Reduced Mass Calculator - mu for Any Two-Body System

A reduced mass calculator takes two masses and returns mu = m1 m2 / (m1 + m2) in kilograms and in unified atomic mass units, alongside the mass ratios that show how light the equivalent one-body picture is. The primary answer drops directly into the radial equation of the hydrogen atom, the harmonic oscillator frequency of a diatomic molecule, and the relative kinetic energy and angular momentum of the two-body Kepler problem.

• • Hydrogen Atom: Compute mu for the electron plus proton system, the mass that shows up in the Bohr radius and Rydberg constant.
• • Diatomic Molecule Vibration: Compute mu for two bonded atoms to feed the harmonic oscillator frequency and rotational constant.
• • Planetary Two-Body Problem: Compute mu for the Earth and the Moon so the relative kinetic energy and angular momentum of the Earth-Moon system can be written with a single effective mass.
• • Equal-Mass and Heavy-Light Limits: Check that equal masses give mu = m / 2 and that a heavy partner leaves mu just below the lighter mass.

The reduced mass is the bridge between a two-body problem and an equivalent one-body problem. Every textbook that derives the Bohr hydrogen energy levels, the vibrational frequency of a diatomic molecule, or the relative kinetic energy of two gravitating bodies starts by replacing the two real masses with one effective mass mu.

When the hydrogen atom is the next step, the Bohr Model Calculator takes the reduced mass output and turns it into the Bohr radius, Rydberg constant, and ground-state energy of the same system.

The reduced mass calculator reads the two masses in kilograms (or fills them from a preset), computes mu = m1 m2 / (m1 + m2), divides by the atomic mass constant 1.66053906660 x 10^-27 kg/u to get the value in unified atomic mass units, and reports the m1 / mu, m2 / mu, and heavier-to-lighter mass ratios.

• m1: Mass of the first body in kilograms.
• m2: Mass of the second body in kilograms.
• mu: Reduced mass of the two-body system in kilograms, or in atomic mass units after division by the atomic mass constant.
• u: Unified atomic mass unit, exactly 1.66053906660 x 10^-27 kg.

Internally the calculator clamps each input at zero, so a missing or negative mass is treated as zero. When m1 + m2 is zero, mu and all the ratios collapse to zero and the preset selector refills the inputs.

The formula mu = m1 m2 / (m1 + m2) is symmetric in m1 and m2, so swapping the two inputs leaves mu unchanged while the m1 / mu and m2 / mu rows swap.

According to NIST CODATA electron mass, the electron rest mass is 9.1093837015 x 10^-31 kg with a relative uncertainty below 1 x 10^-10

According to NIST CODATA proton mass, the proton rest mass is 1.67262192369 x 10^-27 kg with a relative uncertainty below 1 x 10^-10

For a planetary two-body system such as the Earth and the Moon, the Orbital Period Calculator applies the two-body form of Kepler's third law and returns the relative orbit period from the semi-major axis and the central-body gravitational parameter.

Four ideas make every reduced mass calculator result easier to read: the reduced mass itself, the center-of-mass frame, the equivalent one-body picture, and the symmetric structure of the formula.

For two equal masses, mu = m / 2 and the equivalent one-body picture is a single mass m orbiting at half the real separation. For m2 >> m1, mu approaches m1 from below and the picture is a particle of mass almost m1 orbiting a much heavier fixed partner.

Because the equivalent one-body picture conserves the same L = mu v r, the Angular Momentum Calculator is the natural place to compute the angular momentum that goes with the reduced mass.

Pick the preset that matches your system, confirm or edit the two masses, and read the reduced mass in kilograms and atomic mass units alongside the three mass ratios.

1. 1 Pick a preset or choose Custom: Select Hydrogen, Deuterium, Proton-proton, Earth-Moon, Heavy-light, or Custom. Choosing a preset fills the two mass fields with the standard values from NIST CODATA and NASA JPL.
2. 2 Enter mass 1 in kilograms: Type the first body mass in kilograms. For subatomic particles use scientific notation such as 9.1093837015 x 10^-31 for an electron. The field accepts both very small and very large values.
3. 3 Enter mass 2 in kilograms: Type the second body mass in kilograms using the same conventions. The two mass inputs are symmetric in the formula, so the order does not affect mu.
4. 4 Read the reduced mass: The primary card shows mu in kilograms. The next row gives mu in unified atomic mass units (u), and the three rows after that show m1 / mu, m2 / mu, and the heavier-to-lighter mass ratio.
5. 5 Switch presets to compare systems: Walk through the preset list to see how mu compares for a hydrogen atom, a proton-proton pair, and an Earth-Moon pair. The mass ratios show why hydrogen is dominated by the proton and Earth-Moon by the Earth.

Choose the Earth-Moon preset and read mu = 7.254 x 10^22 kg, m1 / mu = 82.35, m2 / mu = 1.012, and a heavier-to-lighter ratio of 81.36.

The reduced mass calculator handles the unit conversion between kilograms and atomic mass units, applies the symmetric formula, and exposes the mass ratios that make the equivalent one-body picture obvious.

• • Two masses, one reduced mass: Skip the hand calculation and get mu = m1 m2 / (m1 + m2) in kilograms and atomic mass units in one step.
• • Built-in preset library: Pick Hydrogen, Deuterium, Proton-proton, Earth-Moon, or Heavy-light to skip looking up CODATA and NASA JPL masses.
• • Mass ratios for context: m1 / mu, m2 / mu, and the heavier-to-lighter ratio show how light the equivalent one-body picture is.
• • SI and atomic units handled: Inputs stay in kilograms and the atomic mass constant 1.66053906660 x 10^-27 kg/u is applied internally.
• • Symmetric formula handling: Swapping the two mass fields leaves mu unchanged while the m1 / mu and m2 / mu rows swap, so the order never hides a mistake.

When the answer disagrees with an expectation, check that the preset selector matches the system, that the masses are in kilograms (not grams or atomic mass units), and that m2 is not so much larger than m1 that mu is hiding under m1.

When one of the masses is an atom or isotope rather than a single particle, the Atomic Mass Calculator looks up the unified atomic mass so the reduced mass can be computed with consistent atomic units.

The reduced mass depends on the two masses in a specific way, so the four cards below show how each input shifts mu, the unit conversion, and the heavier-to-lighter mass ratio.

• • The calculator reports the reduced mass for a single pair of masses and assumes the two-body problem. Systems of three or more gravitating bodies do not collapse to a single mu.
• • Reduced mass is a classical and quantum-mechanical concept. For relativistic speeds or strong fields, the same mu still appears in the two-body Dirac equation and post-Newtonian expansion, but mu = m1 m2 / (m1 + m2) is only the leading-order result.

For very unequal masses such as an electron orbiting a proton, mu is dominated by the electron mass even though the proton is about 1836 times heavier. For two equal masses such as a proton-proton pair, mu is exactly half of one proton mass.

According to NIST Guide for the Use of the SI, the kilogram is the SI base unit of mass and the unified atomic mass unit u is defined as exactly 1.66053906660 x 10^-27 kg

For the diatomic molecule limit where the reduced mass sets the vibrational frequency, the Angular Frequency Calculator converts between hertz, period, and rad/s for the same omega.