Parallax Distance: Measuring the Stars
Cause: Orbit Geometry (Top View)
Observable: Detector/Sky Shift
Measured shift $\Delta\theta$
mas
Signed: mas
Effective baseline $B_{\rm eff}$
AU
Chord AU, $\Delta\phi =$ $^{\circ}$
Inferred parallax $\hat p \pm \sigma_{\hat p}$
mas
arcsec | Equivalent Jan-Jul shift $2\hat p$ (derived): mas
True distance (set) $d_{\rm true}$
pc
ly
Inferred distance (measured) $\hat d \pm \sigma_{\hat d}$
pc
ly
Signal-to-noise $\hat p/\sigma_{\hat p}$ (inferred)
Measurement quality:
- Cause: Earth moves continuously, so the line-of-sight changes.
- Observable: the target shifts against fixed background stars on the detector.
- Inference: capture A and B, read $\Delta\theta$, then infer $\hat p$ and $\hat d$.
- Distance relation: $d\,(\mathrm{pc}) = 1/p\,(\mathrm{arcsec})$.
- General capture inference uses projected shift and effective baseline $B_{\rm eff}$.
- The target's apparent shift is opposite to Earth's displacement along the parallax axis, the way each Earth-to-star sightline leans.
- Signed $\Delta\theta$ is positive when B appears right of A on the detector.
- The orbit view is not to scale: the star is drawn just beyond Earth's orbit, but at $10\,\mathrm{pc}$ it is about $2\times10^{6}$ times farther from the Sun than Earth is.
- Detector exaggeration changes only visualization, not computed $\hat p$ or $\hat d$.
- Parallax is not star motion; observer baseline changes drive the apparent shift.
- Small effective baseline captures can make inference unstable even if the chord is large.
- Larger measurement uncertainty $\sigma_{\rm meas}$ can dominate tiny shifts, weakening distance inference.