Fitting

  • Gaussian:
\begin{equation} G(\lambda;\lambda_0,\sigma) = \frac{1}{\sqrt{2\pi} \sigma^2} e^{\frac{-(\lambda - \lambda_0)^2}{2 \sigma^2}} \end{equation}
  • Lorentz:
 \begin{equation} L(\lambda;\gamma) = \frac{1}{\pi} \frac{\gamma/2}{(\lambda)^2 + (\gamma / 2)^2} \end{equation}
  • Circle:
 \begin{equation} C(\lambda; r) = \frac{2 \sqrt{r^2-(\lambda)^{2}}}{\pi r^2}. \end{equation}
  • Background:
 \begin{equation} B(\lambda) = b_{0}+b_{1}(\lambda-\lambda_0)+b_{2}(\lambda-\lambda_0)^{2} \end{equation}

 \begin{equation}f(\lambda;A,\lambda_0,\sigma,\gamma,b_0,b_1,b_2) = A((G \mathop{*} L) \mathop{*} C)(\lambda) + B(\lambda)\end{equation}

  • Now the equation were normalised correctly we used the discrete convolution,
 \begin{equation} (G \mathop{*} L \mathop{*}C) (\lambda) = \sum_j{\sum\limits_{i}{G(\lambda-j,\lambda_0,\sigma)L(j-i,\gamma)C(i,r)}} \end{equation}

Parameters $\rm{H}_{\epsilon}$ $\rm{H}_{\delta}$ $\rm{H}_{\gamma}$ $\rm{H}_{\beta}$ $\rm{O}_2$ $\rm{H}_{\alpha}$
Amp -1.0126 0.1667 -1.2144 0.0785 -1.2313 0.0664 -5.9029 0.0742 -0.4894 0.0296 -2.59 0.0644
$\lambda_0$ 399.6 0.0301 413.1 0.0185 435.9 0.0159 486.5 0.0051 630.1 0.0206 658.3 0.0098
$\sigma$ 0.2319 0.3356 0.203 0.1328 0.0196 0.0785 0.0157 0.0082 0.0102 0.0053 0.012 0.0165
$\gamma$ 2.6072 0.7141 2.041 0.288 2.1042 0.2034 2.3018 0.0321 0.9542 0.0821 2.1649 0.0626
$b_0$ 3.0666 0.0115 3.3082 0.0052 3.2936 0.005 5.543 0.0068 5.3445 0.0043 4.4195 0.0057
$b_1$ 0.0219 0.0003 0.0225 0.0002 0.0184 0.0002 0.0104 0.0003 -0.0201 0.0004 -0.0102 0.0004
$b_1$ -0.0012 0.0002 -0.0001 0.0001 0.0005 0.0001 -0.001 0.0001 0.0012 0.0001 0.0 0.0001
$\chi^2$ 1071.8 1252.0 1237.5 748.5 722.8 788.3
$N_\mathrm{dof}$ 143 203 183 153 157 157
$\chi^2/n_{\rm{dof}}$ 7.495 6.167 6.762 4.892 4.604 5.021
$\lambda_0 - \lambda_{\rm{true}}$ 2.6 2.9 1.8 0.4 2.44 2.0

  • The plots were:

Hepsilon.png

Hdelta.png

Hgamma.png
br /> Hbeta.png

O2.png

Halpha.png

  • Tried running the discrete convolution with more points, now that it is normalised the amplitude stays the same.
  • The $\sigma$ however seems to decrease with the increased number of points

Parameter 100 200 300 400 500 600
Amp -5.505 0.0662 -5.9177 0.0748 -6.0392 0.0697 -6.0839 0.0701 -6.1048 0.0717 -6.1207 0.0705
$\lambda$ 0.1717 0.0052 0.1717 0.0051 0.1716 0.0051 0.1717 0.0051 0.1716 0.0051 0.1716 0.0051
$\sigma$ 0.0484 0.0049 0.0158 0.0085 0.0053 0.0006 0.0028 0.0002 0.0028 0.0038 0.0014 0.0002
$\gamma$ 2.2283 0.0277 2.3013 0.0325 2.3283 0.0246 2.3355 0.0245 2.3368 0.0263 2.3406 0.0245
$b_0$ 5.5362 0.0066 5.5429 0.0068 5.5454 0.0065 5.546 0.0065 5.5462 0.0066 5.5466 0.0065
$b_1$ 0.0104 0.0003 0.0104 0.0003 0.0104 0.0003 0.0104 0.0003 0.0104 0.0003 0.0104 0.0003
$b_1$ -0.0009 0.0001 -0.001 0.0001 -0.001 0.0001 -0.001 0.0001 -0.001 0.0001 -0.0011 0.0001
$\chi^2$ 100792.4 100646.9 100588.6 100589.4 100566.6 100564.5
$N_{\rm{dof}}$ 153 153 153 153 153 153
$\chi^2/n_{\rm{dof}}$ 658.774 657.823 657.442 657.447 657.298 657.284

Full Spectrum

  • This is the full spectrum stitched together and smoothed by convoluting a gaussian with a width of 2nm :

  • FullSpectrum.png
  • The blue line is the spectrum of Vega from elodie, and the red line is the elodie spectrum convoluted with our instrumental width.

  • ElodieSpectrum.png
  • This is sections of the spectrum normalised to the continuum.

  • Norm.png

Errors

  • RMS is falling at higher means
  • Thought it could be that the pixels are reaching saturation however it is happening at lower mean pixel values as well.

  • RMS.png
-- JosephBayley - 11 Feb 2016

Physics WebpagesRHUL WebpagesCampus Connect • Royal Holloway, University of London, Egham, Surrey TW20 0EX; Tel/Fax +44 (0)1784 434455/437520

Topic revision: r2 - 11 Feb 2016 - JosephBayley

 
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