Synchrotron Radiation Frequency Calculator

Enter magnetic field strength, electron Lorentz factor, and pitch angle to compute peak synchrotron frequency, wavelength, power loss, and cooling time.

⚡ Synchrotron Radiation Frequency Calculator
Magnetic Field B
Lorentz Factor γ
dimensionless
Pitch Angle α90°
deg
1°90°
Critical Frequency νc
—
Emission Wavelength
—
EM Band
—
Gyration Frequency νB
—
Synchrotron Power Loss
—
Cooling Timescale
—

⚡ What is Synchrotron Radiation?

Synchrotron radiation is electromagnetic emission produced when relativistic electrons spiral along magnetic field lines. Because the electrons travel near the speed of light, special relativistic effects transform the cyclotron emission pattern from a simple dipole into a narrow beam swept forward in the direction of motion. This beaming, combined with the Doppler boost experienced by an observer in the beam, shifts the emitted frequencies from the non-relativistic gyration frequency up to a critical frequency that scales with the square of the Lorentz factor gamma. The result is broadband emission spanning radio waves through X-rays and beyond, depending on the electron energy and magnetic field strength.

Synchrotron radiation is one of the dominant emission processes in high-energy astrophysics. It powers the radio and X-ray emission of supernova remnants and pulsar wind nebulae, the jets of active galactic nuclei, radio galaxy lobes, the diffuse radio emission of galaxy clusters, and the cosmic-ray-illuminated interstellar medium. The non-thermal power-law radio spectra observed from these sources (flux ~ nu^-alpha, typically alpha = 0.5 to 1) arise directly from the power-law energy distribution of the underlying relativistic electron population.

This calculator implements the standard Rybicki and Lightman formula for the synchrotron critical frequency nu_c = (3eBgamma^2 sin alpha)/(4*pi*m_e*c) in CGS units, where B is the magnetic field in Gauss, gamma is the electron Lorentz factor, and alpha is the pitch angle between the electron velocity and the field. The power loss formula P = (4/3)*sigma_T*c*gamma^2*U_B uses the Thomson cross-section sigma_T = 6.652e-25 cm^2 and the magnetic energy density U_B = B^2/(8*pi). The synchrotron cooling time is t_cool = gamma*m_e*c^2/P, the time for the electron to radiate away its kinetic energy.

Four preset buttons demonstrate real astrophysical environments: the ISM Radio preset loads typical Galactic interstellar medium conditions (B = 5 uG, gamma = 500, producing MHz-frequency radio), the Radio Lobe preset shows a hotspot in a powerful radio galaxy (B = 100 uG, gamma = 5000, GHz radio to microwave), the PWN X-ray preset reproduces Crab Nebula-like conditions (B = 200 uG, gamma = 10^8, extreme ultraviolet to soft X-ray), and the AGN Jet preset models the base of a blazar jet (B = 1 mG, gamma = 10^6, hard X-ray band).

📐 Formula

νc  =  (3eBγ² sinα) / (4πmec)  ≈  4.199 × 106 × BG × γ² × sinα   [Hz]
νc = synchrotron critical (peak) frequency (Hz)
e = electron charge = 4.803 × 10−10 esu (CGS)
B = magnetic field strength (Gauss; 1 μG = 10−6 G, 1 T = 104 G)
γ = electron Lorentz factor = E / (mec2)
α = pitch angle between velocity and field line
Psync = (4/3) σT c γ² UB  [erg/s]  —  power loss per electron (UB = B2/8π)
tcool = γmec2 / Psync  —  synchrotron cooling time
Example: B = 5 μG, γ = 500, α = 90°: νc = 5.248 × 106 Hz (5.248 MHz), wavelength = 57.1 m (radio band)

📖 How to Use This Calculator

Steps

1
Enter magnetic field B - Type the field strength and select units. ISM values are typically 5 to 10 uG; radio galaxy hotspots reach 100 to 200 uG; AGN jet bases can be 1 mG or more.
2
Enter Lorentz factor gamma - Type the electron Lorentz factor. Radio-emitting electrons have gamma ~ 100 to 10000; X-ray-emitting electrons in PWNe have gamma ~ 10^7 to 10^9.
3
Set pitch angle - Enter the angle between the electron velocity and the magnetic field (1 to 90 degrees). Use the slider for quick adjustment. 90 degrees maximises emission frequency and power.
4
Click Calculate or use a preset - Press Calculate to see critical frequency, wavelength, EM band, power loss, and cooling time. Presets load real astrophysical environments instantly.

💡 Example Calculations

Example 1 - Interstellar Medium Radio Electrons

ISM: B = 5 μG, γ = 500, α = 90°

1
Convert: B = 5 × 10−6 G. Compute νc = 4.199 × 106 × 5 × 10−6 × 5002 × 1 = 5.248 × 106 Hz
2
Wavelength: λ = c/νc = 3 × 1010 cm/s ÷ 5.248 × 106 Hz = 57.1 m (radio band)
3
Synchrotron power: P = (4/3) × 6.652 × 10−25 × 3 × 1010 × 5002 × (5e−6)2/(8π) = 6.628 × 10−25 erg/s
νc = 5.248 × 106 Hz (5.248 MHz) | λ = 57.1 m | Band: Radio
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Example 2 - Radio Galaxy Lobe Hotspot

Radio lobe: B = 100 μG, γ = 5000, α = 90°

1
Compute νc = 4.199 × 106 × 10−4 × 50002 × 1 = 1.050 × 1010 Hz = 10.50 GHz
2
Wavelength: λ = 3 × 1010 / 1.050 × 1010 = 2.856 cm (microwave/radio boundary)
3
Cooling time: U_B = (10−4)2/(8π) = 3.979 × 10−10 erg/cm3; P = (4/3) × σT × c × 50002 × U_B = 2.645 × 10−16 erg/s; t_cool ~ 490 kyr
νc = 1.050 × 1010 Hz (10.50 GHz) | λ = 2.856 cm | tcool = 490 kyr
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Example 3 - Pulsar Wind Nebula X-ray Synchrotron

PWN (Crab-like): B = 200 μG, γ = 108, α = 64° (sin = 0.9)

1
Compute νc = 4.199 × 106 × 2 × 10−4 × (108)2 × 0.9 = 7.558 × 1018 Hz (soft X-ray, EHz band)
2
Wavelength: λ = 3e10 / 7.558e18 = 3.967 × 10−9 cm = 0.0397 nm (soft X-ray range)
3
Cooling time: P = 4.232 × 10−7 erg/s; E_kin = 108 × mec2 = 8.187 × 10−7 erg; t_cool = 8.187e−7 / 4.232e−7 / 3.156e7 = ~6.1 yr
νc = 7.558 × 1018 Hz (X-ray) | λ = 0.0397 nm | tcool = ~6.1 yr
Try this example →

❓ Frequently Asked Questions

What is the synchrotron critical frequency formula?+
The peak synchrotron frequency is nu_c = 3*e*B*gamma^2*sin(alpha) / (4*pi*m_e*c). In convenient units, nu_c = 4.199e6 Hz times B[Gauss] times gamma^2 times sin(alpha). For a 5 uG field and gamma = 500, this gives nu_c = 5.25 MHz, firmly in the radio band.
What is the Lorentz factor gamma in synchrotron calculations?+
The Lorentz factor gamma = E / (m_e c^2) measures how relativistic an electron is. At gamma = 1 the electron is at rest; at gamma = 1000 its kinetic energy is 512 MeV. Cosmic-ray electrons in pulsar wind nebulae reach gamma ~ 10^8 to 10^9, producing synchrotron emission at X-ray frequencies. Radio-emitting electrons in galaxy lobes have gamma ~ 10^3 to 10^4.
What is the pitch angle in synchrotron radiation?+
The pitch angle alpha is the angle between an electron's velocity vector and the local magnetic field direction. At 90 degrees (sin = 1) the electron gyrates perpendicular to the field and emits at maximum frequency and power. At smaller angles both nu_c and power are reduced by sin(alpha). For a random distribution of pitch angles, averaging over all directions gives the standard isotropic synchrotron power formula.
How is synchrotron power loss calculated per electron?+
Total synchrotron power is P = (4/3) sigma_T c gamma^2 U_B, where sigma_T = 6.652e-25 cm^2 is the Thomson cross-section and U_B = B^2/(8*pi) is the magnetic energy density in erg/cm^3. This is the isotropic pitch-angle average. The formula is structurally identical to inverse Compton loss with U_B replaced by photon energy density, reflecting the common origin in photon-electron scattering.
What is the synchrotron cooling time and why does it matter?+
The synchrotron cooling time t_cool = E_kin / P = gamma*m_e*c^2 / P_sync is the time for an electron to radiate away its kinetic energy. It decreases as gamma^-1 at fixed B. X-ray electrons in the Crab Nebula (gamma ~ 10^8) cool in about 6 years, producing time-variable wisps near the pulsar. Radio electrons (gamma ~ 10^3) in galaxy lobes can survive for hundreds of millions of years without significant cooling.
What electromagnetic band does synchrotron radiation cover?+
Synchrotron spans the full EM spectrum. ISM cosmic-ray electrons (gamma ~ 500, B ~ 5 uG) radiate at MHz radio. Radio galaxy hotspots (gamma ~ 5000, B ~ 100 uG) emit at GHz microwave. Pulsar wind nebulae (gamma ~ 10^8, B ~ 200 uG) produce EHz-frequency soft X-rays. AGN blazar jets (gamma ~ 10^6 to 10^7, B ~ 1 mG) can produce X-ray and even gamma-ray synchrotron.
How does synchrotron radiation differ from cyclotron radiation?+
Cyclotron radiation is from non-relativistic electrons (gamma ~ 1) and is nearly monochromatic at the gyration frequency nu_B = eB/(2*pi*m_e*c). Synchrotron radiation is from relativistic electrons and produces a broadband spectrum extending to nu_c ~ gamma^2 * nu_B, far above the gyration frequency. The forward-beamed emission and relativistic Doppler shift are responsible for the frequency enhancement.
Why is synchrotron radiation important in radio astronomy?+
Most non-thermal radio emission from astrophysical sources is synchrotron radiation, including emission from supernova remnants, active galactic nuclei, radio galaxies, galaxy clusters, and the Galactic plane. The power-law radio spectrum directly traces the energy distribution of relativistic electrons. Measuring the spectral index constrains particle acceleration mechanisms and magnetic field strengths in these environments.
Can I solve for the magnetic field from an observed synchrotron frequency?+
Yes. Rearranging gives B = nu_c * 4*pi*m_e*c / (3*e*gamma^2*sin_alpha). If you have an observed spectral break frequency and an independent constraint on gamma (e.g., from X-ray inverse Compton emission or particle simulations), you can solve for B. This is one of the standard methods for measuring magnetic fields in pulsar wind nebulae and extended radio sources.
What is the minimum energy condition for synchrotron sources?+
The minimum energy or equipartition condition assumes that the total energy in relativistic electrons plus magnetic field is minimized when U_electrons ~ U_B = B^2/(8*pi). Setting these equal gives the equipartition magnetic field, often used to estimate B in radio lobes, supernova remnants, and other extended synchrotron sources where no direct B measurement exists.
What is synchrotron self-absorption?+
Synchrotron self-absorption occurs when a synchrotron source is optically thick to its own radiation at low frequencies. Below the self-absorption frequency nu_SSA, the spectrum rises as nu^(5/2) instead of falling. Self-absorption is common in AGN cores, compact galactic radio sources, and young supernova remnants. It sets the low-frequency turnover seen in many compact radio spectra.
Tags: Synchrotron Radiation Calculator Synchrotron Frequency Relativistic Electron Magnetic Field Astrophysics Pulsar Wind Nebula Calculator Formula Free Online

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What is the synchrotron critical frequency formula?

The peak synchrotron emission frequency is nu_c = 3*e*B*gamma^2*sin(alpha) / (4*pi*m_e*c) in CGS units, where e is the electron charge (4.803e-10 esu), B is the magnetic field in Gauss, gamma is the Lorentz factor, alpha is the pitch angle, m_e is the electron mass (9.109e-28 g), and c is the speed of light (3e10 cm/s). The coefficient evaluates to 4.199e6 Hz per Gauss per gamma^2.

What is the Lorentz factor gamma in synchrotron calculations?

The Lorentz factor gamma = E / (m_e c^2) is the ratio of a particle's total energy to its rest energy. For electrons at rest, gamma = 1. A Lorentz factor of 1000 means the electron has 511 MeV total energy. Cosmic-ray electrons in pulsar wind nebulae can reach gamma ~ 10^8 to 10^9, producing X-ray and gamma-ray synchrotron emission.

What is the pitch angle in synchrotron radiation?

The pitch angle alpha is the angle between the electron's velocity vector and the local magnetic field direction. At alpha = 90 degrees (sin = 1), the electron gyrates in a plane perpendicular to B and emits at maximum frequency and power. At smaller pitch angles the emission frequency is reduced by sin(alpha). For an isotropic electron distribution, averaging over all pitch angles produces the standard synchrotron spectrum.

How is synchrotron power loss calculated?

The total synchrotron power radiated by a single electron is P = (4/3) sigma_T c gamma^2 U_B, where sigma_T = 6.652e-25 cm^2 is the Thomson cross-section and U_B = B^2/(8*pi) is the magnetic energy density in erg/cm^3. This formula assumes an isotropic pitch-angle distribution. For a single pitch angle alpha, replace U_B with U_B*sin^2(alpha).

What is the synchrotron cooling time?

The synchrotron cooling timescale is t_cool = E_kin / P = gamma*m_e*c^2 / P_sync. It decreases rapidly with increasing gamma: t_cool scales as gamma^-1 for fixed B. X-ray-synchrotron electrons in the Crab Nebula (gamma ~ 10^8) cool in about 6 years, while radio-emitting electrons (gamma ~ 10^3) survive for millions of years.

What electromagnetic band does synchrotron radiation produce?

Synchrotron radiation spans the entire electromagnetic spectrum depending on B and gamma. Radio galaxies produce radio-frequency synchrotron from electrons with gamma ~ 10^3 to 10^4 in fields of 10 to 100 uG. Pulsar wind nebulae produce X-ray synchrotron from electrons with gamma ~ 10^8 in fields of 100 to 300 uG. Blazar jets produce gamma-ray synchrotron at the highest gamma values.

What is the difference between synchrotron radiation and cyclotron radiation?

Cyclotron radiation is emitted by non-relativistic electrons (gamma ~ 1) gyrating in a magnetic field; the spectrum is nearly monochromatic at the gyration frequency nu_B = eB/(2pi m_e c). Synchrotron radiation is emitted by relativistic electrons (gamma much greater than 1); the beamed emission produces a broad spectrum extending to nu_c ~ gamma^2 * nu_B, far above the gyration frequency.

Why is synchrotron radiation important in radio astronomy?

Most non-thermal radio emission from astrophysical sources, including supernovae remnants, AGN jets, radio galaxies, galaxy clusters, and the interstellar medium, is synchrotron radiation. The power-law radio spectrum (flux ~ nu^-alpha, typically alpha = 0.5 to 1) directly traces the power-law energy distribution of relativistic electrons, making synchrotron observations a tool for mapping magnetic fields and particle acceleration sites.

Can I compute the magnetic field from the observed synchrotron frequency?

Yes, by rearranging: B = nu_c * 4*pi*m_e*c / (3*e*gamma^2*sin_alpha). If you measure a synchrotron cutoff frequency and have an independent estimate of gamma (e.g., from inverse Compton X-rays), you can solve for B. This is one of the primary methods for measuring magnetic fields in pulsar wind nebulae and radio galaxies.

What is the minimum energy condition for synchrotron sources?

The minimum energy (or equipartition) condition assumes that the total energy of relativistic electrons and magnetic field is minimized when they are roughly equal: U_electrons ~ U_B = B^2/(8*pi). This gives the minimum-energy magnetic field and is widely used to estimate B in extended radio sources like radio lobes and supernova remnants.

How does synchrotron self-absorption affect the spectrum?

At low frequencies, a synchrotron source can become optically thick to its own radiation through the inverse process of synchrotron absorption. Below the self-absorption frequency nu_SSA, the spectrum rises as nu^(5/2) instead of falling as nu^(-alpha). Self-absorption is common in compact radio sources such as AGN cores, young supernova remnants, and compact Galactic radio emitters.

📌 Quick Tips

💡The critical frequency nu_c scales as gamma^2, so doubling the Lorentz factor quadruples the emission frequency. This extreme sensitivity makes synchrotron radiation the dominant high-energy process in pulsar wind nebulae and AGN jets.
💡Pitch angle of 90 degrees (sin = 1) gives the maximum emission frequency and power for a given B and gamma. Random pitch-angle distributions produce the standard synchrotron power-law spectrum.
💡The ISM has typical B ~ 5 to 10 uG. Radio galaxies and pulsar wind nebulae have B ~ 100 to 200 uG. AGN jets can reach B ~ 1 mG or higher near the base.
💡Synchrotron cooling time decreases as gamma^-1 for fixed B. X-ray-emitting electrons in the Crab Nebula (gamma ~ 10^8, B ~ 200 uG) cool in just a few years, explaining the wisp-like variability.
💡The gyration frequency nu_B = eB/(2pi m_e c gamma) gives the fundamental orbital rate. The critical frequency is (3/2) gamma^2 sin(alpha) times nu_B, reflecting the strong forward beaming of relativistic emission.