20160111MSciPhotometryAnalysisOb1part5 < Public < RHUL Physics Department TWiki

Comparing stars between different filters

In order to compare the quality of our data, we looked at the data with the largest signal to noise ratio, in this case data taken with a green filter.
For this, we found the largest and smallest stars, as determined by the amount of pixels which constituted them.
The coordinates of the stars were recorded, so they can later be compared to the same stars in other filters.
For the fitting the backgrounds were removed, therefore the error on the no background data is the following (assuming error is Poissonian):

$\begin{equation} \rm{Data \ with \ background \ removed} = \sqrt{(\Delta \rm{Original \ data})^2 + (\Delta \rm{Background})^2}\end{equation} \begin{equation} \rm{Data \ with \ background \ removed} = \sqrt{ \rm{Original \ data} + \rm{Background}}\end{equation} <br />$

Below fits can be seen for the maximum and minimum size stars for all filters:

The plots are projections (they are summed in rows and columns), so to plot the errors the errors were summed in the following way:

$\begin{equation} \rm{Error \ projection} = \sqrt{\sum_i^{Number \ of \ pixels} \Delta p_i^2} \end{equation}$

where the $\Delta p_i$ is the error on the data point (pixel value).

Introductory $\chi^2$ estimation

Then set about finding $\chi^2$ for the above stars for all filters.

	Visual Brightest Star	Visual Dimmest Star	Green Filter Brightest Star	Green Filter Dimmest Star	Red Filter Brightest Star	Red Filter Dimmest Star}	Blue Filter Brightest Star	Blue Filter Dimmest Star
$\rm{Degrees \ of \ Freedom}$
$\chi^2$
$\rm{reduced} \chi^2$
$\rho$	$0.107 \pm 0.00518$	$0.118 \pm 0.0316$	$0.0286 \pm 0.00530$	$0.106 \pm 0.0485$	$0.0631 \pm 0.00473$	$0.0715 \pm 0.0294$	$0.00456 \pm 0.00625$	$0.0423 \pm 0.0358$
	$9399820\pm 43459$	$18450 \pm 441$	$9242014 \pm 42579$	$14155 \pm 522$	$12354155 \pm 51521$	$25730 \pm 573$	$12945440 \pm 70680$	$20255 \pm 547$
	$30.1 \pm 0.0201$	$30.3 \pm 0.0867$	$30.7 \pm 0.0248$	$31.0 \pm 0.158$	$30.5 \pm 0.0195$	$30.0 \pm 0.0855$	$30.3 \pm 0.0263$	$30.4 \pm 0.108$
	$31.7 \pm 0.0218$	$29.9 \pm 0.0902$	$31.7 \pm 0.021$	$30.4 \pm 0.130$	$30.8 \pm 0.0204$	$30.4 \pm 0.0904$	$31.6 \pm 0.0246$	$30.1 \pm 0.104$
$\sigma_x$	$4.18 \pm 0.0162$	$3.76 \pm 0.0870$	$5.16 \pm 0.020$	$4.51 \pm 0.160$	$4.51 \pm 0.0157$	$4.01 \pm 0.0861$	$4.66 \pm 0.0215$	$4.19 \pm 0.109$
$\sigma_y$	$4.51 \pm 0.0178$	$3.91 \pm 0.0907$	$4.41 \pm 0.0178$	$3.71\pm 0.131$	$4.69 \pm 0.0166$	$4.25 \pm 0.0909$	$4.34 \pm 0.0205$	$4.06 \pm 0.105$
	$159 \pm 8.36$	$-4.96 \pm 0.373$	$140 \pm 8.68$	$-3.69 \pm 0.416$	$173 \pm 9.505$	$-2.30 \pm 0.448$	$255 \pm 12.2$	$1.52 \pm 0.430$

In general, a good $\chi^2$ should be around equal to the number of degrees of freedom,
From the plots it is obvious that the Poisson errors are not correct as they are too small for the brightest star.
Also the brightest stars are poorly fitted and therefore have high $\chi^2$ in comparison to dimmest stars.

Testing curve_fit for better error estimation

In order to better obtain errors from curve_fit (as it tends to give outlandish errors), we set out to create a Gaussian of known parameters, and then fit a general Gaussian curve to this.

	Fractional Error When Guessing Incorectly	Fractional Error When Guessing Exactly
Correlation Coefficient		0
Amplitude	$1.87 \times 10^{-10}$	0
X Centre		0
Y Centre		0
$\sigma_x$	$1.25 \times 10^{-10}$	0
$\sigma_y$	$1.25 \times 10^{-10}$	0
Offset	$1.66 \times 10^{-10}$	0

Setting the threshold

We decided to use our mode image to set the threshold.
The mode image is shown below:

We took the Poissonian error (the square root) of each section.
We used the error as our threshold by setting the threshold:

$\begin{equation} \rm{threshold} = 3 \times \rm{Poisson \ error}\end{equation} <br />$

Standard deviation of the stars

We previously tried to show that all stars in the image have the same standard deviations.
To find our standard deviations we did the following:

$\begin{equation} \sigma_x = \sqrt{\frac{\sum_{i = 1}^N (x_i - x_{\rm center})^2 p_i}{\sum_{i=1}^N p_i}}\end{equation} \begin{equation} \sigma_y = \sqrt{\frac{\sum_{i = 1}^N (y_i - y_{\rm center})^2p_i}{\sum_{i=1}^N p_i}}\end{equation}$

We cropped an area of 20x20 pixels around each star and found the standard deviations within the area.
Histograms below are of the ratio of $\frac{\sigma_x}{\sigma_y}$ for Location 1 images:

for Location 1 images:

Ratio of $\frac{\sigma_x}{\sigma_y}$ for Location 2 images:

for Location 2 images:

Therefore for both locations we decided to set the requirement that an objects is only a star if its ratio:

$\begin{equation} 0.9 &lt; \frac{\sigma_x}{\sigma_y} &lt; 1.1 \end{equation} <br />$

Monte Carlo simulation of a 2D star field

We have done some prelimenary work on the simulation.
To generate the positions of stars we uniformly generated x and y pixel positions.
To create a background we generated Poisson distributed random variables around the mean of 160, which corresponded to a typical value for images of Location 1.
For each position we wanted to create a Gaussian distribution that represents a star centered on the star's x,y position.

$\begin{equation} f(x,y,\mu_x,\mu_y,\sigma_x,\sigma_y,\rho,a,d) = \frac{a}{2\pi\sigma_x\sigma_y\sqrt{(1-\rho^2)}} exp\Bigg\{-\frac{1}{2(1-\rho^2)}\Bigg[\bigg(\frac{x-\mu_x}{\sigma_x}\bigg)^2 + \bigg(\frac{y-\mu_y}{\sigma_y}\bigg)^2 -2\rho\bigg(\frac{x-\mu_x}{\sigma_x}\bigg)\bigg(\frac{y-\mu_y}{\sigma_y}\bigg)\Bigg]\Bigg\} \end{equation} <br />$

As the standard deviations should be the same for each star we decided to generate only one value for $\sigma$ for all stars. We also assumed that the standard deviations are the same in both directions so we generated only one $\sigma$ for $\sigma_x$ and $\sigma_y$ .
To experiment we uniformly generated values for and $\rho$ for each individual star.
We then for each pixel in the image calculated the Gaussian value at that particular pixel for each star.
Resulting image for 50 stars is shown below:

-- ElenaCukanovaite - 11 Jan 2016

Attachments

Topic attachments
I	Attachment	History	Action	Size	Date	Who
png	30_by_30.png	r1	manage	52.9 K	13 Jan 2016 - 11:31	ElenaCukanovaite
png	None_1_max.png	r1	manage	113.1 K	13 Jan 2016 - 13:12	DavidHadden
png	None_1_min.png	r1	manage	109.8 K	13 Jan 2016 - 13:13	DavidHadden
png	blue_1_max.png	r1	manage	120.1 K	13 Jan 2016 - 13:11	DavidHadden
png	blue_1_min.png	r1	manage	115.9 K	13 Jan 2016 - 13:12	DavidHadden
png	elena_b_b.png	r1	manage	104.1 K	14 Jan 2016 - 19:51	ElenaCukanovaite
png	elena_b_d.png	r1	manage	104.0 K	14 Jan 2016 - 19:51	ElenaCukanovaite
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png	elena_g_d.png	r1	manage	98.1 K	14 Jan 2016 - 19:40	ElenaCukanovaite
png	elena_no_br.png	r1	manage	98.9 K	14 Jan 2016 - 19:12	ElenaCukanovaite
png	elena_no_d.png	r1	manage	100.1 K	14 Jan 2016 - 19:12	ElenaCukanovaite
png	elena_r_b.png	r1	manage	103.5 K	14 Jan 2016 - 19:57	ElenaCukanovaite
png	elena_r_d.png	r1	manage	105.1 K	14 Jan 2016 - 19:58	ElenaCukanovaite
png	green_1_max.png	r1	manage	121.4 K	13 Jan 2016 - 13:12	DavidHadden
png	green_1_min.png	r1	manage	107.0 K	13 Jan 2016 - 13:12	DavidHadden
png	location_2.png	r1	manage	50.5 K	14 Jan 2016 - 16:16	ElenaCukanovaite
png	mode_sec.png	r1	manage	46.3 K	13 Jan 2016 - 11:42	ElenaCukanovaite
png	monte_carlo.png	r1	manage	278.0 K	13 Jan 2016 - 13:38	ElenaCukanovaite
png	red_1_max.png	r1	manage	119.8 K	13 Jan 2016 - 13:12	DavidHadden
png	red_1_min.png	r1	manage	117.4 K	13 Jan 2016 - 13:12	DavidHadden
png	sigma_x_loc_1.png	r1	manage	66.3 K	14 Jan 2016 - 16:13	ElenaCukanovaite
png	sigma_x_loc_2.png	r1	manage	46.8 K	14 Jan 2016 - 16:17	ElenaCukanovaite
png	stan_dev_n_t.png	r1	manage	70.1 K	13 Jan 2016 - 16:13	ElenaCukanovaite

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Topic revision: r11 - 14 Jan 2016 - ElenaCukanovaite