---
title: "Binary-Star Mass Laboratory Interactive Lab | Sandbox Physics"
description: "Can motion weigh stars without knowing the viewing angle? Only partly. The two stars' line-of-sight velocities give their mass ratio and a projected mass, which is only a lower limit because of the unknown tilt. The true total mass needs the inclination; masses from the stars' motion across the sky need the distance as well. A fit that matches the data closely cannot tell you whether those assumed values are right."
canonical_url: "https://sandboxphysics.com/labs/binary-star-mass/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/binary-star-mass/"
last_modified: "2026-10-22"
---

M073 · Orbital observation / inference

# Binary-Star Mass Laboratory

**Can motion weigh stars without knowing the viewing angle?** Only partly. The two stars' line-of-sight velocities give their mass ratio and a projected mass, which is only a lower limit because of the unknown tilt. The true total mass needs the inclination; masses from the stars' motion across the sky need the distance as well. A fit that matches the data closely cannot tell you whether those assumed values are right.

Watch a Kepler binary on the sky and through the radial velocities of both stars. Tilt the observer, take repeatable noisy records and fit the amplitudes. Then compare the projected mass, the inclination-corrected mass and an astrometric estimate that depends on distance.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

The stars are point masses in an isolated Newtonian binary. Units are AU, Julian years and solar masses, with the teaching value of the gravitational constant. Eclipses, spectra, light-travel time and relativity are not modeled. The fits take the eccentricity, periapsis direction and epoch as known from elsewhere, and they use explicitly adopted values for the period, viewing inclination and distance. Amplitude errors from the residuals are conditional and do not cover the full mass uncertainty.

Physics tutorial

## Measuring a mass means measuring a geometry

**Background** An isolated pair of stars orbits its barycenter under Newtonian gravity. Spectroscopy sees motion along the line of sight; astrometry sees the two projected sky coordinates. Each one hides a different part of the same three-dimensional orbit.

**Why it matters** The Newtonian two-body dynamics and projected-velocity geometry follow NASA's educational construction for binary masses. This Lab adds a record of the conditional regression and states its adopted geometry explicitly; it is not a general orbit-fitting pipeline.

Start with the essentials

- **Focus question**: Which observations determine mass, and which assumptions still remain?
- **One-sentence intuition**: Fitting both velocity curves together measures the amplitudes and the mass ratio, but the total mass also needs the viewing geometry. Angular motion needs the distance as well. Clean residuals alone cannot confirm either outside assumption.

### Core mathematical model

#### Two bodies, one relative orbit

$$
\mathbf r_1=-\frac{M_2}{M_1+M_2}\mathbf r,\qquad\mathbf r_2=\frac{M_1}{M_1+M_2}\mathbf r
$$

The heavier star moves on the smaller orbit around the barycenter. The relative orbit shown is the separation vector between the stars.

#### Period and total mass

$$
P^2=\frac{4\pi^2a^3}{G(M_1+M_2)}
$$

In the teaching units of AU, years and solar masses, the gravitational constant equals four pi squared. The semimajor axis is that of the relative orbit.

#### Projected sky motion

$$
X=x\cos\omega-y\sin\omega,\quad Y=(x\sin\omega+y\cos\omega)\cos i,\quad Z=(x\sin\omega+y\cos\omega)\sin i
$$

Tilting the orbit shrinks one sky coordinate and creates velocity along the line of sight. The angular separation also depends on distance.

#### Fitting the velocities of both stars

$$
v_1=\gamma-K_1[\cos(\nu+\omega)+e\cos\omega],\quad v_2=\gamma+K_2[\cos(\nu+\omega)+e\cos\omega]
$$

The linear regression fits a shared systemic velocity and two amplitudes. The orbit's shape and phase are taken from independent information, and a wrong adopted period changes the basis functions of the fit.

#### What spectroscopy determines

$$
\frac{M_2}{M_1}=\frac{K_1}{K_2},\qquad (M_1+M_2)\sin^3i=\frac{P(K_1+K_2)^3(1-e^2)^{3/2}}{2\pi G}
$$

Without the viewing inclination, the projected mass is only a lower bound, not the true mass. Seen nearly face-on, the radial signal fades away; the Lab requires both amplitudes to exceed three conditional standard errors.

#### Distance enters cubed

$$
a_{\rm fit}=\theta_{a,\rm fit}d_{\rm fit},\quad M_{\rm ast}=\frac{4\pi^2a_{\rm fit}^3}{GP_{\rm fit}^2},\quad M_{\rm RV}=\frac{M_{\rm proj}}{\sin^3i_{\rm fit}}
$$

The fit of the angular semimajor axis assumes the adopted projection geometry. If the distance is off by some factor, the astrometric mass is off by that factor cubed, even when the angular residuals are tiny.

### Common difficulties

#### Projection is not mass

**Typical misconception** A fitted velocity amplitude determines the mass by itself.

**Better mental model** Radial velocities give the mass times the cube of the sine of the inclination. Look at the whole family of masses that fits.

#### A vanishing signal does not reveal the masses

**Typical misconception** Face-on motion means the stars have no mass or do not orbit.

**Better mental model** The orbit is still there on the sky. Both radial signals vanish together, so the mass ratio cannot be determined.

#### Good residuals do not verify assumptions

**Typical misconception** A tiny residual proves the adopted distance and inclination.

**Better mental model** Different physical masses and viewing geometries can produce the same velocity record. The distance comes from outside the angular fit.

#### Standard errors are conditional

**Typical misconception** The amplitude standard error is a complete error bar on the mass.

**Better mental model** It leaves out the uncertainty in period, orbital shape, phase, distance and inclination. Close to face-on, the inversion is poorly determined.

### Run the experiment

1. 01

   #### Recover a clean binary

   Choose Recover the masses, press Acquire full record and compare both mass estimates with the true mass.

   **What to observe:** The amplitudes give the mass ratio, and the correct adopted inclination gives back the total mass.

2. 02

   #### Separate viewing from analysis

   Drag the observer ring, then set the adopted inclination to the same value.

   **What to observe:** The projected mass changes with viewing angle. The corrected mass matches only when the adopted angle is right.

3. 03

   #### Diagnose a bad period

   Compare the Wrong period and Noisy observation record presets, then press New observation record.

   **What to observe:** A wrong period leaves a pattern in the residuals, while seeded Gaussian errors scatter around the conditional fit.

4. 04

   #### Question a clean-looking result

   Select Wrong adopted distance, then Face-on weak signal.

   **What to observe:** A clean angular fit can still give a biased mass. Near face-on, spectroscopy cannot resolve the masses even though the orbit is visible on the sky.

Physical conclusion

Look at the observations, the conditional fits and the conservation checks together.

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)