Fitting

  • Gaussian:
\begin{equation} G(x;\mu,v) = \frac{1}{\sqrt{2\pi}V} e^{\frac{-(x - \mu)^2}{2 V}} \end{equation}
  • Lorentz:
 \begin{equation} L(x;\gamma) = \frac{\gamma}{4\pi^2} \frac{1}{(x)^2 + (\gamma / 4\pi)^2} \end{equation}
  • Circle:
 \begin{equation} C(x; r) = \frac{2 \sqrt{r^2-(x)^{2}}}{\pi r^2}. \end{equation}
  • Background:
 \begin{equation} B(x) = b_{0}+b_{1}(x-\mu)+b_{2}(x-\mu)^{2} \end{equation}

 \begin{equation}f(x;A,\mu,V,\gamma,r,b_0,b_1,b_2) = A((G \mathop{*} L) \mathop{*} C)(x) + B(x)\end{equation}

  • We changed how we were convoluting the functions so we are now using a discrete convolution:
 \begin{equation} (G \mathop{*} L \mathop{*}) (x) = \sum_j{\sum_i{G(x-j,\mu,v)L(j-i,\gamma)C(i,r)}} \end{equation}

  • For these plots the sums ran from -10 to 10 with 30 intervals.
  • The data ran from -10 to 10,

Parameter 1.5mm 2mm 2.5mm1 2.5mm2 3mm 3.5mm 5.5mm
Amp -0.4602 0.0268 -0.6582 0.4833 -0.3821 0.0146 -1.0856 0.0219 -0.7308 0.0652 -2.6147 0.4622 -1.0784 0.0091
$\mu$ 0.411 0.0191 0.2848 0.0179 0.2568 0.0166 2.3219 0.0075 0.284 0.0069 0.2254 0.0106 1.3405 0.0036
$V$ 0.151 0.0824 0.0251 0.0208 0.0743 0.0188 0.132 0.0392 0.0358 0.0067 0.0292 0.0078 0.0289 0.0008
$\gamma$ 2.3535 0.2115 2.1662 0.0692 2.1336 0.0921 2.032 0.0865 2.146 0.054 2.2882 0.026 2.1029 0.0496
$b_0$ 3.3185 0.006 3.2992 0.0033 3.1705 0.0033 4.4888 0.0041 3.5619 0.004 5.5419 0.0066 4.4172 0.0057
$b_1$ 0.022 0.0002 0.0181 0.0002 0.0179 0.0002 0.0101 0.0002 0.0081 0.0002 0.0103 0.0003 -0.0102 0.0004
$b_1$ -0.0003 0.0001 0.0003 0.0 0.0002 0.0 -0.0004 0.0 -0.0003 0.0001 -0.001 0.0001 0.0001 0.0001
$\chi^2$ 1065 1320 1779 1038 1031 743 794
$N_{\rm{dof}}$ 163 203 203 237 173 153 157
$\chi^2/n_{\rm{dof}}$ 6.53 6.5 8.76 4.38 5.96 4.86 5.05


1.5mmL.png

2.0mmL.png

2.51mmL.png

2.52mmL.png

3.0mmL.png

3.5mmL.png

5.5mmL.png
 

  • The number of number of times the sum was completed was changed and the range of the o sum was changed to see how the parameters errors changes however by increasing the number of times the loop was run the value of $\sigma$ decreased and the amplitude increased

  • In the range of -10 to 10:

# elements in sum 40 80 100 150 200 400
Amp -11.9882 0.1643 -3.7725 0.0438 -2.354 0.0301 -1.0657 0.0123 -0.6032 0.0069 -0.1517 0.0017
$\mu$ 0.1722 0.0052 0.1718 0.0051 0.1717 0.0051 0.1716 0.0051 0.1718 0.0051 0.1716 0.0051
$V$ 0.0862 0.012 0.0176 0.0014 0.0161 0.0091 0.0052 0.0006 0.0027 0.0002 0.0008 0.0002
$\gamma$ 1.3238 0.0246 1.4549 0.0159 1.4641 0.0213 1.4823 0.0157 1.4872 0.0156 1.4913 0.0156
$b_0$ 5.5229 0.007 5.5414 0.0065 5.5428 0.0069 5.5455 0.0065 5.5461 0.0065 5.5468 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
$b_1$ -0.0008 0.0001 -0.001 0.0001 -0.001 0.0001 -0.001 0.0001 -0.001 0.0001 -0.0011 0.0001

  • In he range -20 to 20:

Parameter 60 80 100 150 200 400
Amp -21.5978 0.3202 -11.7596 0.1602 -8.6932 0.1045 -5.4388 0.0636 -2.3362 0.0295 -0.6005 0.0069
$\mu$ 0.1745 0.0053 0.1721 0.0052 0.1717 0.0052 0.1716 0.0051 0.1717 0.0051 0.1717 0.0051
$V$ 0.1486 0.0154 0.0839 0.0117 0.0484 0.0049 0.023 0.0036 0.0158 0.0085 0.0028 0.0002
$\gamma$ 1.1582 0.0285 1.3298 0.0243 1.4186 0.0177 1.4929 0.0166 1.4651 0.0207 1.4868 0.0156
$b_0$ 5.5009 0.0071 5.5237 0.007 5.5362 0.0066 5.5471 0.0066 5.5429 0.0068 5.546 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
$b_1$ -0.0005 0.0001 -0.0008 0.0001 -0.0009 0.0001 -0.0011 0.0001 -0.001 0.0001 -0.001 0.0001

  • In the range -30 to 30:

Parameter 100 150 200 400
Amp -16.2015 0.2188 -8.657 0.104 -6.1456 0.1007 -1.325 0.016
$\mu$ 0.1729 0.0052 0.1717 0.0052 0.1717 0.0051 0.1717 0.0051
$V$ 0.113 0.0108 0.0481 0.0049 0.0371 0.0194 0.0091 0.0059
$\gamma$ 1.2454 0.0234 1.4196 0.0176 1.4638 0.0345 1.4727 0.0183
$b_0$ 5.5124 0.0068 5.5363 0.0066 5.5428 0.008 5.5441 0.0067
$b_1$ 0.0104 0.0003 0.0104 0.0003 0.0104 0.0003 0.0104 0.0003
$b_1$ -0.0006 0.0001 -0.0009 0.0001 -0.001 0.0001 -0.001 0.0001

  • In the range -40 to 40:

Parameter 100 150 200 400
Amp -16.2015 0.2188 -8.657 0.104 -6.1456 0.1007 -1.325 0.016
$\mu$ 0.1729 0.0052 0.1717 0.0052 0.1717 0.0051 0.1717 0.0051
$V$ 0.113 0.0108 0.0481 0.0049 0.0371 0.0194 0.0091 0.0059
$\gamma$ 1.2454 0.0234 1.4196 0.0176 1.4638 0.0345 1.4727 0.0183
$b_0$ 5.5124 0.0068 5.5363 0.0066 5.5428 0.008 5.5441 0.0067
$b_1$ 0.0104 0.0003 0.0104 0.0003 0.0104 0.0003 0.0104 0.0003
$b_1$ -0.0006 0.0001 -0.0009 0.0001 -0.001 0.0001 -0.001 0.0001

  • Also tried with the Integral method which used:
 \begin{equation} (G \mathop{*} L) (x) = \int^{\infty}_{-\infty} G(t)L(x-t)dt\end{equation}
  • This gave parameters of :
Parameter 1.5mm 2mm 2.5mm1 2.5mm2 3mm 3.5mm 5.5mm
Amp -14.0804 1.0931 -12.6003 0.6665 -10.659 0.6431 -32.8673 0.6888 -17.9033 0.7534 -58.1549 1.2883 -23.0793 0.8149
$\mu$ 0.4084 0.0186 0.3252 0.0159 0.2636 0.018 2.3199 0.0074 0.3074 0.0116 0.1712 0.0051 1.3656 0.0096
$\sigma$ 0.0802 0.1333 0.0431 0.0778 0.0607 0.0889 0.0921 0.0383 0.0521 0.0603 0.0654 0.0313 0.0156 0.043
$\gamma$ 1.5931 0.2005 1.3613 0.1279 1.2989 0.1467 1.3527 0.0573 1.354 0.1004 1.417 0.0523 1.3504 0.0767
$b_0$ 3.321 0.0077 3.2939 0.0049 3.1652 0.0047 4.493 0.0044 3.5581 0.0056 5.5259 0.0098 4.4057 0.0068
$b_1$ 0.022 0.0002 0.0183 0.0002 0.0179 0.0002 0.0096 0.0002 0.0081 0.0002 0.0104 0.0003 -0.0103 0.0004
$b_1$ -0.0003 0.0001 0.0004 0.0001 0.0003 0.0001 -0.0005 0.0 -0.0003 0.0001 -0.0007 0.0002 0.0002 0.0002
$\chi^2$ 1064.4 1238.7 1692.6 1028.49 1027.96 749.86 821.39
$N_rm{dof}$ 163 183 183 237.0 173.0 153.0 157.0
$\chi^2/n_{\rm{dof}}$ 6.530 6.768 9.249 4.34 5.942 4.901 5.232

  • This is just the Voigtian fit as adding in the circle took too long.
  • Mosty parameters look ok only $\sigma$
  • Could this be because is it much less that $r = 0.4$
  • also this fit gave odd features towards the edges:
    5.5N.png

  • Fit parameters physical relation:

  • For thermal broadening:
  • For a particle of mass m and Temperature T, the thermal width in the line of sight is:
\begin{equation} v_{\rm{th}} = \sqrt{\frac{2kT}{m}}\end{equation}
  • where $k$ is the Boltzmann constant.
  • Then in units of wavelength:
\begin{equation} \Delta \lambda_{\rm{th}} = \frac{\lambda_0}{c}\sqrt{\frac{2kT}{m}}\end{equation}

  • for Vega the thermal broadening should be:
Parameter $T$ [K] $k$ [$\rm{m}^2 \rm{kg} \rm{s}^{-2} \rm{K}^{-1}$] $m$ [kg]
Value 9602 $1.38 \times 10^{-23}$ $1.66 \times 10^{-27}$

  • The thermal width should then be:
$\lambda_0$ 410.2 434.1 486.1 656.93
$\Delta \lambda$ 0.01 0.0182 0.021 0.0277
$\sigma$ 0.387 0.158 0.363 0.167
    0.272 0.230  
      0.171  

  • The pressure broadening follows a Lorentzian distribution:
 \begin{equation} L(\lambda;\gamma) = \frac{\Gamma}{4 \pi^2} \frac{1}{(\lambda - \lambda_0)^2 + (\gamma/4\pi)^2} \end{equation}
  • Where
 \begin{equation} \gamma = \Gamma + 2\nu_{\rm{col}}\end{equation}
  • and $\Gamma$ is the Lorentzian width from the natural linewidth and $\nu_{\rm{col}}$ is the frequency of collision.

Errors

  • Took 10 frames at exposures of 0.1,0.3 and 0.5s for a line in Helium.
  • This was repeated for 0, -5,-10 and -15 degrees.
  • However we do not believe that we waited long enough for the lamp to warm up therefore the rms plot. *This plot is a -5 degrees

  • RMS1.png
  • This plot is at -15 degrees

  • RMS15.png
-- JosephBayley - 30 Jan 2016

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Topic revision: r14 - 20 Feb 2016 - AshleaKemp

 
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