Reduced Mass Calculator
Turn a two-body problem into a one-body problem: reduced mass, the missing mass, the orbital period and the hydrogen atom correction.
⚛️ What is Reduced Mass?
The reduced mass of two bodies is the effective inertial mass that appears when their relative motion is treated as the motion of a single particle: μ = m1 m2 / (m1 + m2). When two bodies attract each other, both move around their common center of mass. The two-body problem looks hard, but the relative position r = r1 − r2 obeys the equation of a single body of mass μ moving in the same force field, while the center of mass simply moves at constant velocity. That simplification, described in the Wikipedia article on reduced mass, is why the quantity appears all over physics and chemistry.
Typical uses cover very different scales. In astronomy, a binary star or a planet and its moon both orbit their shared barycenter, and the reduced mass sets the energy and angular momentum of the relative orbit. In molecular spectroscopy, a diatomic molecule vibrates as if it were one particle of mass μ on a spring, so the vibrational frequency is ω = √(k / μ) and isotope substitution shifts the lines. In atomic physics, the hydrogen atom is a two-body problem whose Bohr energy levels scale with the reduced mass of the electron and proton, which is why deuterium and hydrogen have slightly different spectra. In collisions, the kinetic energy in the center of mass frame is half of μ times the relative speed squared.
A few properties are worth remembering. The reduced mass is always smaller than the lighter of the two bodies, and for a large mass ratio it is almost equal to the lighter body. For equal masses it is exactly half of either one. It is symmetric, so swapping m1 and m2 changes nothing. It is not the quantity that sets gravity: the force still uses both real masses, and the orbital period depends on the total mass m1 + m2. Mixing up the reduced mass and the total mass is the most common error, along with entering the two masses in different units.
This calculator covers four jobs. Reduced mass gives μ and the center of mass position with a chart of μ against the second mass. Missing mass reverses the formula. Orbit gives the period, speed, orbital energy and each body's distance from the center of mass. Hydrogen-like atom gives μ, the Rydberg constant, the ionization energy and the H-alpha line for hydrogen, deuterium, tritium, He+ and positronium. For the underlying gravity see the gravitational force calculator, for orbits around one dominant mass the orbital period and velocity calculator, and for the hydrogen levels the hydrogen atom energy levels calculator.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — A Binary Star
Two stars of 1 and 0.5 solar masses
Example 2 — A Carbon Monoxide Molecule
Carbon-12 at 12 u and oxygen-16 at 15.995 u
Example 3 — The Hydrogen Atom
An electron (1 electron mass) and a proton (1836.15267343 electron masses)
Example 4 — Finding a Missing Mass
A system with a reduced mass of 0.3 solar masses containing a 1 solar mass star
Example 5 — The Earth and Moon
1 Earth mass and 0.0123 Earth masses, separated by 384400 km
Example 6 — Two Sun-Like Stars at 1 AU
Two 1 solar mass stars separated by 1 AU
Example 7 — Deuterium Against Hydrogen
An electron around a deuteron of 3670.48296788 electron masses, Z = 1
❓ Frequently Asked Questions
🔗 Related Calculators
What is the formula for reduced mass?
mu = m1 x m2 / (m1 + m2). It is the product of the two masses divided by their sum. For 2 kg and 3 kg the reduced mass is 6 / 5 = 1.2 kg.
What is reduced mass used for?
It reduces a two-body problem to a one-body problem. The relative motion of two bodies that attract each other behaves like a single particle of mass mu, which simplifies binary stars, planets and moons, molecular vibrations and the hydrogen atom.
Why is the reduced mass smaller than both masses?
Because mu = m1 / (1 + m1/m2) and the bracket is greater than 1, mu is always smaller than m1, and by symmetry smaller than m2. It approaches the lighter mass as the heavier one grows.
What is the reduced mass of two equal masses?
It is half of one mass, mu = m / 2. Two 1 kg masses have a reduced mass of 0.5 kg, and positronium, an electron bound to a positron, has half the electron mass.
What is the reduced mass of the hydrogen atom?
Using the electron mass me and the proton mass mp, mu = me mp / (me + mp) = 0.99945568 me, or about 9.1044e-31 kg. It is 0.0544 percent smaller than the electron mass, and that is enough to shift hydrogen's spectral lines.
How do I find the reduced mass of a diatomic molecule?
Use the atomic masses of the two atoms in the same unit. For carbon monoxide with 12 u and 15.995 u it is 6.856 u. The vibrational frequency depends on the reduced mass through omega = sqrt(k / mu).
Does the orbital period depend on the reduced mass?
The period of a two-body orbit depends on the total mass: T = 2 pi sqrt(a cubed / (G (m1 + m2))). The reduced mass shows up in the energy E = -G m1 m2 / (2a) and in the angular momentum, not in the period.
How do I find a missing mass from the reduced mass?
Rearrange the formula to m2 = mu m1 / (m1 - mu). It only works when mu is smaller than m1. With mu = 0.3 and m1 = 1 the missing mass is 0.3 / 0.7 = 0.4286.
Where is the center of mass of two bodies?
It lies on the line joining them, at a distance m2 / (m1 + m2) times the separation from body 1. In the Earth and Moon system it sits about 4671 km from Earth's center, inside the Earth.
What is the difference between reduced mass and total mass?
Total mass m1 + m2 sets the gravitational pull on the pair and the orbital period. Reduced mass is the effective inertia of the relative motion. For a small planet around a huge star, mu is close to the planet's mass while the total mass is close to the star's.
Why does deuterium have a different spectrum from hydrogen?
The deuteron is about twice as heavy as the proton, so the reduced mass is slightly larger (0.99972763 me against 0.99945568 me). That raises the Rydberg constant and moves H-alpha from 656.470 nm to 656.291 nm in vacuum.