flowchart LR
O["🔭 Observable\nWhat we measure"]
M["📐 Model\nWhat we assume"]
I["💡 Inference\nWhat we derive"]
O --> M --> I
style O fill:#dbeafe,stroke:#2563eb
style M fill:#fef3c7,stroke:#d97706
style I fill:#dcfce7,stroke:#16a34a
Step
This lecture
Observable
Position shifts, brightness
Model
Parallax geometry, ISL
Inference
Distance, luminosity
Good scientists always ask: what am I assuming, and what would break it?
Angular Measure
The language of the sky
Degrees, Arcminutes, Arcseconds
Unit
Size
\(1^\circ\)
Full Moon width
\(1'\) (arcmin)
\(1/60\) of a degree
\(1''\) (arcsec)
\(1/3600\) of a degree
Credit: cococubed.com
Stellar parallax happens at the arcsecond scale — tiny shifts against distant background stars.
Radians — The Natural Unit
Definition: 1 radian is the angle subtended by an arc whose length equals the radius.
Gaia’s \(100\times\) better precision directly translates to \(100\times\) better luminosity measurements for every star observed.
Active Learning Break 1: Think-Pair-Share
Problem: A star has parallax \(p = 0.050''\).
(a) What is its distance in parsecs?
(b) Convert to light-years.
(c) Could Hipparcos have measured this parallax? Could Gaia?
(d) Estimate the parallax of a star at the center of the Milky Way (\(\sim 8\) kpc away).
Credit: cococubed.com
Reference: parallax geometry
Answers: (a) \(d = 1/0.050 = 20\) pc. (b) \(20 \times 3.26 = 65\) ly. (c) Yes (Hipparcos limit \(\sim 1\) mas \(= 0.001''\), this is much larger). (d) \(p = 1/8000 = 0.000125'' = 125\,\mu\)as — detectable by Gaia.
The Inverse-Square Law
How light spreads through space
Predict First
If you move twice as far from a light source, by what factor does the brightness change?
Think about why before we derive it.
Geometric Derivation: Concentric Spheres
A star emits luminosity \(L\) uniformly in all directions.
At distance \(d\), this energy is spread over a sphere of area \(4\pi d^2\).
Flux = energy per time per area:
\[\boxed{F = \frac{L}{4\pi d^2}}\]
Credit: Course illustration (A. Rosen)
Double the distance → flux drops by 4×. Ten times farther → flux drops by 100×.
Quick Check 4: Inverse-Square Law
You move 3× farther from a lamp. By what factor does the brightness drop?
A. 3× dimmer
B. 6× dimmer
C. 9× dimmer
D. 27× dimmer
Dimensional Analysis Check
Assume \(F \propto L^a \, d^b\)
Quantity
Dimension
\(F\) (flux)
\([M\,T^{-3}]\)
\(L\) (luminosity)
\([M\,L^2\,T^{-3}]\)
\(d\) (distance)
\([L]\)
Match exponents:
Mass: \(1 = a\) → \(a = 1\)
Time: \(-3 = -3a\) ✓
Length: \(0 = 2a + b\) → \(b = -2\)
\[F \propto \frac{L}{d^2}\]
The \(4\pi\) comes from geometry — DA gives the scaling.
Where Does the 4π Come From?
Credit: Fundamentals of Astrophysics (Owocki)
A full sphere subtends \(4\pi\) steradians of solid angle.
Light from an isotropic source spreads evenly over this solid angle. At distance \(d\), the sphere has area \(4\pi d^2\).
So \(F = L/(4\pi d^2)\) simply says: total luminosity ÷ total area of the sphere you’re standing on.
Flux vs. Luminosity — Don’t Confuse Them!
Flux \(F\) (what you measure)
Depends on where you stand
Units: erg cm\(^{-2}\) s\(^{-1}\)
Changes with distance
Luminosity \(L\) (what the star emits)
Intrinsic property of the star
Units: erg s\(^{-1}\)
Does not depend on distance
A dim red dwarf nearby can have the same observed flux as a luminous blue giant far away. You cannot determine luminosity from flux alone — you always need distance.
A Useful Surprise: Surface Brightness Is Distance-Independent
Credit: Fundamentals of Astrophysics (Owocki)
Flux drops as \(1/d^2\), but angular area also drops as \(1/d^2\).
This star at \(L \approx 0.08\,L_\odot\) sits on the lower main sequence — a late K or early M dwarf. These small, cool, red stars dominate the Galaxy by number.
Active Learning Break 2: Synthesis Problem
A star has parallax \(p = 0.1''\) and flux \(F = 10^{-11}\) erg cm\(^{-2}\) s\(^{-1}\).
Distance? (\(d = 10\) pc \(= 3.09 \times 10^{19}\) cm)
In solar luminosities? (\(L/L_\odot \approx 3 \times 10^{-5}\))
What kind of star? (Very faint red dwarf — \(\sim 30{,}000\times\) dimmer than the Sun!)
Credit: ESO
Where does your answer land on the HR diagram?
Quick Check 5: Putting It Together
Two stars have identical flux. Star A is at 10 pc, Star B at 100 pc. Which has higher luminosity, and by what factor?
A. Star A, by 10×
B. Star B, by 10×
C. Star A, by 100×
D. Star B, by 100×
Beyond Parallax
Standard candles and the distance ladder
The Parallax Horizon
Parallax works out to:
\(\sim 100\) pc from ground-based telescopes
\(\sim 10\) kpc with Gaia
But the Milky Way is \(\sim 30\) kpc across, and Andromeda is \(\sim 770\) kpc away.
How do we measure distances beyond parallax?
Standard Candles: Using the ISL Backwards
If we know a star’s luminosity \(L\), we can measure its flux \(F\) and infer distance:
\[\boxed{d = \sqrt{\frac{L}{4\pi F}}}\]
Objects with known or determinable luminosity are called standard candles.
Credit: Course illustration (A. Rosen)
Cepheids and Type Ia Supernovae
Cepheid Variables
Pulsating stars; period → luminosity
Discovered by Henrietta Leavitt (1912)
Reach: \(\sim 30\) Mpc
Hubble used them to prove Andromeda is a separate galaxy (1920s)
Type Ia Supernovae
Thermonuclear explosions of white dwarfs
Peak luminosity: ~few \(\times 10^9 L_\odot\)
Reach: \(\sim 10\) Gpc (cosmological!)
Led to the discovery of dark energy (2011 Nobel Prize)
Credit: cococubed.com
Massive, luminous stars are rare and distant — we need long-reach methods to study them. Standard candles solve this.
The Cosmic Distance Ladder
Credit: Course illustration (A. Rosen)
Each rung is calibrated by the one below. The entire cosmic distance scale rests on getting nearby parallaxes right — which is why Gaia matters for all of astronomy.
Wrapping Up
Summary: Key Equations
Equation
What it does
\(\alpha \approx s/d\)
Angular size ↔︎ true size + distance
\(d\,(\text{pc}) = 1/p\,('')\)
Parallax → distance
\(F = L/(4\pi d^2)\)
Inverse-square law
\(L = 4\pi d^2 F\)
Luminosity from flux + distance
\(d = \sqrt{L/(4\pi F)}\)
Standard candle distance
Credit: cococubed.com
flowchart LR
P["Position shift\n(Observable)"] --> G["Parallax geometry\n(Model)"]
G --> D["Distance d\n(Inference)"]
F["Flux F\n(Observable)"] --> ISL["Inverse-square law\n(Model)"]
D --> ISL
ISL --> L["Luminosity L\n(Inference)"]
L --> HR["HR Diagram"]
style P fill:#dbeafe,stroke:#2563eb
style F fill:#dbeafe,stroke:#2563eb
style G fill:#fef3c7,stroke:#d97706
style ISL fill:#fef3c7,stroke:#d97706
style D fill:#dcfce7,stroke:#16a34a
style L fill:#dcfce7,stroke:#16a34a
style HR fill:#f3e8ff,stroke:#9333ea
The Takeaway
If you forget everything else from today, remember this:
Distance is the master key.
Measure distance, and luminosity becomes knowable — the first step to understanding any star.
Questions?
Common questions at this point:
“Why can’t we just use radar to measure star distances?”
“What if dust absorbs the light — doesn’t that break the inverse-square law?”
“How do we know the period-luminosity relation for Cepheids is correct?”
Looking Ahead
Next time (Lecture 2): Surface Flux & Colors of Stars
Stefan-Boltzmann law: \(L = 4\pi R^2 \sigma T^4\)
Wien’s law: peak wavelength → temperature
Stellar radii from luminosity + temperature
Before then:
Read: Lecture 1 reading (distance & parallax)
Try: Practice problems 3–5 (parallax conversions & luminosity calculations)
