Duration of Synchrotron Radiation in a Typical Supernova RemnantJSFHLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic=
<Text-field style="Heading 1" layout="Heading 1">Problem</Text-field> Find an estimate for the length of time a typical supernova remnant can radiate, given its magnetic field strength. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic= Hints: Find the total number of electrons in the ejecta (N). Integrate the power-law energy distribution formula from 1 to 106 (over all frequencies) to find particle density (nn). Calculate the total synchrotron power of electrons in a magnetic field at frequency \316\275. Find the total luminosity of the remnant by integrating the total synchrotron power over frequency range from \316\275L to \342\210\236. Calculate the magnetic energy. Calculate the particle energy. Find the total energy content of the remnant by adding the magnetic energy (UB times V) and particle energy. Divide the total energy by the luminosity to find the duration of the energy.
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<Text-field style="Heading 1" layout="Heading 1">Data</Text-field> Number Density of Electrons 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the Remnant 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 Speed of Light 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Thomson Cross-Section 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Magnetic Field Strength 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 Permeability of Free SpaceJSFH 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Magnetic Energy Density 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Atomic Charge Unit 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 Electron Mass 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 Critical Frequency 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 Power Law Exponent for Synchotron Power of Electrons in Magnetic Field 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
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic=
<Text-field style="Heading 1" layout="Heading 1">Useful Equations</Text-field> Total Number of Particle Species in Volume V LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYzLUkjbWlHRiQ2JVEiTkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSsmSW50ZWdyYWw7RidGOUY7Rj4vRkFGMUZCL0ZFRjFGRkZIL0ZLUSYwLjBlbUYnL0ZORlUtRiw2JVEibkYnRi9GMi1GNjYtUSJ+RidGOUY7Rj5GQEZCRkRGRkZIRlRGVi1GNjYtUTAmRGlmZmVyZW50aWFsRDtGJ0Y5L0Y8USZ1bnNldEYnL0Y/RltvL0ZBRltvL0ZDRltvL0ZFRltvL0ZHRltvL0ZJRltvRlRGVi1GLDYlUSJWRidGL0YyRlpGNUZaLUYsNiVRI25WRidGL0YyRlpGWkZaRlovJStleGVjdXRhYmxlR0Y9Rjk= Particle Density of Supernova Remnant 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 Synchrotron Power of Electrons in Magnetic Field 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 Total Luminosity LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYsLUkjbWlHRiQ2JlEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JKG1zdWJzdXBHRiQ2Jy1GOTYuUSsmSW50ZWdyYWw7RidGMkY8Rj5GQC9GQ0YxRkQvRkdGMUZIRkovRk1RJjAuMGVtRicvRlBGWi1GIzYmLUYsNiZRJSZudTtGJy9GMEY0RjJGPC1JKG1mZW5jZWRHRiQ2Jy1GIzYlRisvJTBmb250X3N0eWxlX25hbWVHUSkyRH5JbnB1dEYnRjxGMkY8LyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnRmFvRjwtRiM2JS1GOTYuUSgmaW5maW47RidGMkY8Rj5GQEZCRkRGRkZIRkpGWUZlbkZhb0Y8LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy8lL3N1YnNjcmlwdHNoaWZ0R0ZhcC1GLDYmUSJQRidGL0YyRjUtRl1vNictRiM2JUZobkZhb0Y8RjJGPEZkb0Znby1GOTYuUSJ+RidGMkY8Rj5GQEZCRkRGRkZIRkpGWUZlbi1GOTYuUTAmRGlmZmVyZW50aWFsRDtGJ0YyRjwvRj9RJnVuc2V0RicvRkFGYnEvRkNGYnEvRkVGYnEvRkdGYnEvRklGYnEvRktGYnFGWUZlbkZobkZhb0Y8 Magnetic Energy 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 Particle Energy LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYyLUkjbWlHRiQ2JlEjbm5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi5RJyZzZG90O0YnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTi1JKG1zdWJzdXBHRiQ2Jy1GOTYuUSsmSW50ZWdyYWw7RidGMkY8Rj5GQC9GQ0YxRkQvRkdGMUZIRkpGTEZPLUYjNiUtSSNtbkdGJDYlUSIxRidGMkY8LyUwZm9udF9zdHlsZV9uYW1lR1EpMkR+SW5wdXRGJ0Y8LUYjNiUtSSVtc3VwR0YkNiUtRmZuNiVRIzEwRidGMkY8LUYjNiUtRmZuNiVRIjZGJ0YyRjxGaW5GPC8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGaW5GPEZpby8lL3N1YnNjcmlwdHNoaWZ0R0ZbcC1GLDYmUSgmZ2FtbWE7RicvRjBGNEYyRjxGOC1GLDYmUSJtRidGL0YyRjVGOC1GX282JS1GLDYmUSJjRidGL0YyRjUtRiM2JS1GZm42JVEiMkYnRjJGPEZpbkY8RmlvRjgtRl9vNiVGXnAtRiM2Ji1GOTYvUSomdW1pbnVzMDtGJ0YyLyU2c2VsZWN0aW9uLXBsYWNlaG9sZGVyR0YxRjxGPkZARkJGREZGRkhGSi9GTVEsMC4yMjIyMjIyZW1GJy9GUEZpcS1GZm42JlEiNUYnRjJGZnFGPEZpbkY8RmlvLUY5Ni9RIn5GJ0YyRmZxRjxGPkZARkJGREZGRkhGSkZMRk8tRjk2LlEwJkRpZmZlcmVudGlhbEQ7RidGMkY8L0Y/USZ1bnNldEYnL0ZBRmVyL0ZDRmVyL0ZFRmVyL0ZHRmVyL0ZJRmVyL0ZLRmVyRkxGT0ZecC1GOTYuRmByRjJGPEY+RkBGQkZERkZGSEZKRkxGT0ZpbkY8
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic=
<Text-field style="Heading 1" layout="Heading 1">Solution</Text-field> First find the total number of electrons in the ejecta (N). N = the integral over the volume of the number density of electrons times the volume:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic= 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 JCIrK0NpVmUiI1k= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic= Calculate the average density of electrons (nn) over all values of \316\263, where \316\263 varies from 1 to 106 in the ejecta by integrating the power-law energy distribution formula from 1 to 106 (over all frequencies). (Use 'x' as the variable of integration for simplicity.) 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 JCIrK2MhNFkiIiNZ Calculate the total synchrotron power of electrons in a magnetic field at frequency \316\275: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic= 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 LyZJIlBHNiI2I0kjbnVHRiUsJCowJCIrciZ5Oj0jIiNEIiIiLUklVW5pdEc2JC9JK21vZHVsZW5hbWVHRiVJJlVuaXRzRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlRjU2IzYjLkkibUdGJUYtLUYvNiM2Iy5JInNHRiUhIiIpLUYvNiM2Iy5JI2NtR0YlIiIjRi0tRi82IzYjLkkjUGFHRiVGLSlGJ0ZHRkAtRi82IyokRj1GQEYtRi0= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic= The total synchrotron power of the ejecta is approximately: 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic= The power emitted at frequency \316\275 is approximately 2.2 * 1034 watts.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic= 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 LCQqJiQiKysrKytBIiNEIiIiLUklVW5pdEc2JC9JK21vZHVsZW5hbWVHNiJJJlVuaXRzRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YtRjA2IzYjLkkiV0dGLUYnRic= To find the total luminosity of the remnant, integrate this power over the range of frequencies, from \316\275L to \342\210\236: 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 LCQqJiQiKz49V3Z4IiNAIiIiLUklVW5pdEc2JC9JK21vZHVsZW5hbWVHNiJJJlVuaXRzRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YtRjA2IzYjLkkiV0dGLUYnRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic= The power-law energy distribution of the electrons is described by: # 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic= The particle energy is 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 %; LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic= Change the variable of integration from \316\263 to x and drop the Lorentz factor: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic= 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 LCQqJiQiK1NRTGVIIiNLIiIiLUklVW5pdEc2JC9JK21vZHVsZW5hbWVHNiJJJlVuaXRzRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YtRjA2IzYjLkkiSkdGLUYnRic= To find the lifetime of the radiation, divide the particle energy by the luminosity: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEjbHNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSomY29sb25lcTtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSlzaW1wbGlmeUYnRi9GMi1JKG1mZW5jZWRHRiQ2JC1GIzYlLUkmbWZyYWNHRiQ2KC1GIzYkLUYsNiVRI3BwRicvRjBGPUY5RjktRiM2Jy1GLDYlUSJMRidGL0YyRi8vJStleGVjdXRhYmxlR0Y9LyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnRjIvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmlvLyUpYmV2ZWxsZWRHRj1GX29GOUY5Rl9vRjk= LCQqJiQiK3paci9RIiIiRiYtSSVVbml0RzYkL0krbW9kdWxlbmFtZUc2IkkmVW5pdHNHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRixGLzYjNiMuSSJzR0YsRiZGJg== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic= 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 LCQqJiQiK2c4azA3ISInIiIiLUklVW5pdEc2JC9JK21vZHVsZW5hbWVHNiJJJlVuaXRzRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YtRjA2IzYjLkkjeXJHRi1GJ0Yn Synchrotron energy of electrons in a magnetic field may enable a supernova remnant to radiate for more than a thousand years, compared with a relatively short period of time without such a magnetic field (See the worksheet "Crab"). Clearly, a synchrotron source is needed. This source is thought to be the pulsar that is found in the centre of a typical supernova remnant.
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