Errors

  • The root mean square was found for each pixel column across 10 frames, for each exposure.
\begin{equation} N_{\mathrm{rms}} = \sqrt{\frac{N_{0}^2 + N_{1}^2 + ... +N_{n}^2}{n}}\end{equation}

  • where $N_{i} = N_{\mathrm{pe},i} - \overline{N_{\mathrm{pe}}}$
  • By then plotting the rms against the average, we could see how the error varies with intensity.
  • This shows a plot of rms against the average for exposure times from 0.05-1s, for all 10 frames.
RMSAll.png
  • This was not what we expected so we then plotted the same plot at one exposure time of 1s, however changes which images we used.
  • This plot shows all 10 images:
RMS010_1s.png
  • This plot shows the first 5 images:
RMS05_1s.png
  • This plot shows the second 10 images
RMS510_1s.png
  • This may be due to the fact that we took the first five frames at a different time to the second five frames.
  • Therefore we then plotted for all exposure times the second five frames:
RMS510.png

Mapping

  • Initially this equation gave the best fit to the data:
\begin{equation} f(\mu,m) = \lambda_{0} + a_{1} \mu + b_{1} m + a_{2} \mu^{2} + b_{2} m^{2} + c_{2} \mu m \end{equation}
  • however there may be additional terms to the equaiton.
  • Looking at the manual for the spectrometer it give the equation for the grating as:
\begin{equation} \sin{\alpha} + \sin{\beta} = 10^{-6} k n \lambda \end{equation}
  • and also:
\begin{equation} \mathrm{D}_{\mathrm{v}} = \mathrm{D}_{\mathrm{v}(x = 0)} + \tan^{-1}\left(\frac{x}{f_{2}}\right) \end{equation}
  • and $\beta = \mathrm{D}_{\mathrm{v}} - \alpha$
  • From which the dispersion can be derived as:
\begin{equation} \frac{\mathrm{d} \lambda}{\mathrm{d} x} = \frac{10^6 \cos{\beta f_{2}}}{kn (f_{2}^2 + x ^2 )}\end{equation}
  • where
Grating.png
  • therefore
\begin{equation} \tan{\gamma} = \frac{m}{L}\end{equation}

  • so would $\gamma$ vary with $\alpha$ as $\gamma = \alpha + const$?
  • If so then from the grating equation an extra term could be added similar to:
\begin{equation} \lambda = 2 \frac{10^6}{k n} \sin{\left( \tan^{-1}\left(\frac{m}{L}\right) + const + \frac{\mathrm{D_{\mathrm{v}}}}{2}\right)} \cos{\left(\frac{\mathrm{D_{\mathrm{v}}}}{2}\right)} \end{equation}
  • However adding this term did not change the fit that much

Vega Lines

  • All of the lines that could be found in vega were fitted with the voigtian:


1.5mm.fit.png

2.0mm.fit.png

2.5mm1.fit.png

2.5mm2.fit.png

3.0mm.fit.png

3.5mm.fit.png

5.0mm.fit.png

5.5mm.fit.png

  • Of which our best fit was at 3.5mm:
3.5mm.fit.png

  • The parameters for the Voigtian fit are shown below:

Parameters 1.5mm 2.0mm 2.5mm1 2.5mm2 3.0mm 3.5mm 5.0mm 5.5mm
Amplitude -1.960.44 -1.940.78 -1.8240 -2.850.94 -2.1229 -96.71102 -1.146000? -2.722410?
Mean 1023 1011 1055 105.40.6 1004 96.50/64 811 85.70.56
Gaussian $\sigma$ 5.192 0.361.5 0.313 1.572.1 0.2298 0.0981 0.1952887? 0.1632
Skew $a_{0}$ -(0.00920.0002) (-9.330.05)e-03 (-9.518)e-03 (-8.958)e-03 (-9.78290)e-03 (10.30.8)e-03 -0.01250 (1.160.09)e-02
Lorentzian width 16.51.4 19.10.8 19.61 22.10.4 20.20.6 21.10.2 8.170.74 22.20.7
Circle radius 4.005.27 5.161.55 2.8637.3 2.610.6 3.812.16 2.475.6 3.973.01 4.001.6
$b_{0}$ 2.210.01 2.2020.001 2.110.01 3.000.001 2.370.003 3.6980.003 3.550.002 2.95323?
$b_{1}$ (1.620.008)e-03 (1.370.01)e-03 (1.360.01)e-03 (7.050.09)e-04 (5.990.17)e-04 (8.70.2)e-04 (-1.290.03)e-03 (-7.410.2)e-04
$b_{2}$ (-2.582)e-07 (3.710.25)e-07 (1.850.18)e-06 (-4.60.2)e-06 (-2.130.22)e-06 (-9.080.43)e-06 (1.170.04)e-05 (1.560.7)e-06
Slice [390:600] [340:550] [60:270] [550:764] [280:480] [10:200] [600:764] [600:764]
$\chi^2$ 809 835 896 750 773 621 626 599
$N_{\mathrm{dof}}$ 201 201 201 205 191 181 155 155
Reduced $\chi^2$ 4.02 4.15 4.46 3.65 4.04 3.4 4.04 3.86

  • Question about skew Gaussian?
\begin{equation} f(x; \mu, \sigma, a0) = \frac{1}{\sqrt{2 \pi} \sigma^2} e^{\frac{-((x-\mu)^2 + a0(x-\mu)^3)}{2 \sigma^2}}\end{equation}

-- JosephBayley - 06 Dec 2015

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Topic revision: r7 - 21 Jan 2016 - JosephBayley

 
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