Synchrotron Critical Frequency Calculator

Introduction: synchrotron radiation and the critical frequency

Synchrotron radiation is produced when a charged particle moves through a magnetic field and is forced to curve. For an ultra-relativistic electron, the emission is strongly beamed into a narrow cone (opening angle ≈ 1/γ) and spans a broad spectrum. A convenient way to summarize that spectrum is the critical frequency ν_c, which marks the characteristic frequency near which the emitted spectral power is concentrated.

This calculator is intentionally simple and focuses on a single electron in a uniform magnetic field. You enter the magnetic field strength B (tesla) and the electron energy E (GeV). The output includes ν_c (Hz), the total radiated power P (W), a coarse electromagnetic band label, and the synchrotron cooling time t = E/P (seconds). These quantities are widely used in accelerator physics (to estimate radiation losses and photon energies) and in astrophysics (to connect observed spectra to electron energies and magnetic fields).

How to use the 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.

Practical tip: if you are working in astrophysical units (e.g., microgauss), convert to tesla first (1 G = 10^-4 T; 1 μG = 10^-10 T). If you have energy in MeV or TeV, convert to GeV (1 TeV = 1000 GeV; 1 MeV = 10^-3 GeV). Keeping units consistent is the most common source of mistakes.

Formulas and assumptions (what the calculator actually computes)

The calculator uses standard classical synchrotron expressions for a relativistic electron in a magnetic field. Internally, 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 (Larmor/Liénard result for transverse acceleration) is computed as:

P = (2 e⁴ B² γ²) / (3 m_e² c³)

Finally, the cooling time is reported as:

t_cool = E/P

Constants used are the electron mass m_e, elementary charge e, and the speed of light c. The band label (radio/IR/visible/UV/X-ray/gamma-ray) is a simple frequency-threshold classification for quick interpretation; it is not a detector model and it does not account for spectral shape.

Units and conversions (quick reference)

The inputs are B in tesla and E in GeV. If your values are in other units, 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 are comparing to textbook expressions that include a pitch angle α, you may see ν_c written with a factor of sin(α). This page does not ask for α; interpret the entered B as the effective perpendicular component relevant for the curvature.

Worked example (step-by-step)

Consider an electron with energy E = 1 GeV in a B = 1 T magnetic field. Enter B = 1 and E = 1, then compute. The calculator returns a critical frequency on the order of 10¹⁴ Hz, which is in the optical/near-IR neighborhood, and a small total power for a single electron.

Now keep B fixed at 1 T and increase the energy to 10 GeV. Because γ increases roughly linearly with energy in the ultra-relativistic regime, ν_c increases approximately with γ², so the characteristic frequency rises by about a factor of 100. The total power P also scales approximately with γ², so it rises by a similar factor.

The scaling is the key takeaway for quick reasoning: ν_c ∝ Bγ² and P ∝ B²γ². Doubling B doubles ν_c but quadruples P. Doubling energy (at high γ) increases both ν_c and P by roughly 4×.

Interpretation notes (what the outputs mean in practice)

The critical frequency is not a sharp cutoff; it is a characteristic scale for the synchrotron spectrum. The single-electron spectrum rises roughly as ν^1/3 at low frequency and falls rapidly above ν_c. In many applications, ν_c is used as a proxy for “where most of the power is” or “where the spectrum turns over,” but the exact peak depends on the definition and on whether you are looking at power per unit frequency, per unit 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 source, the total luminosity is obtained by summing over many electrons and, typically, over a distribution of energies and pitch angles. Even if P is tiny for one electron, a macroscopic beam current or a large astrophysical electron population can produce substantial radiation.

The cooling time t_cool = E/P is a simple estimate of how quickly an electron would lose its energy if synchrotron losses were the only process at work. In real systems, additional processes can matter: inverse Compton scattering, adiabatic expansion, bremsstrahlung, Coulomb losses, re-acceleration, and escape. Still, E/P is a useful first diagnostic and often sets the scale for spectral aging.

Limitations and modeling choices

• Single-particle model: Real sources contain distributions of electron energies and pitch angles; observed spectra are integrals over those distributions.
• Pitch angle not included: Many textbook formulas include a sin(α) factor (α = pitch angle). This calculator assumes the effective perpendicular component is represented by B as entered.
• Classical regime: Quantum corrections can matter when the emitted photon energy becomes a significant fraction of the electron energy or in extremely strong fields. For most laboratory and typical astrophysical fields, the classical approximation is adequate.
• Uniform field assumption: Spatially varying fields, curvature radiation, and complex trajectories are not modeled.
• Band labels are approximate: The boundaries are simple thresholds and do not represent detector response, atmospheric transmission, or detailed spectral peaks.

Background: why ν_c depends on γ² and B

In a uniform magnetic field, an electron follows a curved trajectory. Relativistic beaming compresses the observed emission into short pulses, which broadens the spectrum. A useful way to see the scaling is to note that the curvature radius decreases as the magnetic field increases, and the time compression increases with γ. Combining these effects yields a characteristic frequency that grows quickly with energy. This is why modest changes in electron energy can shift emission from radio to optical to X-ray in high-energy environments.

In accelerators, synchrotron radiation is both useful (bright photon beams for imaging and spectroscopy) and limiting (energy loss and heat load). In astrophysics, synchrotron emission traces relativistic electrons and magnetic fields in supernova remnants, jets, pulsar wind nebulae, and galaxy clusters. The same basic physics underpins both contexts; only the typical values of B, E, and particle density differ.

Reference table (example inputs)

The following table provides representative parameter choices and typical outputs. Your computed values may differ slightly due to rounding and the exact constants used.

These examples highlight the strong dependence on energy and field strength. At fixed energy, increasing B by 10× increases ν_c by 10× and P by 100×. At fixed B, increasing energy increases both ν_c and P approximately with γ². If you are trying to match a specific reference, check whether that reference includes pitch-angle factors or uses a different definition of “critical” frequency.

FAQ

Is the energy E kinetic energy or total energy?

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

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

The band label is a quick interpretation aid based on frequency thresholds. It helps you sanity-check whether your inputs are producing radio-like, optical-like, or X-ray-like characteristic emission. It does not mean the emission is monochromatic; synchrotron radiation is broadband.

Can I use this for protons or ions?

The formulas here are parameterized for an electron (m_e and e). Heavier particles radiate much less power at the same γ and B because the power scales strongly with mass. If you need other particles, 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 usually indicates very large B or very large E. Extremely small values often come from entering astrophysical fields without converting units (for example, typing “10” for 10 μG instead of 10×10^-10 T). Double-check unit conversions and try a known case such as B = 1 T and E = 1 GeV.

Does this include radiation reaction or quantum synchrotron effects?

No. The calculator uses classical expressions and reports E/P as a simple cooling-time estimate. In extreme fields or at ultra-high energies, quantum corrections can modify the spectrum and the effective power. For most everyday accelerator and astrophysical parameter ranges, the classical approximation is a good starting point.