Binary Orbits: Dynamical Reasoning Lab

Name: ________________________________ Section: __________ Date: __________

Station: __________ Group members: ________________________________________________

Goal: Use the demo to build a constraint-based explanation (prediction + invariants + observable).

Station card: Binary Orbits (8–10 minutes) Controls: $M_2/M_1$ ($0.01\rightarrow1$), separation $a$ (AU, log scale), inclination $i$ (deg) Readouts: $a_1$, $a_2$, $v_1$, $v_2$, $\omega$, momentum check, $K_1$, $K_2$, $K$, $U$, $E$, period $P$

Your station artifact (fill in):

1. Capture + predict: click Capture current state, then lower $M_2/M_1$ and predict whether $P$, $v_1$, and $a_1$ increase/decrease/same.
2. Compare against the live update: keep watching the live orbit/readouts, then use Compare with current system to record which trends matched.
3. Invariant check: use “What must be true?” to select all must-hold statements and avoid distractors.
4. Observable link: set two inclinations (e.g., $i=20^\circ$ and $i=80^\circ$) and compare $K_1$, $K_2$.
5. Energy check: switch to Energy view, change separation, and describe how $E$ moves with $a$.
6. RV inversion challenge: measure both amplitudes from the RV panel, compute inferred $q=K_1/K_2$, reveal, and record error.
7. Connection sentence: “From this instrument, we can infer stellar masses from spectra because …”
8. Integrity check: use the live Physics integrity card to verify $a_1+a_2=a$, $M_1a_1=M_2a_2$, and $K_1/K_2=q$. Only the unrevealed RV challenge should lock snapshots/Copy Results.

Word bank + sanity checks Word bank:

• Barycentric frame: center of mass at rest; total linear momentum is zero.
• Momentum balance: $M_1v_1 = M_2v_2$.
• Shared angular frequency: both bodies have one $\omega$, so they share period $P$.
• Projected velocity: radial-velocity amplitude scales as $K = v\sin i$.
• Orbital energy (circular): $E = -\dfrac{G M_1 M_2}{2a}$.

Must-hold relationships:

$$a_1 + a_2 = a, \qquad M_1a_1 = M_2a_2, \qquad \frac{v_1}{v_2} = \frac{M_2}{M_1}, \qquad P_1 = P_2.$$

Sanity checks:

• If $M_1=M_2$, then $a_1=a_2$ and $v_1=v_2$.
• If $M_2 \ll M_1$, then $a_1 \to 0$ (planet limit).
• If $i\to0^\circ$, then $K_1, K_2 \to 0$ even though the orbit still exists.