Errors

  • Took 10 frames of one spectral line from Helium lamp at exposure times between 0-0.5s in 0.05s intervals.
  • 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 plotting the RMS of each pixel column against the mean of each pixel column the plot below was obtained.
    RMS.png
  • A straight line equation of the form $y = a + b x$ was fit to this with parameters: $a = 271.82$ and $b = 0.003356$
  • This equation can now be used to propagate errors through in the Vega fits, as we found that poisson errors were incorrect.

Mapping

  • For the mapping the quadratic equating was replaced with a trigonometric one
\begin{equation}10^{-6} k n \lambda = \sin{\alpha} + \cos{\beta} = 2\sin{\frac{\beta+\alpha}{2}}\cos{\frac{\beta-\alpha}{2}} \end{equation}
  • From the spectrometers manual we know that:
\begin{equation} \rm{D}_v = \beta-\alpha = \rm{D}_{x = 0} + \tan^{-1}{\left(\frac{x}{f_2}\right)} \end{equation}
  • As the position is measured in pixels this need to be in mm and zero should be at the center. The manual for the camera says that each pixel is 9x9 µm, therefore to get in mm:
\begin{equation} x = (p-382)(9 \times 10^{-3}) [\rm{mm}] \end{equation}
  • to relate the angle $\alpha$ to the micrometer position the figure below was used:

  • Grating.png
  • here the angle $\gamma$ and $\alpha$ were related by:
\begin{equation} \gamma = \alpha + \alpha_0\end{equation}
  • This can then be related to the micrometer position by:
\begin{equation} \tan{\gamma} = \tan{\left( \alpha + \alpha_0 \right)} = \frac{m}{l} \end{equation}
  • however the micrometer scale for the grating moved in the opposite direction to the length $m$, therefore:
\begin{equation}m = m_{0} - m_{\mathrm{mic}}\end{equation}
  • where $m_{\mathrm{mic}}$ is the micrometer setting.
  • These can then all be put into the initial equation:
\begin{equation} \lambda = 2\frac{10^6}{kn} \sin{\left[ \tan^{-1}{\left( \frac{m_0 - m}{l} \right)} -\alpha_0+ \frac{1}{2} \left(\rm{D}_{x = 0} + \tan^{-1}{\left(\frac{(p-382)(9 \times 10^{-3})}{f_2}\right)} \right) \right]} \cos{\left[\frac{1}{2} \left( \rm{D}_{x = 0} + \tan^{-1}{\left(\frac{(p-382)(9 \times 10^{-3})}{f_2}\right)}\right)\right]} \end{equation}
  • We know that its first order, so $k=1$, the groove density is $n = 1200 \mathrm{gr/mm}$ and the constant $\rm{D}_{x = 0} = 30$.
  • We initally guessed the parameters by measuring the angles and distances inside the spectrometer. By then fitting this equation to the data we get the parameters:

Parameter Guess Trig Fitted Trig Guess Polynomial Fitted Polynomial
$l$ 30 20.24±0.099 mm    
$\alpha_0$ 10 11.37±0.038 degrees    
$m_0$ 7 4.87±0.79 mm    
$f_2$ 10 67.4±1.17 mm    
$\lambda_0$     278 264±3
$a_1$     0.1 0.12±0.01
$b_1$     50 62±1
$a_2$     0 (-1±0.1)e-05
$b_2$     0 (-0.68±0.01)
$c_2$     0 (-1±0.1)e-03
$\chi^2/\mathrm{DOF}$   10.61   9.68


  • both.png

Vega Fitting


V1.5mm.png
V2mm.png

V2.51mm.png
V2.52mm.png

V3mm.png
V3.5mm.png

V5mm.png
V5.5mm.png

Parameters 1.5mm 2.0mm 2.5mm1 2.5mm2 3.0mm 3.5mm 5.0mm 5.5mm
Amplitude -0.69±36460 -1.17±3252 -0.53±35121 -1.97±1.19 -1.08±8.5 -4.21±38 -0.52±0.77 -16.6±1188411
Mean 397±0.1 427±0.7 458±0.1 458±0.01 486±1 515±1 597±0.5 627.3±0.02
Gaussian $\sigma$ 0.05±2459 0.06±100 0.03±771

0.16±0.3

 0.16±4.1 0.18±6 0.22±2.8 0.046±615
Lorentzian width 7.82±0.31 6.73±0.77 6.19±0.3 7.31±0.29 7.06±0.8 7.53±0.18 3.51±1.3 10.1±0.2
$b_{0}$ 3.31±0.01 3.31±0.01 3.16±0.01 4.49±0.01 3.55±0.006 5.52±0.01 5.33±0.005 4.42±0.01
$b_{1}$ (6±0.05)e-03 (4.9±0.6)e-03 (5.12±0.04)e-03 (2.71±0.05)e-03 (2.17±0.05)e-03 (1.2±0.01)e-03 (-3.96±0.01)e-03 (-2.03±0.06)e-03
$b_{2}$ (-5.67±2)e-06 (2.72±0.3)e-05 (2.16±0.29)e-06 (-3.7±0.4)e-06 (-1.38±0.4)e-06 (-6.71±0.49)e-06 (7.17±0.05)e-06 (8.83±0.5)e-06
Slice [390:600] [340:550] [60:270] [550:764] [280:480] [10:200] [600:764] [600:764]
$\chi^2$ 384 394 430 340 367 299 270 270
$N_{\mathrm{dof}}$ 203 203 203 207 193 183 157 157
Reduced $\chi^2$ 1.89 1.94 2.12 1.64 1.9 1.63 1.72 1.72


  • V3.5mmL.png
  • However the mean from the fit is correct, the vertical line in the image shows 517 however the mean from the fit is 515.
  • This is to do with how we are convoluting the functions, the mean in both convolution below is 545

  • convolution.png


  • Zconv.png
  • The mode 'full' seems to convolute them with the mean in the correct place, however I cannot fit this as it is a different length array
  • The mode 'same' is the right length array, however the way that it is adjusted changes the position of the mean.
-- JosephBayley - 11 Jan 2016
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Topic revision: r8 - 24 Feb 2016 - AshleaKemp

 
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