Synchrotron Critical Frequency Calculator

Introduction: synchrotron radiation and the critical frequency

In synchrotron radiation, a charged particle is forced to curve by a magnetic field, so the emitted light is not confined to one line but spread across a broad spectrum. For an ultra-relativistic electron, the beam of emitted radiation is tightly beamed into a narrow cone (opening angle ≈ 1/γ), and the spectrum is usually summarized with the critical frequency ν_c, the characteristic scale near which the spectral power is concentrated.

This page is built for a single electron in a uniform magnetic field, which keeps the calculation focused and easy to interpret. Enter the magnetic field strength B in tesla and the electron energy E in GeV, and the calculator returns ν_c in hertz, the total radiated power P in watts, a coarse electromagnetic band label, and the synchrotron cooling time t = E/P in seconds. Those outputs are useful in accelerator work when you are estimating radiation loss and photon energies, and in astrophysics when you are connecting observed spectra to the electrons and fields that produce them.

How to use the synchrotron critical frequency calculator

1. Enter the magnetic field strength B in tesla (T).
2. Enter the electron energy E in gigaelectronvolts (GeV).
3. Select Compute Frequency to calculate ν_c, P, the emission band, and E/P.
4. Use Copy Result to copy the output table for notes, lab books, or reports.

When you use this synchrotron calculator, the most common mistake is entering a field value in the wrong unit. If your magnetic field is in gauss, microgauss, or milligauss, convert it before you enter it; if your energy is in MeV or TeV, convert that to GeV first. Consistent units matter because ν_c and P both rise very quickly as the field or the electron energy increases, so a simple unit slip can move the answer by many orders of magnitude.

Formulas and assumptions for synchrotron critical frequency

The synchrotron critical frequency calculator uses the standard classical expressions for a relativistic electron in a magnetic field. Internally, the electron energy is converted from GeV to joules and the Lorentz factor is computed as:

γ = E/(m_ec²) + 1

The critical frequency is computed from:

ν_c = (3/2) γ² (eB) / (2π m_e)

The total radiated power for the electron is computed as:

P = (2 e⁴ B² γ²) / (3 m_e² c³)

Finally, the cooling time is reported as:

t_cool = E/P

The constants in the synchrotron formulas are the electron mass m_e, the elementary charge e, and the speed of light c. The band label is just a quick frequency-threshold guide for interpretation, so it should be treated as a convenience label rather than a full spectral model.

Units and conversions for synchrotron inputs

For this synchrotron critical frequency calculator, the inputs are B in tesla and E in GeV. If your values arrive in a different unit system, use the following conversions before entering them:

• Magnetic field: 1 T = 10⁴ G; 1 G = 10^-4 T; 1 mG = 10^-7 T; 1 μG = 10^-10 T.
• Energy: 1 GeV = 10⁹ eV; 1 MeV = 10⁶ eV = 10^-3 GeV; 1 TeV = 10¹² eV = 10³ GeV.
• Frequency to photon energy: E_γ = hν. For rough intuition, 10¹⁴ Hz corresponds to optical photons (a few eV), while 10¹⁸ Hz corresponds to X-rays (keV scale).
• Frequency to wavelength: λ = c/ν. For example, ν = 3×10⁸ Hz corresponds to λ ≈ 1 m (radio), and ν = 3×10¹⁴ Hz corresponds to λ ≈ 1 μm (near infrared).

If you compare this page with textbook formulas that include a pitch angle α, you may see ν_c written with an extra sin(α) factor. This calculator does not ask for α, so interpret the magnetic field you enter as the effective perpendicular component that sets the curvature for the electron.

Worked example for synchrotron critical frequency (step-by-step)

For a simple synchrotron worked example, take an electron with energy E = 1 GeV in a B = 1 T field. Enter B = 1 and E = 1, then compute the result. The output should land in the optical or near-infrared neighborhood, with a small total power for a single electron.

To see how strongly the synchrotron critical frequency depends on electron energy, keep B fixed at 1 T and raise the energy to 10 GeV. In the ultra-relativistic regime, γ grows roughly in proportion to energy, so ν_c rises approximately with γ². That means the characteristic frequency moves upward by about a factor of 100, and the total power P rises by a similar factor because it also scales approximately with γ².

The main scaling idea is the one worth remembering when you use the calculator on your own numbers: ν_c ∝ Bγ² and P ∝ B²γ². If you double the magnetic field, ν_c doubles but P quadruples. If you double the energy at high γ, both ν_c and P increase by roughly four times. That is why small changes in the magnetic field or electron energy can shift synchrotron emission from radio to optical, or from optical to X-ray, very quickly.

Interpretation notes for synchrotron critical frequency results

For synchrotron critical frequency results, the critical frequency is a characteristic scale rather than a sharp cutoff. The single-electron spectrum rises roughly as ν^1/3 at low frequency and then falls rapidly above ν_c. In practice, ν_c is often used as a compact way to describe where the emission is concentrated, but the exact peak depends on the definition being used and on whether you are looking at power per unit frequency, power per logarithmic frequency, or a population-averaged spectrum.

The total radiated power P is the instantaneous power emitted by one electron. In a storage ring or an astrophysical plasma, the total luminosity comes from many electrons, often with a spread of energies and pitch angles. Even if a single electron emits only a modest amount of power, a beam with large current or a large particle population can radiate strongly.

The cooling time t_cool = E/P is a simple estimate of how long synchrotron losses would take to remove the electron’s energy if no other process were acting. Real systems can also involve inverse Compton scattering, adiabatic expansion, bremsstrahlung, Coulomb losses, re-acceleration, and particle escape. Even so, E/P remains a useful first diagnostic because it gives a quick sense of whether synchrotron cooling is slow, moderate, or fast compared with the scale of your problem.

Limitations and modeling choices for this synchrotron calculator

• Single-particle model: Real sources contain distributions of electron energies and pitch angles, so observed spectra are integrals over a broader population.
• Pitch angle not included: Many textbook formulas include a sin(α) factor, where α is the pitch angle. This calculator assumes the effective perpendicular component is represented by the B value you enter.
• Classical regime: Quantum corrections can matter when the emitted photon energy becomes a significant fraction of the electron energy or when fields are extremely strong. For many laboratory and astrophysical situations, the classical approximation is a good first pass.
• Uniform field assumption: Spatially varying fields, curvature radiation, and more complex trajectories are not modeled here.
• Band labels are approximate: The frequency bands are simple thresholds for quick reading and do not represent detector response, atmospheric transmission, or a full spectral fit.

Background: why ν_c depends on γ² and B in synchrotron radiation

In a uniform magnetic field, an electron follows a curved path, and relativistic beaming compresses the emitted radiation into short pulses. Those two effects broaden the spectrum and create a characteristic frequency that scales rapidly with energy. As the field gets stronger, the curvature radius shrinks; as γ rises, the pulse compression becomes more extreme. Together, those trends explain why a modest change in energy can move synchrotron emission across large stretches of the electromagnetic spectrum.

In accelerators, synchrotron radiation is both useful and costly: it can provide bright photon beams for imaging and spectroscopy, but it also creates energy loss and heat load. In astrophysics, synchrotron emission is a major clue to the presence of relativistic electrons and magnetic fields in supernova remnants, jets, pulsar wind nebulae, and galaxy clusters. The same basic physics applies in both cases, even though the typical field strengths, electron energies, and particle densities can be very different.

Reference table of synchrotron example inputs

The following table shows representative synchrotron parameter choices and typical outputs. Your own result may differ slightly because of rounding and the exact constants used in the calculation.

These examples make the scaling easy to see: at fixed energy, a tenfold increase in B produces a tenfold increase in ν_c and a hundredfold increase in P. At fixed B, increasing the electron energy raises both ν_c and P approximately with γ². If you are checking a textbook or paper, make sure you know whether that source includes pitch-angle factors or uses a different definition of “critical” frequency.

Synchrotron critical frequency FAQ

Is the energy E kinetic energy or total energy?

In this synchrotron critical frequency calculator, the energy input is treated as the electron’s kinetic energy in GeV and then converted to joules. The Lorentz factor is computed as γ = E/(m_ec²) + 1, which adds the rest-mass contribution. At GeV energies, the kinetic term dominates, so the distinction is usually small in relative terms.

Why does the calculator show a “band” like Visible or X-ray?

The band label is a quick reading aid based on simple frequency thresholds. It helps you see whether your synchrotron inputs are pushing the characteristic emission into radio-like, optical-like, or X-ray-like territory. It does not mean the emission is monochromatic, because synchrotron radiation is broadband by nature.

Can I use this for protons or ions?

The formulas on this page are written for an electron, using the electron mass m_e and the elementary charge e. Heavier particles radiate much less power at the same γ and B because the power depends strongly on particle mass. If you need another particle species, the structure is similar, but the constants must be changed.

What if I get an extremely large or small number?

Extremely large ν_c or P values usually mean the magnetic field or electron energy is very large. Extremely small values often come from entering a field in astrophysical units without converting it first, such as typing “10” for 10 μG instead of entering 10×10^-10 T. If you are unsure, re-check the units and try a familiar reference case such as B = 1 T and E = 1 GeV.

Does this include radiation reaction or quantum synchrotron effects?

No. This calculator uses classical synchrotron expressions and reports E/P as a simple cooling-time estimate. In extreme fields or at ultra-high energies, quantum corrections can alter the spectrum and the effective power. For many ordinary accelerator and astrophysical parameter ranges, the classical approximation is the right place to start.