<Text-field style="Heading 1" layout="Heading 1">Problem</Text-field>

Calculating and Graphing the Bremsstrahlung Emission Over All Frequencies for the Orion Nebula

JSFHJSFH

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Calculate and graph the bremsstrahlung emission over all frequencies for the Orion Nebula. (The data are from Anderson and Eastwood (2007).)

Hints:

Use cgs units.

Begin by calculating the Planck function for frequency.

Calculate the free-free emission coefficient.

Calculate the optical depth.

Calculate the specific intensity.

Plot specfic intensity as a function of frequency.

<Text-field style="Heading 1" layout="Heading 1">Data</Text-field>

Temperature in Kelvins

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUkjbW5HRiQ2JFElODAwMEYnRjktRjY2LVEiOkYnRjlGO0Y+RkBGQkZERkZGSEZKRk0vJStleGVjdXRhYmxlR0Y9Rjk=

Electron Density in 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

Line-of-Sight Thickness in Parsecs

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

Cm per PC

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

Speed of Light in cm/s

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYtLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUkjbW5HRiQ2JFEnMi45OTc5RidGOS1GNjYtUScmc2RvdDtGJ0Y5RjtGPkZARkJGREZGRkgvRktRJjAuMGVtRicvRk5GVy1JJW1zdXBHRiQ2JS1GUDYkUSMxMEYnRjktRiM2J0ZmbkYvLyUrZXhlY3V0YWJsZUdGPS8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJ0YyLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy1GNjYtUSJ+RidGOUY7Rj5GQEZCRkRGRkZIRlZGWEZjby1GNjYtUSI6RidGOUY7Rj5GQEZCRkRGRkZIRkpGTUZjb0Zbb0Y5

Planck Constant

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

Boltzmann Constant

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

Gaunt Factor: Free-Free

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

Atomic Number for Hydrogen

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiWkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUkjbW5HRiQ2JFEiMUYnRjktRjY2LVEiOkYnRjlGO0Y+RkBGQkZERkZGSEZKRk0vJStleGVjdXRhYmxlR0Y9Rjk=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 1" layout="Heading 1">Useful Equations</Text-field>

Planck's Function

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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

Optical Depth

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

Specific Intensity

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

<Text-field style="Heading 1" layout="Heading 1">Solution</Text-field>

Calculate the Planck function:

JSFH

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

LyZJIkJHNiI2I0kjbnVHRiUsJCooJCIrTCEzWFoiISNjIiIiKUYnIiIkRi0sJi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjLCQqJiQiKzBjWSgqZiEjQ0YtRidGLUYtRi1GLSEiIkY8Ri0=

Calculate the optical depth (\317\204), using (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEjbmVGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiM2Jy1JI21uR0YkNiRRIjJGJy9GNlEnbm9ybWFsRidGMi8lK2ZvcmVncm91bmRHUSxbMjAwLDAsMjAwXUYnLyUscGxhY2Vob2xkZXJHRjRGNS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGPg==), where (ni*ne) is considered to be equal to LUklbXN1cEc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEjbmVGJy8lJ2l0YWxpY0dRJXRydWVGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JJW1yb3dHRiQ2Ji1JI21uR0YkNiVRIjJGJ0YyL0Y2USdub3JtYWxGJ0YyLyUwZm9udF9zdHlsZV9uYW1lR1ElVGV4dEYnRj8vJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYn:

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

LCQqKCQiKzM2ZHI+IiNDIiIiLCZGJ0YnLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMsJComJCIrMGNZKCpmISNDRidJI251R0YuRichIiJGNkYnKUY1IiIkRjZGJw==

Calculate J, the specific intensity, and plot it as a function of \316\275:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

JSFH

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

L0kiSkc2IiomSSJCR0YkIiIiLCZGJ0YnKUkiZUdGJCwkKigkIiszNmRyPiIjQ0YnLCZGJ0YnLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMsJComJCIrMGNZKCpmISNDRidJI251R0YkRichIiJGPUYnKUY8IiIkRj1GPUY9Ric=

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

LCQqKiQiK3AnNFdaIiEjYyIiIilJI251RzYiIiIkRicsJkYnRictSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRio2IywkKigkIitjNSI+RCMiI0NGJywmRidGJy1GLjYjLCQqJiQiKy9jWSgqZiEjQ0YnRilGJyEiIkZARidGKEZARkBGQEYnLCYtRi42IywkKiYkIiswY1koKmZGP0YnRilGJ0YnRidGJ0ZARkBGJw==

Then plot J(\316\275):

JSFH

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRJnBsb3RzRidGL0YyLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjNRJ25vcm1hbEYnRkAtSSNtb0dGJDYtUSI6RidGQC8lJmZlbmNlR0Y/LyUqc2VwYXJhdG9yR0Y/LyUpc3RyZXRjaHlHRj8vJSpzeW1tZXRyaWNHRj8vJShsYXJnZW9wR0Y/LyUubW92YWJsZWxpbWl0c0dGPy8lJ2FjY2VudEdGPy8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlZGPUZA

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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The curve begins on the left as a Planck function (\317\204>>1). At the top, where \317\204<<1 (optically thin gas), the emission is no longer a function of \316\275, and the curve levels off. This is the bremsstrahlung portion of the plot. At high frequency, the emission is a function of both frequency and temperature and falls exponentially. The specific intensity (J) is given in units of erg s-1 cm-2 sr-1. To convert to SI units (W m-2 sr-1), multiply the cgs values by 10-3.

<Text-field style="Heading 1" layout="Heading 1">References</Text-field>

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Anderson, M. and Eastwood, M. (2013). Problem Set 5. AY 121 Radiative Processes. http://www.astro.caltech.edu/~mmanders/ay121/solutions/ps5solutions.pdf. (Accessed: 2016-03-09.)

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