The ETC for spectroscopic observations is currently being developed. In the meantime, users can use the following spectra to estimate exposure times.

SW Spectra

The star HD1160 (B9V) was observed near Zenith in slitless mode. The spectra have been normalised to zeroth magnitude. The count rates are inelectrons/second/wavelength element. The wavelength element is the dispersion,ie 1 pixel.

Mode (FITS)	Wavelength Element (Angstroms)	Spatial Pixel Scale (arcseconds)	Number of Spatial Pixels	Slit Area (86mas/172mas)	Slit Factor (86mas/172mas)	Average sky background* (86mas/172mas)	RON (e-) FowlerNSamp/Double_RdRstRd
S54_2_SK	9.7	0.0543	3	0.0139 / 0.0278	0.93 / 1.86	1.5 / 3.0	20/46
S54_3_SK	10	0.0543	3	0.0139 / 0.0278	0.93 / 1.86	2.1 / 4.2	20/46
S54_4_SK	19.6	0.0543	3	0.0139 / 0.0278	0.93 / 1.86	3.8 / 7.6	20/46
S54_3_H	6.9	0.0543	2	0.0093 / 0.0186	1.09 / 2.19	0.6 / 1.2	20/46
S54_3_SH	6.9	0.0543	2	0.0093 / 0.0186	1.09 / 2.19	0.6 / 1.2	20/46
S54_4_SJ	19.8	0.0543	2	0.0093 / 0.0186	2.05 / 4.10	0.8 / 1.6	20/46
S27_2_SK	4.9	0.0270	6	0.0139 / 0.0278	0.93 / 1.86	0.9 / 1.8	20/46
S27_3_SK	5.0	0.0270	6	0.0139 / 0.0278	0.93 / 1.86	1.2 / 2.4	20/46
S27_3_SH	3.4	0.0270	4	0.0093 / 0.0186	1.09 / 2.19	0.3 / 0.6	20/46
S27_4_SH	9.8	0.0270	4	0.0093 / 0.0186	1.09 / 2.19	1.2 / 2.4	20/46
S27_4_SJ	9.5	0.0270	3	0.0070 / 0.0140	1.54 / 3.06	0.5 / 1.0	20/46

The fourth column is the number of pixels in the spatial direction overwhich S/N ratios are computed. This number is computed by rounding up 2*1.25 Lambda/D  to the nearest integer, where Lambda is the wavelength of observation and D is the diameter of M1 (8m).

The fifth column is the number of pixels along the spatial direction(column 4) times the pixel scale times the slit width. It is in square arcseconds.

The slit factor is the ratio between the sit area and the area within adisk of radius 1.25 Lambda / D. It is a crude approximation of the fraction of light entering the slit relative to the encircled energy.

The second last column gives a crudeestimate of the backgound counts over a region which equals the slit area. The units are electrons/second/ wavelength element. The estimated sky background is an average and does not include any wavelength dependence.

The last column gives the RON. The first value corresponds to the readout noise if the readmode is set to Fowler Nsamp and the second corresponds to Double_RdRstRd. Since the pixel scale is so fine, the RON is an important part of the S/N calculation.

The wavelength dependence of the night sky spectrum is not taken intoaccount in these calculations.

The dichroics modify the flux reaching the detector and the backgound. The values reported in the above table are valid for the Visual dichroic. Forother dichroics, the flux and background have to modified with the following factors.

To calculate how many electrons you are getting from the object through theslit.

1. Use the FITS file in the first column in the first table to determine the number of electrons/second/wavelength element your object would give if it were observed without the slit.
2. Multiply this number by the transmission factor for the dichroic that will be used in NAOS
3. Use the exposure time calculator (or PS) to work out the encircled energy (For this, you will need to know the details of the reference source)
4. Muliply 1 with 2, 3 and the slit factor (column 6 in first table) to work out how many electrons/second/wavelength element your object will give with the slit in place.
5. Work out the background counts (column 7 in the first table).
6. Multiply 5 by the sky factor in the second table to get the correct sky flux for the appropriate dichroic.

Here is an example.

• Detector Settings: DIT=6, NDIT=10, Double_RdRstRd
• CONICA Setting: S54_2_SK, 86mas slit
• NAOS Setting: N20C80
• Target and Reference: K=10.

Object flux with the VIS dichroic at 21000 Angstroms is then: 2374electrons/sec/9.7 Angstrom.

Object flux with the N20C80 dichroic is then 2374 * 0.8 = 1899electrons/sec/9.7 Angstrom.

The ETC gives an encircled energy of 0.43. The slit factor is 0.92.

The flux with the 86 mas slit is then 1899 * 0.43 * 0.93 = 759electrons/s/9.7 Angstrom

The background flux with the visual dichroic is 1.5 e/second/9.7Angstromsover the slit area

The background flux with the N20C80 dichroic is then 1.05 * 1.5e/second/9.7Angstroms

The S/N ratio is then Object flux / sqrt ( Object flux + Sky flux +Detector Noise)

Object flux = 6 * 10 * 759 = 4.55e4 electrons/9.7Angstroms

Sky flux = 6 * 10 * 1.58 = 95 electrons/9.7Angstroms

Detector noise = 10 * 3 * 46 * 46 = 6.3e4

S/N ratio = 138 per wavelength element or per pixel along the dispersiondirection.

LW Spectra

The star Hip038316 (B9.5V) was observed near Zenith in slitless mode. The spectra have been normalised to zeroth magnitude. The count rates are inelectrons/second/wavelength element. The wavelength element is the dispersion,ie 1 pixel.

To calculate how many electrons you are getting from the object through theslit.

1. Use the FITS file in the first column in the first table to determine the number of electrons/second/wavelength element your object would give if it were observed without the slit.
2. Use the exposure time calculator (or PS) to work out the encircled energy (For this, you will need to know the details of the reference source)
3. Muliply 1 with 2 and the slit factor (column 6 in first table) to work out how many electrons/second/wavelength element your object will give with the slit in place.
4. Work out the background counts (column 7 in the first table).

For LW spectroscopic observations the preformance of the 'VIS' and 'JHK'dichroics are similar.

Here is an example.

• Detector Settings: DIT=1, NDIT=60, Double_RdRstRd
• CONICA Setting: L54_2_LP, 86mas slit
• Target and Reference: L=8.

Object flux with the VIS dichroic at 38000 Angstroms is then: 10280electrons/sec/32 Angstrom.

The ETC gives an encircled energy of 0.73. The slit factor is 0.50.

The flux with the 86 mas slit is then 10280 * 0.73 * 0.50 = 3750electrons/s/32 Angstrom

The background flux is 38000 e/second/32Angstroms over the slit area

The S/N ratio is then Object flux / sqrt ( Object flux + Sky flux +Detector Noise)

Object flux = 60 * 1 * 3750 = 2.25e5 electrons/32Angstroms

Sky flux = 60 * 1 * 38000 = 2.28e6 electrons/32Angstroms

Detector noise = 60 * 5 * 46 * 46 = 6.3e5

S/N ratio = 127 per wavelength element or per pixel along the dispersiondirection.