- Found previously that errors in background of frame are not poisson - wanted to investigate how points fluctuate in peaks as well as background.
- Plotted the spectrum of Helium lamp at micrometer position 6.5mm, exposure time 0.5s, to find pixel positions of background and peaks

- Chose 3 random pixel positions of background: 150, 350, 550
- Chose 3 pixel positions where the two peaks is present: 245 (1st peak), 255 (1st peak), 655 (2nd peak)
- Analysed the point at these pixel positions to see how they fluctuated in each of the 20 frames

- For each frame, calculated the mean of the 20 data points , the calculated standard deviation of the data points , and the expected poisson error of the data points,

Pixel Position | ||||

150 (back) | 520.96 | 2.34 | 1.43 | 1.64 |

350 (back) | 538.51 | 2.09 | 1.45 | 1.44 |

550 (back) | 560.17 | 2.42 | 1.48 | 1.64 |

245 (peak) | 1928.15 | 18.34 | 2.75 | 6.67 |

255 (peak) | 4016.91 | 31.86 | 3.97 | 8.03 |

655 (peak) | 3964.86 | 28.23 | 3.94 | 7.16 |

- Found that for calibration lamps, standard deviation not equivalent to . Wanted to find a way to assign error bars
- Using Helium lamp, micrometer position at 4.0mm, took 10 frames at each exposure time, starting at 0.05s up to 1s in 0.05s increments. For each of the 10 frames of each exposure time, chose 5 points close together that were all in background (pixel numbers: 170,180,190,200,210) and 5 points close together in the peak (pixel numbers: 185,190,195,200,205). Found the of the 10 frames for each of these points, then found for that exposure time
- Plotted as a function of exposure time. Expected to be able to fit a function to these points, such that for a specific exposure time, can find the appropriate error.
- However, the plot was not what we expected - the points are random, a function cannot be fitted.
- Unsure what is the best method in assigning error bars?

- Firstly a Gaussian convoluted with a circular function was fitted to a spectral line
- Then as towards the edges of the frame the line appeared to become skewed therefore a skewed Gaussian was convoluted with the circular function and fitted

where

- Finally a quadratic term was fitted for the background of the image, which has the equation:

- These are plotted below

- The quadratic can be seen to improve the fit to the background of the data as shown below:

- The three different fits produced parameter values of:

Parameters | Gaussian + circle | Skewed Gaussian + Circle | Skewed Gaussian + circle + Quadratic background |

Gaussian amplitude | 369 ±(7.8e6) | 417±(7.7e05) | 447±(4.7e06) |

Gaussian mean | 33.34±0.001 | 14.2±(5e04) | 13.3±(1e04) |

Gaussian Sigma | 1.99±0.001 | 2.19±(2e04) | 3.14±(2.06e04) |

Circle radius | 4.16±0.01 | 4.46±0.01 | 4.43±0.00001 |

Circle mean | 33.34±0.001 | 58.6±0.02 | 58.4±0.01 |

Circle amplitude | 365±(7.7e6) | 422±(4.2e06) | 532±(5e06) |

Background | 624.7±0.2 | 618±0.2 | |

8.18±(5e04) | 4.23±(3e04) | ||

0.034±308 | 0.264±1742 | ||

0.077±705 | 0.008±581 | ||

612±0.3 | |||

-3.54±0.003 | |||

-0.069±0.001 | |||

191158 | 21259 | 14404 | |

64 | 60 | 58 | |

per DOF | 2987 | 354 | 248 |

- The line chosen was at a micrometer setting of 3.5mm, the exposure was 120s,
- Initially the convolution of the skewed Gaussian and circle plus the quadratic background was fitted to the Vega line, this had the equations same as above.
- Then to get a better fit the skewed Gaussian was convoluted with a Lorentz distribution so that a Voigt profile could be fit. The Lorentz had the equation:

- Both of these fits are shown with the data in the plot below:

Paramater | Skewed Gaussian + Circle + Quadratic Background | Voigt + Circle + Quadratic Background |

Gauss Amp | 1.077 | -1.48±(1.39e05) |

Gauss mean, | 130 | 138±(4.2e06) |

Gauss | 9.33 | 19.7±(3.1e05) |

4.3 | -15.6±(1.6e06) | |

0.26 | -5.44±(3.93e04) | |

0.088 | 0.0982±1226 | |

Lorentz Amp, | -7.43±(6.8e05) | |

Lorentz width, | 18.69±0.29 | |

Lorentz center, | 204±1.86 | |

Circle radius | 5.98 | 2.39±286 |

Circle mean | 130 | 128±231 |

Circle Amplitude | -21.7 | -30.2±(7.1e05) |

3.64 | 3.68±0.19 | |

0.00088 | (8.42±14.7)e-04 | |

-8.49e-07 | (-3.18±0.07)e-06 | |

1369 | 1115 | |

288 | 285 | |

per dof | 4.75 | 3.91 |

- The three different equations that were fitted were:

- These were then plotted for each micrometer setting with the data.
- Ask about what errors to put in??
- The parameters were found to be:

Parameter | Linear | Squared | Cubed |

278±2 | 265±2.8 | 275±7 | |

0.105±0.002 | 0.122±0.008 | 0.074±0.028 | |

55.64±0.26 | 62.07±1.08 | 56.75±4.45 | |

(-1.19±0.89)e-05 | (6.73±6.16)e-05 | ||

-0.689±0.104 | 0.08±0.94 | ||

(-1.61±0.98)e-03 | 0.011±0.007 | ||

(-4.62±4.81)e-08 | |||

(-3.35±6.44)e-02 | |||

(-7.59±6.11)e-06 | |||

(-7.63±5.07)e-04 | |||

109524 | 63944 | 60107 | |

69 | 66 | 62 | |

per dof | 1587 | 968 | 969 |

- Some of the grating setting are shown below

- For the halogen lamp the above fit parameters were used to set the wavelength of each frame, these were then plotted as shown below:
- The below plot shows the linear and squared plots and it can clearly be seen to improve at larger and smaller wavelengths, however not much change in the center.

- When the cubed parameters were used, the plot did no seem to improve, as the image shows

- This may be due to an incorrect point which the cubed plot fits better to however is further from the true value

I | Attachment | History | Action | Size | Date | Who | Comment |
---|---|---|---|---|---|---|---|

png | 150.png | r2 r1 | manage | 36.3 K | 27 Nov 2015 - 18:09 | AshleaKemp | Pixel position 150 |

png | 255.png | r1 | manage | 33.4 K | 27 Nov 2015 - 18:11 | AshleaKemp | Pixel position 255 |

png | errorback.png | r2 r1 | manage | 47.3 K | 29 Nov 2015 - 17:47 | AshleaKemp | |

png | errorpeak.png | r1 | manage | 41.0 K | 29 Nov 2015 - 17:46 | AshleaKemp | |

png | Helium.png | r2 r1 | manage | 20.0 K | 27 Nov 2015 - 17:56 | AshleaKemp | |

png | sigma.png | r1 | manage | 15.6 K | 01 Dec 2015 - 11:23 | AshleaKemp | |

png | sigmapeak.png | r1 | manage | 17.7 K | 01 Dec 2015 - 11:57 | AshleaKemp | |

png | spectrum.png | r2 r1 | manage | 29.1 K | 29 Nov 2015 - 18:01 | AshleaKemp |

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Topic revision: r7 - 01 Dec 2015 - JosephBayley

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