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
	3.0666 ± 0.0115	3.3082 ± 0.0052	3.2936 ± 0.005	5.543 ± 0.0068	5.3445 ± 0.0043	4.4195 ± 0.0057
	0.0219 ± 0.0003	0.0225 ± 0.0002	0.0184 ± 0.0002	0.0104 ± 0.0003	-0.0201 ± 0.0004	-0.0102 ± 0.0004
	-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:


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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
	5.5362 ± 0.0066	5.5429 ± 0.0068	5.5454 ± 0.0065	5.546 ± 0.0065	5.5462 ± 0.0066	5.5466 ± 0.0065
	0.0104 ± 0.0003	0.0104 ± 0.0003	0.0104 ± 0.0003	0.0104 ± 0.0003	0.0104 ± 0.0003	0.0104 ± 0.0003
	-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 :
The blue line is the spectrum of Vega from elodie, and the red line is the elodie spectrum convoluted with our instrumental width.
This is sections of the spectrum normalised to the continuum.

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.

-- JosephBayley - 11 Feb 2016

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Topic revision: r2 - 11 Feb 2016 - JosephBayley

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