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<h1>Measuring the Reddening of M42 using the Balmer Decrement</h1>
<p class="author">
  Eric Cai<sup>1</sup><br>
  <em>1</em> Rice University, ericwiththec@rice.edu<br>

  <em>Date:</em> May 2, 2023<br>
  <em>Department:</em> Physics and Astronomy
</p>

<hr>

<h2>Introduction</h2>
<p class="paragraph">
This paper looks into the Balmer lines of Orion Nebula, M42, and attempted to measure the reddening of this emission nebula. The paper utilizes the Balmer decrement to get a relative ratio between the Balmer lines in the nebula, and further derived an elegant way to describe reddening with the Balmer Decrement
</p>
<h2>Balmer Decrement and Interstellar Reddening</h2>

<p class="paragraph">
In H2 regions, which is basically ionized hydrogen regions, exist free electrons and ionized hydrogens, or to say, hydrogen nuclei. Sometime a electron will fell into this hydrogen nuclei, and release some energy. If the electron drop to a layer of n > 2, and then drop to n=2, it releases a Balmer line, that is a line that can be observed in the opticals.

</p>
<hr color="transparent">
<div class="image-container">
  <img src="balmerDecrement.png" alt="Description of the image" width="800" >
</div>
<hr color="transparent">
<p class="paragraph">

Then, according to a certain energy distribution, the intensity of different balmer lines will be fixed, according to their energy. A calculated intensity table, presented by calculating the intensity over the H-beta lines, is as follows:
</p>
<hr>
<div class="image-container">
  <img src="theoreticalDerement.png" alt="Description of the image" width="500" >
</div>
<hr>
<p class="paragraph">
  However, when the light passes through interstellar medium, that is the dust between the observer and the source, shorter wavelength got scattered more than the longer ones, and thus, it reddens.
</p>
<hr>
<div class="image-container">
  <img src="reddeningWhy.png" alt="Description of the image" width="800" >
</div>
<hr>

<p class="paragraph">
To measure reddening, several different methods are developed and they are:
</p>
<hr>
<div class="image-container">
  <img src="redenEquations.png" alt="Description of the image" width="800" >
</div>

<hr>
<h2>Measurements</h2>
<p class="paragraph">
Following is the location of M42 I measured:
</p>
<hr>
<div class="image-container">
  <img src="whereM42.png" alt="Description of the image" width="800" >
</div>
<hr>
<p class="paragraph">
  The spectrum image I got
  </p>
<hr>
<div class="image-container">
  <img src="spectrumResult.png" alt="Description of the image" width="800" >
</div>
<hr>
<p class="paragraph">
  Then, I subtracted dark, bias, devided the flat. For both images, I took a image of Regulus to calibrate my flux. I was able to find the Regulus calibration in the red range of the spectrum so the following is my calibrated spectrum across the entire slit.
</p>
<hr>
<div class="image-container">
  <img src="RedSpectrum.png" alt="Description of the image" width="800" >
</div>
<hr>
<p class="paragraph">
  The blue spectrum didn't went as smoothly. It didn't have a calibration file stored in Iraf for blue part of Regulus. Thus, I need to make one myself. I first got rid of all absorbtion and emission lines on the regulus, and fit a polynomial to it:
</p>
<hr>
<div class="image-container">
  <img src="redRegulusObserved.png" alt="Description of the image" width="800" >
</div>
<hr>
<p class="paragraph">
  Then assuming its a black body I generated a black body fit:
</p>
<hr>
<div class="image-container">
  <img src="blueRegulusTheoretical.png" alt="Description of the image" width="800" >
</div>
<hr>
<p class="paragraph">
  Finally getting a calibration data (you time the observation to this and get the real data) (this is basically inverse sensitivity)
</p>
<hr>
<div class="image-container">
  <img src="calibration.png" alt="Description of the image" width="800" >
</div>
<hr>
<p class="paragraph">
  the result is:
</p>
<hr>
<div class="image-container">
  <img src="finalblue.png" alt="Description of the image" width="800" >
</div>
<hr>
<p class="paragraph">
  Interestingly there is a little bump in the blue spectrum, which is additive in it's nature (found by comparing before and after applying flat). This should then be something living in the Houston sky.
</p>
<hr>
<div class="image-container">
  <img src="bump.png" alt="Description of the image" width="800" >
</div>
<hr>

<h2>Results</h2>
<p class="paragraph">
  The first thing I did is to measure the intensity / H-beta for the total slits. The result is:
</p>
<hr>
<div class="image-container">
  <img src="extinctTable.png" alt="Description of the image" width="500" >
</div>
<hr>
<p class="paragraph">
  I then looked at the extinction by location on the slit, and the following is my result:
</p>
<hr>
<div class="image-container">
  <img src="byYExtinction.png" alt="Description of the image" width="800" >
</div>
<hr>

<h2>
  Analysis
</h2>

<p class="paragraph">
  According to the previous work on this subject, I tried first to fit the A values for the individual points, and try to get A(H-beta). I used a linear function to fit A(lambda), and I failed by getting negative A(H-beta) since a negative value meaning that its not extinction but rather got brigher after passing thourgh the gas clouds, which is impossible:
</p>
<hr>
<div class="image-container">
  <img src="resultFit.png" alt="Description of the image" width="1000" >
</div>
<hr>
<p class="paragraph">
  Then, I tried to fit another function, which is simply the magnitude of the observed over intrinsic, setting the magnitude of Beta to be 0. There is still a problem of not able to seperate observed and intrinsic, that in the two values I am trying to compare, beta intensity is not standardized. This will cause the two value to be some what different, but might not able to provide quantitative information. The following is the fit.
</p>
<hr>
<div class="image-container">
  <img src="workedFit.png" alt="Description of the image" width="1000" >
</div>
<hr>
<p class="paragraph">
  The result can also be seen in the table, and it's not looking very well, as expected considering the math and the non-standardized H-beta.
</p>
<hr>
<div class="image-container">
  <img src="finalDatas.png" alt="Description of the image" width="1000" >
</div>
<hr>
<h2>Conclusion</h2>
<p class="paragraph">
  Throughout the investigation of the spectrum of M42, I succesfully obtained the Balmer lines, and the Intensity/H-beta values for different locations throughout the slit. Quantitatively by observation, we can see that the H-alpha is higher than the standard Balmer-decrement distribution, H-beta stayed the same as expected, H-gamma and H-delta are smaller. These results shows that the nebula has higher relative intensity in the Red, lower in the Blue, which indeed is our expectation for the effect of interstellar reddening.
</p>
<hr>
<p class="paragraph">
  However, when trying to quantitatively measure the reddening, I run into mathematical problems of deriving the relationship between the data I retrived and the standard B-V measurement that is often and more widely adopted by most astronomers throughout the history. The fitting of my model to try to get E(B-V) failed brutaley as it saying that the interstellar medium is not causing extinction but rather enhanced the light.
</p>
<hr>
<p class="paragraph">
  I finally tried to fit a function of my choice, a simple magnitude-ratio of before and after passing through the interstellar medium. However, it still failed to show significant consistency throughout the measurments.
</p>
<hr>
<p class="paragraph">
  In the end, although the math is not very succesfull, the measurement it self can be treated as a great achievement, as it was done in the middle of Houston, and it quantitatively found the reddening effect. Also, according to Professor Patrick M.Hartigan, I might be the first person to find that little bump in the blue spectrum, which indicate a continuous spectrum in the Houston sky, possibly someone's security light.
</p>

<h2>References</h2>
<p class="paragraph">
  Domínguez, A., et al. “DUST EXTINCTION from BALMER DECREMENTS of STAR-FORMING GALAXIES at 0.75 ⩽Z⩽ 1.5 withHUBBLE SPACE TELESCOPE/WIDE-FIELD-CAMERA 3 SPECTROSCOPY from the WFC3 INFRARED SPECTROSCOPIC PARALLEL SURVEY.” The Astrophysical Journal, vol. 763, no. 2, 17 Jan. 2013, p. 145, https://doi.org/10.1088/0004-637x/763/2/145. Accessed 1 May 2023.
</p>
<p class="paragraph">
  Gontcharov, G. A. “Interstellar Extinction.” Astrophysics, vol. 59, no. 4, 21 Nov. 2016, pp. 548–579, https://doi.org/10.1007/s10511-016-9457-2. Accessed 1 May 2023.
</p>
<p class="paragraph">
  Mathis, John S. “Interstellar Dust and Extinction.” Annual Review of Astronomy and Astrophysics, vol. 28, no. 1, Sept. 1990, pp. 37–70, https://doi.org/10.1146/annurev.aa.28.090190.000345. Accessed 1 May 2023.

</p>
<p class="paragraph">
  Momcheva, Ivelina G., et al. “NEBULAR ATTENUATION in Hα-SELECTED STAR-FORMING GALAXIES atZ= 0.8 from the NewHα SURVEY.” The Astronomical Journal, vol. 145, no. 2, 15 Jan. 2013, p. 47, https://doi.org/10.1088/0004-6256/145/2/47. Accessed 1 May 2023.

</p>
<p class="paragraph">
  Morisset, C., et al. “The Computation of Interstellar Extinction in Photoionized Nebulae.” Astronomy & Astrophysics, 4 Apr. 2023, https://doi.org/10.1051/0004-6361/202346296. Accessed 1 May 2023.


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