QUESTION:  Will I see a line with flux = 1e{-17} W/m2 superimposed onto
a stellar continuum from a bright, 7.5 mag star?  Both are spatially 
unresolved...

LINE				CONTINUUM
====                            =========

1. Flux density from the line and from the stellar continuum

                                7.5mag
                                Flux at H from a zeroth-mag star is 
                                   1.07e(-23) W/m2/Hz             
                                Spec resln, 4-pix slit, R = 1500
                                Wavelength resln (1.64um) = 1.09e(-9) m 
                                Freq.      resln (1.64um) = 1.21e(11) Hz

Flux(line) = 1e{-17} W/m2       Flux(cont) = 1.3e(-15) W/m2/resln element       

  ... The line flux is spatially and spectrally unresolved, so with a 
      4-pix slit it will be spread over 4 spectral pixels.  Likewise, the 
      continuum flux is "per resolution element", so also over 4 spectral 
      pixels. Both will be spread out along the slit by the seeing, over 
      5 spatial pixels (0.6"), say.  So must divide by 4x5...


2. Flux from line and continuum landing on ONE PIXEL:

Flux(line) = 1e(-17)/(4x5)      Flux(cont) = 1.3e(-15)/(4x5)
           = 5e(-19) W/m2/pix               = 6.5e(-17) W/m2/pix


3. UKIRT's effective size (undersized cold stop) is 3.5m, so multiply by 
pi.r.r = 9.6 m2:

Flux(line) = 4.8e{-18} W/pix    Flux(cont) = 6.2e(-16) W/pix
           = 4.8e(-18) J/s/pix             = 6.2e(-16) J/s/pix 
 
4. At 1.64 um, photon energy = hc/lambda = 1.21e(-19) J
Therefore, the number of photons from the line and continuum per second
landing on each pixel is:

Num-phot(line) =                Num-phot(cont) =
         40 photons/s/pix               5,124 photons/s/pix


5. Finally, correct for instrumental/telescope throughput (20% at H) and 
assume 1 photon = 1 electron:

Num-phot(line - corrected)      Num-phot(cont - corrected) 
      =  8.0 elec/sec/pix             =  1025 elec/sec/pix
         ================                =================
                                              
Note that for the Poissonian statistics below, you have to work with
photons or electrons, not counts on the array (so don't convert the
above numbers to ADU by multiplying by the gain).  

The right-hand number seems reasonable; the gain is 15 electons, so
we're getting ~70 counts per second on the star PER PIXEL.  An
optimally extracted spectrum (extracted over ~5 rows, as already
assumed above) would give ~5x70= 350 counts in 1 sec, or 1000 counts
in a few seconds. 

In any case, in terms of signal/noise.

In 10mins :  Line              8.0x60x10
             -----      =   -------------    =    6.1
             Noise         sqrt(1025x60x10)
               

In 30mins:   Line              8.0x60x30
             -----      =   -------------    =   10.6
             Noise         sqrt(1025x60x30)
                                                                                


If you spot an error in the above, please let me know!
Thanks, 
Chris (c.davis at jach.hawaii.edu).