SCIENCE · INVESTIGATE AND EXPLAIN

Combine size and reflex motion without dropping finite mass

A transit constrains a size ratio under photometric and geometric assumptions, while periodic stellar reflex motion constrains masses with an inclination dependence. Combining independent signals can constrain density, but only after checking that they describe the same physical system and accounting for stellar properties.

Learning outcome: Invert the reflex relation, reconstruct its amplitude, then combine it with two different supported size ratios. Retain the stellar normalization and explain why a composition claim remains nonunique.

Starting recap

Keep interior, thermal, surface-history, orbital and detection assumptions distinct before drawing a planetary inference.

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Try

At K=0.002, the exact q is slightly above 0.002 because the denominator exceeds one. With k=0.1, mean density ratio is 1000q and gravity ratio 100q. Reconstructing K from the solved q checks the inversion independently of the transit depth.

Open the experiment

Predict

Why does the exact finite-mass solution exceed the small-planet estimate q≈K?

Your prediction and reason

Experiment

Invert the reflex relation, reconstruct its amplitude, then combine it with two different supported size ratios. Retain the stellar normalization and explain why a composition claim remains nonunique.

Combine a radius signal and an exact circular reflex mass function

Worked teaching model · synthetic values · Original edge-on circular system G=1, stellar mass 1, stellar radius 1 and period 2pi. K=q/(1+q)^(2/3), q=planet/star mass ratio; solve exact finite mass function. Transit depth supplies k² for uniform unblended full overlap. Mean density ratio=q/k³, gravity ratio=q/k². Physical interpretation still depends on stellar parameters and genuine signal validation.

Stellar reflex velocity amplitude: 0.002 normalized speed Transit radius-ratio case: 0.05,0.10,0.15: 1 case

Reset keeps your prediction, captures and practice history.

Independent radius and reflex-motion information can constrain density, but not a unique interior.

Stellar mass/radius, inclination, eccentricity, dilution and activity matter. These are exact supplied circular signals; a real candidate requires false-positive tests and uncertainty propagation before interpreting its bulk density as composition.

Original conditional model ledger
Quantity	Value	Unit
Exact planet/star mass ratio	0.002002669	ratio
Transit radius ratio	0.1000000	ratio
Uniform unblended transit depth	0.01000000	fraction
Planet/star mean-density ratio	2.002669	ratio
Planet/star surface-gravity ratio	0.2002669	ratio
Relative circular semimajor axis	1.000667	normalized length
Reconstructed stellar reflex amplitude	0.002000000	normalized speed
Small-mass estimate K	0.002000000	approximate mass ratio

Original supplied models, not actual planet observations or a unique interior, atmosphere or discovery claim.

Independent radius and reflex-motion information can constrain density, but not a unique interior.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

1. Define the planetary evidence

2. Connect mechanism and inference

The original circular edge-on system uses G=1, stellar mass and radius one, and period 2π. If q is planet-to-star mass ratio, Kepler’s law and the barycentric orbit give stellar velocity amplitude K=q/(1+q)^(2/3). The workbench solves this relation rather than replacing the total system mass by the stellar mass without disclosure.

For an unblended uniform fully transited star, depth gives k². Mean density relative to the star is q/k³, and surface gravity ratio is q/k². The cubic size dependence makes radius uncertainty important. Neither a high nor a low density identifies a unique composition once compression, temperature and layered materials are considered.

The small-planet approximation q≈K is displayed as an approximation, not the exact solved value. Eccentricity, non-edge-on inclination, stellar activity and blended spectra would require additional modeling. These normalized records are original controlled examples, not actual radial-velocity measurements or newly validated planet discoveries.

3. Work the supplied case

4. Predict and investigate

Why does the exact finite-mass solution exceed the small-planet estimate q≈K?

Apply the evidence to a changed case

At the same K compare k=0.05 and k=0.1. How do inferred density and gravity ratios change?

Retain the evidence frame, calculations and assumptions before opening public feedback.

Compare a worked response

Doubling k at fixed solved mass divides density by eight and surface gravity by four. Transit depth rises by four. These different powers prevent substituting one signal ratio for another.

Source and boundaries

Primary source. Checked 31 October 2026. Original diagrams and supplied model records are not actual planetary observations. No direct solar viewing, instrument operation or physical experiment is requested.

Apply

Public practice checks a specific claim. It does not establish independent mastery.

At the same K compare k=0.05 and k=0.1. How do inferred density and gravity ratios change?

Your explanation and limitations

Continue

Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.

Skipping retains experiments and awards no mastery.

Previous: Separate a transit radius signal from blended light Next: Distinguish detected planets from intrinsic occurrence

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