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# Synchrotron Radiation: the Crab Nebula

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 Problem About 6500 years ago, a star in the constellation of Taurus exploded as a supernova. On July 4, 1054 CE, light from this explosion reached the Earth and was observed by Chinese astronomers, who made a record of their observation. The Crab Nebula has been identified as the detritus from the explosion; and in the approximately 960 years since the sighting of the supernova, the nebula has been expanding at a rate that roughly corresponds to its age. Determine whether or not the synchrotron radiation of the nebula itself is sufficient to explain the duration of its visibility. (Crab Nebula data from Aleksić et al, 2015.)   Hints:   Calculate the Lorentz factor, based on data given below. Calculate the energy of a relativistic electron. Calculate the power. Divide the energy by the power to get the time.
 Data Magnetic Field Strength in G   Mass of Electron in g   Peak Frequency in Hz   Speed of Light in Cm/s   Electron Charge in ESU   Thomsen Cross Section

 Useful Equations Lorentz Factor       Relativistic Energy Synchrotron Formula for Power

Solution

Calculate the Lorentz factor for a relativistic electron in a magnetic field B in Hz, where n is the peak frequency, m is the mass of the electron (g), c is the speed of light (cm/s), e is the electron charge. All units are in cgs.

 (4.1)

With such a large Lorentz factor, v is essentially equal to c, so that

 (4.2)

Calculate the energy of a relativistic electron:

 (4.3)

Calculate the synchrotron power:

 (4.4)

Divide energy by power to determine the time of cooling in seconds. Convert to years:

 (4.5)

 (4.6)

34 years is much shorter than the nebula's age. An additional source of power must be feeding into the nebula to maintain the synchrotron radiation over the requisite period. This is thought to be the pulsar at the centre of the nebula. (And see the worksheet "synchrotron".)

 References -------------------------------------------------------------------------------   Aleksić et al. (2015). Measurement of the Crab Nebula spectrum over three decades in energy with the MAGIC telescopes. Journal of High Energy Astrophysics, 5, 30-38.