AST-15

Stellar parallax

d(pc) = 1 / p("). The definition of the parsec.

Reading speed

StarsParallax

Governing equation

d=1/p

$p \in \left[0.01, 800\right]$

where

p: Parallax (mas)
d: Distance (pc)
\mu: Distance modulus (mag)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

00:0 / 00:08

①

Governing law — red

d=1/p

$p \in \left[0.01, 800\right]$

$p = 0.01$ $d = 1\times 10^{5}$

Along the curve

②

Domain of the variable — blue

p_{\min} = 0.01,\quad p_{\max} = 800

$p \in \left[0.01, 800\right]$

$p = 0.01$ $p = 0.01$

Along the curve

③

Response curve — green

d=1/p

$p \in \left[0.01, 800\right]$

$p = 0.01$ $d = 1\times 10^{5}$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}d}{\mathrm{d}p}

$p \in \left[0.01, 800\right]$

$p = 5.041$ $d^{\prime} = -9927.8$

Along the curve

Lecture brief

Historical brief

Hubble expansion, Jeans collapse, Eddington luminosity, Bondi accretion and Stefan–Boltzmann stars are the first astrophysical budgets. The sheets scale a star, a cloud and a horizon. This sheet (AST-15 — Stellar parallax) is the form associated with Parallax. Working symbols: $p$ $\rightarrow$ $d$ , $\mu$ . Gaia measures p to tens of µas. Distance modulus μ = 5 log₁₀ d_pc − 5.

Purpose

Purpose: compute

d

\mu

from

p

in Astrophysics via

d=1/p

d(pc) = 1 / p("). The definition of the parsec. Use it when a real astrophysics question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

p = 10.000\,\mathrm{mas}

, the governing relation

d=1/p

yields

d = 100.00\,\mathrm{pc}

\mu = 5.00\,\mathrm{mag}

. A baseline AU, a tiny angle, a star. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Parallax p10 mas

Outputs

Distance d100.00 pc
Distance modulus \mu5.00 mag

Reading speed

Watch on YouTube

Free library

Full library

Free PDF / open book

YouTube channels

00:0 / 00:08

Narration of this film

A baseline AU, a tiny angle, a star.

Gaia measures p to tens of µas. Distance modulus μ = 5 log₁₀ d_pc − 5.

Reading speed

Watch on YouTube