Relative Depths Vega

  • To find a measure of the strength of a spectral line the area underneath the spectral line is found.

  • Equivalent width is a measure of the strength of a line and is defined as:
 \begin{equation} W = \int^{\lambda_1}_{\lambda_2} \frac{F_c - F_{\lambda}}{F_c} d\lambda, \end{equation}
  • where $F_c$ is a continuum function and $F_{\lambda}$ is the line profile.
 \begin{equation} F_c(\lambda) = b_{0}+b_{1}(\lambda-\lambda_0)+b_{2}(\lambda-\lambda_0)^{2} \end{equation}

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

Line $H_{\epsilon}$ $H_{\delta,1}$ $H_{\gamma,1}$ $H_{\gamma,1}$ $H_{\beta,1}$ $H_{\beta,1}$ $H_{\beta,1}$ $O_{2,1}$ $H_{\alpha,1}$
Area 0.938 1.209 1.212 1.049 3.482 1.937 6.311 0.533 2.454
$W$ 0.3068 0.3653 0.3677 0.3307 0.7762 0.5438 1.139 0.0953 0.5547


HepsilonNorm.png

HdeltaNorm.png

HgammaNorm.png

HbetaNorm.png
 
HalphaNorm.png

Quasar

  • To find the wavelength compared to position, the equation below was used,
 \begin{equation} \lambda = \frac{1}{n}\sin{\left\{ \tan^{-1}{\left( \frac{0.036p}{l}\right)}\right\}}. \end{equation}

  • First the four visible lines within Denebolas line were fit, the parameters for the fit are shown below,

DenebHGam.png

DenebHBet.png

DenebHAlp.png

DenebO2.png

Parameter $H_{\gamma}$ $H_{\beta} $ $O_{2}$ $H_{\alpha}$
$A$ $-634 \pm 112$ $-568113 \pm 4376$ $-355583 \pm 24289 $ $-581 \pm 76$
$\mu$ $64.63 \pm 0.72$ $83.17 \pm 0.22$ $97.78 \pm 0.18$ $114.1 \pm 0.5$
$\sigma$ $0.65 \pm 0.83$ $0.0188 \pm 0.0059$ $0.675 \pm 0.335$ $0.836 \pm 0.166$
$\gamma$ $0.0003 \pm 0.038$ $21.84 \pm 1.25$ $0.006 \pm 0.049$ $0.0002 \pm 0.029$
$b_0$ $50702\pm 4493$ $98530 \pm 550$ $77395 \pm 257$ $30471 \pm 2782$
$b_1$ $6223v\pm 1605$ $469.8 \pm 6.1$ $-3266 \pm 22$ $-2069 \pm 1325$
$b_2$ $111 \pm 39$ $-10.71 \pm 0.27$ $-29.59 \pm 12.1$ $49.39 \pm 0.62$
$\chi^2_{\rm{red}}$ 12.9 9195.6 10.18 16.545

  • The positions of each of these lines, $\mu$, could then be used with their actual wavelengths and fit with the grating equation above.
  • This gave a plot:

  • Map.png:
    Map.png

  • The parameters n and l were found to be n = 209.5 and l to be 25.44 mm.
  • where n in the slit density and l is the distance of the grating to the CCD.
  • however the grating is known to have a slit density of 200
  • therefore is this is set then the distance l is found to be 26.67 mm

Scan of peak

  • trying to find another function to fit to the scan of the peak,
  • Initially a parabola was used with the function,
 \begin{equation} f(x) = a_{0}+a_{1}(x-x_0)+a_{2}(x-x_0)^{2} \end{equation}

  • then a cosh function was used,
 \begin{equation} f(x) = a_{0}+a_{1}(\cosh{\{a_2(x-x_0)\}}) \end{equation}

-- JosephBayley - 03 Mar 2016

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Topic revision: r6 - 04 Mar 2016 - JosephBayley

 
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