the AO-corrected PSF - the user is left to set an appropriate FWHM.
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### Slit width

The slit-width is now used both to determine the intensity of the signal from (uniformly) extended objects, including the sky, and to calculate vignetting (for point sources).

For fibre or integral-field spectroscopy (e.g. WYFFOS, OASIS), the fibre or lenslet is assumed to be circular with the specified diameter.

For imaging calculations, this parameter is ignored.
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### Airmass

Airmass = 1/sqrt[1 - 0.96×sin2(ZD)] approximately, i.e. approx sec(ZD).
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### Extinction

The default values are taken from La Palma Technical Note 45 and the WHIRCAM Users Manual. Visit the online ING technical notes and manuals for more information, and the site quality web pages for more information about dust on La Palma.
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### Sky brightness

SIGNAL's default optical sky-brightness setting D is for typical dark-of-moon conditions. For sky brightness typical of grey- or bright-of-moon, select G or B respectively. For the darkest sky seen on La Palma, i.e. median at dark-of-moon, solar minimum, at high galactic and ecliptic latitude and in the absence of twilight and moonlight, select 0.

The BVR values are accurate to +- 0.1 mag and are similar to those measured at other dark sites (Chile, Hawaii etc.). U and I are accurate to 0.5 mag. R and I sky brightnesses vary randomly by several tenths of a mag with variations in the OH airglow. The V and R sky brightness include a contribution of about 0.1 mag due to NaD light pollution. Light pollution is negligible in other bands.

The sky brightness values used by SIGNAL refer to low spectral resolution. Between the OH lines in the red, the sky is 1 - 2 mag darker.

The sky is markedly brighter (several tenths of a mag) under very dusty conditions (> 0.3 mag extinction).

The sky is 0.4 mag brighter at solar maximum. Recent minima were in 1986.8, 1996.5, ~2007. Last maximum was 2000.4. Variation is approximately sinusoidal with time.

The sky is 0.4 mag brighter on the ecliptic than at the poles, varying as sine(b) approximately.

The airglow contribution (typically about 70% of the total in V) brightens approximately as airmass. The sky is 0.3 mag brighter at airmass 1.5.

Stars fainter than apparent magnitude 20 contribute negligibly to the total brightness of the sky. Starlight scattered by interstellar dust contributes about 5% of the total, rising to about 30% on the galactic plane. The extragalactic contribution is negligible (< 1%). The brightness of the sky does not vary with time after astronomical twilight.

For further details of the calculations of moonless sky brightness, see La Palma technical note 115 (Benn & Ellison 1998).

SKY BRIGHTNESS WITH MOON UP:
When the moon is 60 deg from zenith, with extinction 0.15 mag, the zenith sky will brighten roughly by M as tabulated below:

```                     New  Crescent  Quarter  Gibbous  Full
Phase angle (deg) 180     135       90        45      0
Approx day:         1       4        8        12     15
D, G or B:          D       G        G         B      B
Illum. frac. %      0      25       50        75    100
M (U, B, V)         0       0.5      2.0       3.1    4.3
M (R)               0       0.3      1.3       2.4    3.5
M (I)               0       0.2      1.1       2.2    3.3
```
Note that the quarter moon (i.e. half disc illuminated) is a factor of 10 (not 2) fainter than full, due to the opposition effect (also responsible for gegenschein on the ecliptic and dry heiligenschein on earth).

Sky brightness for other values of lunar phase, lunar zenith angle, sky position and extinction, can be estimated with SIGNAL's sky-brightness calculator (see the interface above). The contribution of moonlight in V has been calculated according to the scattering formula of Krisciunas & Schaefer (1991, PASP, 103, 1033), normalised (multiplied by a factor of 2.4) to agree with measurements of sky brightness made at the JKT on a dust-free night in 7/98. The moonlight contribution in the other bands is calculated according to the U-B, B-V, V-R, R-I colours of moonlight measured on the same night in 7/98. These values agree +-40% with measurements made by DHPJ in 9/89, but the contribution of moonlight probably depends strongly on local conditions (e.g. dust, telescope baffling), and with current data, the contribution by moonlight on La Palma can probably only be predicted within a factor ~2.

Ian Steele (LJMU) has found that the background brightness at the JKT rises dramatically (factor >~5 brighter than the above numbers) if moonlight falls on the telescope structure (scattering within the telescope).

For further information on optical sky brightness, see the ING site quality web page .

The J, H and K sky brightnesses are taken from bright-of-moon INGRID commissioning observations (Mar 2000). They probably depend little on lunar phase (particularly in K, in which band observing can often continue until after sunrise).
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### Output format

The default format is a text listing of input parameters and results. 'Graph' format gives the text listing plus a choice of graphs e.g. S:N vs exposure time, S:N vs magnitude, optionally for different sky brightness, airmasses etc (for parameters other than sky brightness, specify the required values in the boxes 'curve 1' etc.). This facility was added by Robert Greimel Aug 2003.

### Fortran source code

For faster turnaround, copy across the the Fortran source code (save as source, not text) and edit out the html tags from start and end. f77 signal.f -o signal compiles and links it. The instrument zeropoints are documented around line 1440. Users on the La Palma unix cluster can type ~crb/ing/sig/signal to run the program.

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