- Synchrotron emission from power Law electron energy distributions

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1 - Spectrum of synchrotron emission from a single electron - Synchrotron emission from power Law electron energy distributions - Synchrotron self absorption - Polarization of synchrotron emission - Synchrotron cooling - Equipartition - Case Study: the Crab pulsar

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3 20 minute oral exam on material covered up to today s lecture Budaeva, Nina B (Gregg) Duggan, Gina Elizabeth (Christian) Gossan, Sarah Elizabeth (Gregg) Hsu, Chen Chih (Christian) Leethochawalit, Nicha (Gregg) Li, Cheng (Christian) Ma, Xiangcheng (Gregg) Mukherjee, Eric Shinjini (Christian) O'Sullivan, Donal Brendan (Gregg) Parihar, Prachi Singh (Christian) Gregg Slots: Thu: 9:00,9:30,10:00,10:30,11:30 Fri: 11:00,11:30,13:00,13:30,14:00,14:30,15:0015:30,16:00,16:30 Christian Slots Thu: 9:00,9:30,10:00,10:30,11:30,13:00,13:30,14:00,14:30,15:00 Fri: 11:00,11:30,13:00,13:30,15:30,16:00,16:30 TAs with slots you are available for and they will circulate the schedule A TA will be present for each exam

4 - Observed radiation distribution can be decomposed by Fourier analysis in sum of dipoles radiating at harmonics of the gyrofrequency - Critical frequency: The maximum Fourier component of the pulse is expected to be: The exact calculation generalized for any pitch angle gives The synchrotron power spectrum of a single electron is Readhead (Fig 2.39) where K 5/3 is a modified Bessel function High energy electrons radiate at higher frequencies!

5 Image credit: Giampaolo Pisano - The observed spectrum is the convolution of the electron energy distribution with the spectrum from a single electron. - What kind of spectrum is typical for an astrophysical source?

6 - Spectral energy of cosmic rays electrons approximated by power-law distribution - See Readhead (Ch. 3) for discussion of why cosmic rays have a power law energy distribution. - Because N(E) is nearly a power law over more than two decades of energy and the critical frequency ν c is proportional to E 2, we expect the synchrotron spectrum to reflect this power law over a frequency range of at least (10 2 ) 2 = Remember... - Let s make a crude approximation: Each electron radiates all its power at a frequency close to the critical frequency - The energy spectrum of cosmic-ray electrons in the local interstellar medium (Casadei, D., & Bindi, V. 2004, ApJ, 612,262). In the energy range above a few GeV, N(E) is a power law with slope p =2.4.

7 - The emission coefficient for an ensemble of electrons is where Differentiating E gives so Eliminating E in favor of ν/ν cyc and ignoring the physical constants in this equation for results in the proportionality

8 - In our galaxy p = Therefore we expect - hence the spectral index should be which is in agreement with observation - This is also the typical spectral index of most optically thin extragalactic radio sources, even radio galaxies and quasars. It reflects the power-law energy distribution of cosmic rays accelerated in shocks, the shocks produced by supernova remnants expanding into the ambient interstellar medium for example. - The energy spectrum of cosmic-ray electrons in the local interstellar medium (Casadei, D., & Bindi, V. 2004, ApJ, 612,262). In the energy range above a few GeV, N(E) is a power law with slope p =2.4.

9 - For every emission process there is an associated absorption process. - For a power-law electron distribution we expect - For optically thick plasmas in LTE with we would expect (Rayleigh-Jeans Law) - Does not apply for a non-thermal electron population! - We can associate a temperature to an electron of a given energy: - but an electron will emit mainly at - The effective temperature of the electrons is frequency-dependent and it is function of their energy For a self-absorbed source, must use

10 Credit: Giampaolo Pisano

11 - Averaged synchrotron emission linearly polarized perpendicular to the projection of B on the plane of the sky - Using a power-law electron energy distribution, with index p, degree of polarization - A typical galaxy, with p = 2.4, can have polarization of 72%

12 - Electrons in a plasma emitting synchrotron radiation cool down. The time scale for this to occur is given by the energy of the electrons divided by the rate at which they are radiating away their energy. - The energy E is given by - Synchrotron power emitted by electron Hence cooling time is given by

13 - The existence of a synchrotron source implies the presence of relativistic electrons with some energy density U e and a magnetic field whose energy density is U B =B 2 /(8π). What is the minimum energy in relativistic particles and magnetic fields required to produce a synchrotron source of a given luminosity? - To estimate U e, we assume a power-law electron energy distribution -p spanning the energy range E min to E max that is needed to produce synchrotron radiation over any frequency range min to max. Then For a given synchrotron luminosity Substituting N(E)=KE p and the synchrotron power emitted per electron gives - p - p - p - p

14 Since electrons with energy E emit most of the radiation seen at frequency E 2 B, the electron energy needed to produce radiation at frequency ν scales as If we consider the energy content of only those electrons that emit in a fixed frequency range (e.g., from ν min ~ 10 7 Hz to ν max ~ 10 7 Hz ), then the energy limits E min and E max are both proportional to B 1/2 and Also remember If we call the ion/electron energy ratio η, then the total energy density in cosmic rays is (1+η)U e. The total energy density U of both cosmic rays and magnetic fields is It follows that at minimum energy, the ratio of cosmic-ray particle energy to field energy is

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16 - Discovered in Taurus by John Bevis in Independently discovered by Charles Messier in 1758 who thought he had found comet Halley in its 1758 return. Crab pulsar - To prevent such further occurrences, he compiled a list of all nebulosities known to him. For this reason, the Crab Nebula is also known as M1. - Given the name Crab Nebula by Lord Rosse in ~ : The realization that the nebula was changing s: Associated with historic bright supernova in 1054 (initially proposed by Hubble in 1928) - Baade and Zwicky had predicted neutron stars would be the remnants of supernova explosions - the search was on!

17 - Radio emission from the Crab nebula was first detected by Bolton, Stanley and Slee (1949). It is one of the most powerful radio sources known, with a flux of 1000 Jy at 1 GHz. Crab pulsar - The Crab nebula was the first known extra-solar X-ray source detected (Bowyer et al. 1964). - Total emitted power at all wavelengths: 100,000 times the solar luminosity!!! - Where was the energy coming from? - Two candidates - referred to in the literature as the "north following" and "south preceding" stars Baade and Minkowski separately favor the south preceding star. -Radio scintillation studies led Hewish and Okoye (1964) to propose the existence of a compact radio source near the center of the nebula, with a steep spectrum. -

18 - Pacini (1967) proposed the existence of a highly magnetized, rapidly spinning neutron star as the power source of the nebula. This would radiate a very powerful EM wave with the rotational frequency of the star. This is below the plasma frequency of the nebula, Crab pulsar therefore all this energy will be absorbed and reradiated by the plasma of the nebula. - this is also apparent from arguments based on synchrotron cooling time - 100,000 solar luminosities -> erg sec-1 - Consider electrons in the Crab Nebula synchrotron radiating at 20 kev (n.b. 1eV = erg) or n = Hz, for a magnetic field strength of order 10-4 Gauss - ϒ > 5 x 107 Crab pulsar - = 30 erg - P = erg s-1 - cooling time is ~20 years! -> something is powering the nebula

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20 1967: Pulsars discovered - Confirmed the existence of neutron stars - The Crab pulsar was discovered soon after! - Astrophysicists had already predicted much, but did not predict the neutron star would pulse detectable radio waves!