Vega Fitting

  • Gaussian:
\begin{equation} G(x;V) = \frac{1}{\sqrt{2\pi \sigma^2}} e^{\frac{-(x - \mu)^2}{2 \sigma^2}} \end{equation}
  • Lorentz:
\begin{equation} L(x;\gamma) = \frac{1}{\pi} \frac{\gamma/2}{(x)^2 + (\gamma/2)^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 was convolution the functions so they were convoluted with:
\begin{equation} (G \mathop{*} L) (x) = \int^{\infty}_{-\infty} G(t)L(x-t)dt\end{equation}

* Initially a Voigtian with a quadratic background was fitted:


1.5mm.png
2mm.png

2.51mm.png
2.52mm.png

3mm.png
3.5mm.png

5mm.png
5.5mm.png

  • The parameters are:
Parameter 1.5mm 2mm 2.5mm1 2.5mm2 3mm 3.5mm 5mm 5.5mm
Amp -1.473 ± 0.0876 -1.372 ± 0.0392 -1.1993 ± 0.0649 3.3295 ± 0.0658 -1.8915 ± 0.0786 -6.3556 ± 0.1028 -0.4566 ± 0.0362 -2.8105 ± 0.131
$\mu$ 411.75 ± 0.2529 435.57 ± 20.6697 435.4 ± 13.8989 486.79 ± 0.0616 487.6 ± 5.8552 485.7 ± 2.9217 630.07 ± 9.6767 656.93 ± 3.9025
$\sigma$ 0.418 ± 0.0185 0.3223 ± 0.0159 0.264 ± 0.0184 2.3217 ± 0.0074 0.3059 ± 0.0116 0.7459 ± 0.0051 -0.0414 ± 0.0179 1.3437 ± 0.0105
$\gamma$ 2.6006 ± 0.2642 2.3261 ± 0.1365 2.3387 ± 0.2186 -2.1226 ± 0.0873 2.2822 ± 0.1567 2.4722 ± 0.0667 0.9703 ± 0.1508 2.5449 ± 0.1722
$b_0$ 3.3207 ± 0.004 3.2967 ± 0.372 3.1693 ± 0.2501 4.4704 ± 0.0035 3.5606 ± 0.0447 5.5482 ± 0.0338 5.3326 ± 0.1933 4.4586 ± 0.0312
$b_1$ 0.0228 ± 0.0002 0.0179 ± 0.012 0.0178 ± 0.0056 0.0121 ± 0.0002 0.0083 ± 0.0034 0.0131 ± 0.0063 -0.0197 ± 0.0343 -0.0098 ± 0.0023
$b_1$ -0.0003 ± 0.0001 0.0003 ± 0.0 0.0002 ± 0.0 -0.0005 ± 0.0 -0.0003 ± 0.0001 -0.0011 ± 0.0001 0.0018 ± 0.0001 -0.0003 ± 0.0002
Balmer Lines $\rm{H}{\delta}$ $\rm{H}{\gamma}$ $\rm{H}{\gamma}$ $\rm{H}{\beta}$ $\rm{H}{\beta}$ $\rm{H}{\beta}$   $\rm{H}{\alpha}$
  410.2 434.1 434.1 486.1 486.1 486.1   656.93

  • then the Circle was convoluted numerically and this was fit:


1.5mmC.png
2mmC.png

2.51mmC.png
2.52mmC.png

3mmC.png
3.5mmC.png

5mmC.png
5.5mmC.png

  • with parameters:

Parameter 1.5mm 2mm 2.5mm1 2.5mm2 3mm 3.5mm 5mm 5.5mm
Amp -0.1557 ± 0.0094 -0.1433 ± 0.0074 -0.143 ± 0.008 -0.4475 ± 0.0101 -0.197 ± 0.0074 -0.7377 ± 0.0122 -0.0417 ± 0.0041 -0.3193 ± 0.0182
$\mu$ 411.37 ± 0.4199 435.57 ± 10.3752 435.40 ± 7.7568 486.49 ± 2.6906 487.61 ± 25.8145 485.72 ± 2.3928 630.07 ± 6.9484 656.94 ± 7.2942
$\sigma$ 0.3642 ± 0.0185 0.2687 ± 0.0159 0.2063 ± 0.0184 2.2772 ± 0.009 0.253 ± 0.0117 0.6885 ± 0.0051 -0.0874 ± 0.0189 1.2608 ± 0.0115
$\gamma$ 2.5797 ± 0.2675 2.2258 ± 0.2075 2.2606 ± 0.2301 3.0416 ± 0.1174 2.1832 ± 0.1464 2.3373 ± 0.0683 0.7997 ± 0.2153 2.9078 ± 0.2311
$b_0$ 3.3116 ± 0.0142 3.2951 ± 0.1889 3.1682 ± 0.141 4.5268 ± 0.0334 3.5583 ± 0.2183 5.5309 ± 0.0268 5.3301 ± 0.1341 4.4988 ± 0.0742
$b_1$ 0.023 ± 0.0003 0.0179 ± 0.0063 0.0178 ± 0.0032 0.0137 ± 0.0049 0.0083 ± 0.0138 0.013 ± 0.0045 -0.0197 ± 0.0254 -0.0087 ± 0.0163
$b_1$ -0.0003 ± 0.0001 0.0003 ± 0.0001 0.0002 ± 0.0 -0.0009 ± 0.0 -0.0003 ± 0.0001 -0.0009 ± 0.0001 0.0018 ± 0.0001 -0.0011 ± 0.0002
Balmer Lines $\rm{H}{\delta}$ $\rm{H}{\gamma}$ $\rm{H}{\gamma}$ $\rm{H}{\beta}$ $\rm{H}{\beta}$ $\rm{H}{\beta}$   $\rm{H}{\alpha}$
  410.2 434.1 434.1 486.1 486.1 486.1   656.93

Errors

  • Looked at multinomial errors:
\begin{equation} V[p_i] = \frac{V[n_i]}{N^2} = \frac{p_i(1-p_i)}{N}\end{equation}
  • Where
\begin{equation} p_i = \frac{n_i}{N}\end{equation}
  • Therefore:
\begin{equation} \sigma_{p_i} = \sqrt{\frac{p_i(1-p_i)}{N}}\end{equation}
\begin{equation} \sigma_{n_i} = N\sigma_{p_i}\end{equation}


  • RMS.png

Other targets

Name Spectral Type Temperature [K] Rotational Velocity [km/s]
Rigel B8 12100 25
$\alpha$ - Cephei A8Vn 7740 246
Bellatrix B2 22 000 46
Betelgeuse M2 3500 5

-- JosephBayley - 24 Jan 2016

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Topic revision: r9 - 19 Feb 2016 - JosephBayley

 
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