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Sandbox Physics

M073 · Orbital observation / inference

Binary-Star Mass Laboratory

Observe a Kepler binary on the sky and in double-lined radial velocity. Tilt the observer, acquire repeatable noisy records and fit amplitudes; compare projected mass, inclination-corrected mass and a distance-dependent astrometric estimate.

Interactive modelBinary-Star Mass Laboratory
Record timePending\text{Pending}
True periodPending\text{Pending}
Adopted periodPending\text{Pending}
Primary fitted amplitudePending\text{Pending}
Secondary fitted amplitudePending\text{Pending}
Primary amplitude standard errorPending\text{Pending}
Fitted systemic velocityPending\text{Pending}
Secondary / primary massPending\text{Pending}
Spectroscopic projected massPending\text{Pending}
Inclination-corrected massPending\text{Pending}
True total massPending\text{Pending}
Astrometric total massPending\text{Pending}
Fitted angular semimajor axisPending\text{Pending}
Velocity residual RMSPending\text{Pending}
Sky residual RMSPending\text{Pending}
Fit interpretationPending\text{Pending}
Sampled energy defectPending\text{Pending}
Sampled momentum defectPending\text{Pending}

Physics tutorial

Measuring a mass means measuring a geometry

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

Why it mattersNewtonian two-body dynamics and projected velocity geometry follow the NASA educational binary-mass construction. This Lab adds a conditional regression record and explicit adopted geometry; it does not claim a general orbit-fitting pipeline.

Start with the essentials

Focus question
Which observations determine mass, and which assumptions still remain?
One-sentence intuition
A shared velocity fit measures amplitudes and mass ratio, while total mass needs viewing geometry. Angular motion needs distance as well. Clean residuals alone cannot establish either external assumption.

Core mathematical model

Two bodies, one relative orbit

r1=−M2M1+M2r,r2=M1M1+M2r\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 barycentric orbit. The displayed relative orbit is the separation vector.

Period and total mass

P2=4π2a3G(M1+M2)P^2=\frac{4\pi^2a^3}{G(M_1+M_2)}

In the educational AU-year-solar-mass convention the gravitational constant is four pi squared. The semimajor axis belongs to the relative orbit.

Projected sky motion

X=xcos⁡ω−ysin⁡ω,Y=(xsin⁡ω+ycos⁡ω)cos⁡i,Z=(xsin⁡ω+ycos⁡ω)sin⁡iX=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

Tilt suppresses one sky coordinate and introduces line-of-sight velocity. Angular separation also depends on distance.

Double-lined velocity fit

v1=γ−K1[cos⁡(ν+ω)+ecos⁡ω],v2=γ+K2[cos⁡(ν+ω)+ecos⁡ω]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 offset and two amplitudes. Orbital shape and phase are conditioned on independent information; a biased adopted period changes the fit basis.

What spectroscopy determines

M2M1=K1K2,(M1+M2)sin⁡3i=P(K1+K2)3(1−e2)3/22πG\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 viewing inclination the projected mass is a lower bound, not the true mass. A near face-on record has vanishing radial signal; the Lab requires both amplitudes above three conditional standard errors.

Distance enters cubically

afit=θa,fitdfit,Mast=4π2afit3GPfit2,MRV=Mprojsin⁡3ifita_{\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 angular-axis fit assumes the adopted projection geometry. A distance error rescales astrometric mass cubically even when angular residuals are tiny.

Common difficulties

Projection is not mass

Typical misconceptionA fitted velocity amplitude determines the mass alone.

Better mental modelRadial velocities yield a mass multiplied by the cube of the sine of inclination; inspect the whole allowed family.

Zero is not an equal-mass conclusion

Typical misconceptionFace-on motion means the stars have no mass or do not orbit.

Better mental modelThe orbit remains present on the sky. Both radial signals disappear together, so the ratio is unresolved.

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 share the same velocity record. Distance is external to the angular fit.

Standard errors are conditional

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

Better mental modelIt omits uncertainty in period, orbital shape, phase, distance and inclination. The near face-on inversion is poorly determined.

Run the experiment

  1. 01

    Recover a clean binary

    Acquire the nominal record and compare both mass estimates with truth.

    What to observe: The amplitudes recover the mass ratio and the correct adopted inclination restores total mass.
  2. 02

    Separate viewing from analysis

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

    What to observe: The projected mass changes with viewing angle; the corrected mass agrees only when the adopted angle is right.
  3. 03

    Diagnose a bad period

    Compare the wrong-period and noisy-record presets. Acquire another noisy record.

    What to observe: A period bias creates structured residuals; seeded Gaussian errors vary 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 have a biased mass; near face-on spectroscopy becomes unresolved despite visible orbital motion.