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.

Record time $\text{Pending}$

True period $\text{Pending}$

Adopted period $\text{Pending}$

Primary fitted amplitude $\text{Pending}$

Secondary fitted amplitude $\text{Pending}$

Primary amplitude standard error $\text{Pending}$

Fitted systemic velocity $\text{Pending}$

Secondary / primary mass $\text{Pending}$

Spectroscopic projected mass $\text{Pending}$

Inclination-corrected mass $\text{Pending}$

True total mass $\text{Pending}$

Astrometric total mass $\text{Pending}$

Fitted angular semimajor axis $\text{Pending}$

Velocity residual RMS $\text{Pending}$

Sky residual RMS $\text{Pending}$

Fit interpretation $\text{Pending}$

Sampled energy defect $\text{Pending}$

Sampled momentum defect $\text{Pending}$

Measure the orbit, then design your own

Can motion weigh stars without knowing the viewing angle?

Primary mass

M_1

1.40\,M_\odot

Secondary mass

M_2

0.70\,M_\odot

Relative semimajor axis

a

1.80\,AU

Eccentricity

e

0.35\,

True viewing inclination

i

60.00\,{}^\circ

Periapsis direction

\omega

35.00\,{}^\circ

Adopted inclination

i_{\rm fit}

60.00\,{}^\circ

Adopted period scale

P_{\rm fit}/P

100.00\,\%

True distance

d

20.00\,pc

Adopted distance scale

d_{\rm fit}/d

100.00\,\%

Velocity noise deviation

\sigma_v

0.00\,km/s

Sky noise deviation

\sigma_\theta

0.00\,mas

Reviewed time

t/t_{\rm obs}

0.00\,\%

The orange ring sets the true viewing inclination, and its slider does the same from the keyboard. The analysis uses its own adopted inclination, set separately. Each setup produces 48 paired observations spread over one and a half true periods. Playback steps through this fixed record, and the fits use every sample. New observation record draws a new Gaussian noise seed. Sky noise is added to the separation between the two stars, and their orientation on the sky is taken as known.

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

BackgroundAn 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 mattersThe 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 misconceptionA fitted velocity amplitude determines the mass by itself.

Better mental modelRadial 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 misconceptionFace-on motion means the stars have no mass or do not orbit.

Better mental modelThe 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 misconceptionA tiny residual proves the adopted distance and inclination.

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

Standard errors are conditional

Typical misconceptionThe amplitude standard error is a complete error bar on the mass.

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

Run the experiment

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.
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.
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.
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.