Comisión mercantil [2]

Star trailing is the obvious consequence of stationary observations. The main effect of the trailing will be to spread out the star photons over a larger area of sky increasing the sky noise that needs to be taken into account when calculating the S/N of an observation. The longer the exposure time, the longer the trail and consequently the more sky noise that is added to the overall noise. The length of a star trail in a certain exposure time depends on where we look on the sky, since stars trail at the fastest rate on the celestial equator and at their slowest rate at the celestial poles.

In this section we try to quantify the effect of star trailing on the S/N of the observations. We assume the values listed in Table 2.1 for the AP10 CCD and 50mm f/1.2 lens that we used for constructing PASS0. We calculate the Vega counts per second from Equation 2.27 and divide by the CCD gain (15.5 e−

/ADU) to get 2.639×105ADU/s. We can then calculate the star ADU per secondCADUpsfor a star of magnitudemusing:

CADUps= 2.639×105×10−0.4m (2.42) To calculate the sky counts in ADU per second per pixelSADUpsppwe use:

SADUpspp =2.639×105×10

−_0.4MSKY

MSKY=18−2.5 log ∆2arc

(2.43) and we note that each pixel is of size 57.8′′

(from Table 2.1).

Now we consider the case of no trailing which corresponds to “normal” observations (by WASP0 for example) and to PASS0 when it is observing towards the celestial pole (δ = 90◦

). We want to calculate the signal-to-noise of an observation, using optimal PSF scaling, and as a function of exposure time, which will involve the use of Equations 2.13, 2.20 and 2.21. To do this we assume that the typical FWHM of the untrailed PSF is 2 pix and use this to generate a normalised Gaussian PSF for the PSF functionP(x, y). We then use the known sky flux and star flux to generate a model image, which becomes the termX(x, y). Finally we take the AP10 CCD readout noise to be 1.3 ADU. The signal-to-noise SNR we calculate as:

SNR= N∗

σN∗

(2.44) In Figure 2.7 we plot the calculated signal-to-noise for a non-trailed image as a function of expo- sure time for three example stars of magnitudes 7, 8 and 9 (solid curves).

To calculate the S/N for a trailed star image, we simulate the trail by summing many Gaussian PSFs shifted relative to each other and along the direction of the trail. The trail image is then renormalised to giveP(x, y)and used together with the sky flux and star flux to calculateX(x, y). The trail lengthLtrail, in pixels, is calculated using the equation:

Ltrail= 360_×60_×60 24×60×60 × ∆tcosδ ∆arc (2.45) whereδis the declination of the observation. There is no trailing at the celestial pole (δ = 90◦

) and the fastest trailing occurs at the celestial equator (δ = 0◦

56 CHAPTER 2. DESIGN OF THE PASS0 EXPERIMENT 0 100 200 300 400 500 600 0 50 100 150 200 Signal-To-Noise Exposure Time (s)

Signal-To-Noise Vs Exposure Time For A Single Exposure

7th Mag 8th Mag 9th Mag 7th Mag 8th Mag 9th Mag

Figure 2.7: The S/N as a function of exposure time for a single image with no trailing (solid curves) and trailing at the celestial equator (dashed curves). Each curve corresponds to a star of a different magnitude (7, 8 or 9) and all the curves correspond to the PASS0 camera.

signal-to-noise for trailed images at the celestial equator for three example stars of magnitudes 7, 8 and 9 (dashed curves). The plot shows that, for a single exposure, the effect of the trailing is that the S/N converges towards a maximum S/N and does not continue increasing with exposure time. It is clear that after about 50 s, the gains in S/N for the trailed images are very small.

Our conclusion from Figure 2.7 is that exposure times for PASS0 should be kept on the short side for efficiency of the observations. However, we cannot have very short exposure times be- cause the CCD has a readout time of ∼10 s. In Figure 2.8, we plot the total S/N over 1 hour of observations as a function of exposure time of an individual exposure, taking into account a read- out time of 10 s. The solid curves correspond to stars of 7th, 8th and 9th magnitude without trailing and the dashed curves to the same stars but with the trailing at the celestial equator (maxima are marked by solid black circles). The curves are calculated by taking the previous calculations of signal-to-noise for one exposure and multiplying this by√NIMwhereNIM is the number of im- ages taken in 1 hour (including the readout time of 10 s). The number of images with exposure time∆ttaken in 1 hour is simply NIM = 3600/(∆t+ 10). We conclude from Figure 2.8 that for trailed images there is an optimal exposure time for the individual images for which the total S/N during an observing run is maximised, and that for PASS0 this optimum exposure time will be about 10 s to 15 s.

0 500 1000 1500 2000 2500 3000 0 20 40 60 80 100 Total Signal-To-Noise

Single Image Exposure Time (s)

Total Signal-To-Noise Vs Single Exposure Time For 1 Hour Of Observations

7th Mag 8th Mag 9th Mag 7th Mag 8th Mag 9th Mag

Figure 2.8: The total S/N as a function of the single image exposure time for 1 hour of observa- tions. No trailing is shown by the solid curves and trailing at the celestial equator is shown by the dashed curves. Maxima are marked by solid black circles. Each curve corresponds to a star of a different magnitude (7, 8 or 9) and all the curves correspond to the PASS0 camera.