---
title: "Motion Revealed"
subtitle: "The Doppler Effect and the Astronomer's Toolkit"
author: "Dr. Anna Rosen"
date: "2026-02-20"
description: "Spectral lines shift when objects move. The Doppler effect reveals velocities from exoplanets to dark matter. Telescopes collect this light."
draft: true

format:
  revealjs:
    theme: [default, ../../../assets/theme/slides/theme.scss]
    css: ../../../slides.css
    footer: "ASTR 101 • Lecture 10 • Dr. Anna Rosen"
    slide-number: true
    transition: fade
    transition-speed: fast
    center: false
    chalkboard: true
    title-slide-attributes:
      data-background-image: "../../../assets/images/common/cosmic-cliffs-jwst.png"
      data-background-opacity: "0.35"
    pointer:
      key: "q"
      color: "#5dbfbf"
      pointerSize: 18
      alwaysVisible: true
    spotlight:
      size: 80
      toggleSpotlightOnMouseDown: true
      presentingCursor: "crosshair"
      fadeInAndOut: 100
    quiz:
      shuffleOptions: true
      defaultCorrect: "Correct!"
      defaultIncorrect: "Not quite — let's discuss."

revealjs-plugins:
  - pointer
  - spotlight
  - attribution
  - quiz

filters:
  - roughnotation

execute:
  echo: false
  warning: false
  message: false
---

## Join iClicker {background-color="black"}

::: {.text-center}
{{< img iclicker-qr-astr101 width="36%" class="img-center" >}}

::: {.text-xl .font-semibold .mt-2}
[join.iclicker.com/ASES](https://join.iclicker.com/ASES)
:::

::: {.text-sm .text-muted .mt-2}
If you can't scan: open the link and enter the course code **ASES**.
:::
:::

::: {.notes}
30–60 sec. Get everyone connected before starting.
:::

---

## {background-color="black"}

::: {.r-fit-text}
The Fingerprints Move
:::

::: {.fragment .text-center .mt-4}
In L9, we learned spectral lines are chemical fingerprints.
:::

::: {.fragment .text-center .mt-4}
But what if those fingerprints are **shifted** from their expected wavelengths?
:::

::: {.notes}
0–1 min. "Shifted lines mean the source is moving."
:::

---

## Motion Causes Shifts {background-image="../../../assets/images/common/cosmic-cliffs-jwst.png" background-opacity="0.2"}

::: {.text-center .text-xl}
Just like an ambulance siren changes pitch as it passes...
:::

::: {.fragment .mt-4}
Light waves get **compressed** or **stretched** depending on motion.

This is the **Doppler effect**.
:::

::: {.notes}
1–2 min. "The Doppler effect for light."
:::

---

## Today's Learning Objectives {background-image="../../../assets/images/common/cosmic-cliffs-jwst.png" background-opacity="0.2"}

::: {.bg-muted .p-2}

By the end of today, you'll be able to:

::: {.incremental .text-sm}
- Apply the **Doppler formula** to calculate radial velocities
- Distinguish **radial** (line-of-sight) from **transverse** motion
- Explain how Doppler reveals **exoplanets** and **dark matter**
- Compare **light-gathering power** and **resolution** for telescopes
- Explain why we observe from **space** at certain wavelengths
:::

:::

::: {.notes}
2–3 min. "Today completes the astronomer's toolkit."
:::

---

## {.center}

::: {.r-fit-text}
Part 1: The Doppler Effect
:::

*Measuring motion from light*

---

## Predict First! {background-color="#1a1a2e"}

You measure a spectral line at a **longer wavelength** than its lab value ($\Delta\lambda > 0$). Is the source moving…

:::: {.columns .mt-4}
::: {.column width="50%"}
A) Toward us
:::

::: {.column width="50%"}
B) Away from us
:::
::::

::: {.notes}
30 s. Commit first. Then reveal: longer wavelength = redshift = moving away along our line of sight.
:::

---

## The Doppler Formula {background-color="#0a2f5c"}

::: {.text-center .text-xl .bg-muted .p-3}
$$\frac{\Delta\lambda}{\lambda_0} = \frac{v}{c}$$
:::

::: {.fragment .mt-4}
| Symbol | Meaning |
|--------|---------|
| $\lambda_0$ | Rest wavelength (lab measurement) |
| $\Delta\lambda$ | Observed shift from rest |
| $v$ | Radial velocity (toward/away) |
| $c$ | Speed of light (300,000 km/s) |
:::

::: {.notes}
3–4 min. "The Doppler formula connects wavelength shift to velocity."
:::

---

## Blueshift and Redshift {background-color="#1a1a2e"}

:::: {.columns}
::: {.column width="50%"}
**Blueshift**

$\Delta\lambda < 0$

Wavelength **decreased**

Source **approaching**
:::

::: {.column width="50%"}
**Redshift**

$\Delta\lambda > 0$

Wavelength **increased**

Source **receding**
:::
::::

::: {.fragment .mt-4 .text-muted}
These names apply to ALL wavelengths — radio, X-ray, everything. "Blue direction" = shorter; "red direction" = longer.
:::

::: {.notes}
4–5 min. "Blueshift = approaching. Redshift = receding."
:::

---

## Worked Example {background-color="black"}

**Problem:** A star's Hα line is observed at 656.50 nm. Rest wavelength = 656.28 nm. Find the velocity.

::: {.fragment}
$$\Delta\lambda = 656.50 - 656.28 = +0.22 \text{ nm}$$
:::

::: {.fragment}
$$\frac{\Delta\lambda}{\lambda_0} = \frac{0.22}{656.28} = 3.35 \times 10^{-4}$$
:::

::: {.fragment}
$$v = c \times 3.35 \times 10^{-4} = 100 \text{ km/s}$$
:::

::: {.fragment .mt-2}
**Positive shift → redshift → moving away at 100 km/s**
:::

::: {.notes}
5–6 min. Walk through the calculation.
:::

---

## Quick Check: Doppler Calculation {background-color="#1a1a2e"}

::: {.quiz}
A star's sodium D line (rest 589.0 nm) is observed at 588.8 nm. The star is:

- [x] Moving toward us at ~100 km/s
- [ ] Moving away from us at ~100 km/s
- [ ] Moving toward us at ~1000 km/s
- [ ] Not moving
:::

::: {.fragment .mt-2 .text-sm}
**$\Delta\lambda = -0.2$ nm (blueshift) → approaching**
:::

::: {.notes}
1–2 min. iClicker poll, then explain.
:::

---

## Critical Distinction: Radial vs. Transverse {background-color="black"}

{{< fig radial-vs-transverse >}}

::: {.fragment .mt-2}
- **Radial:** Toward/away → Doppler shift detected
- **Transverse:** Across sky → **No Doppler shift!**
:::

::: {.notes}
6–7 min. "Doppler only measures the radial component."
:::

---

## Quick Check: What Doppler Measures {background-color="#1a1a2e"}

::: {.quiz}
A star moves at 100 km/s exactly across the sky (perpendicular to our line of sight). Its Doppler shift is:

- [ ] Large redshift
- [ ] Large blueshift
- [x] Zero
- [ ] Cannot determine
:::

::: {.notes}
1 min. iClicker poll.
:::

---

## {.center}

::: {.r-fit-text}
Exoplanet Detection
:::

*The radial velocity method*

---

## Stars Wobble Too {background-color="black"}

::: {.text-center}
A planet doesn't just orbit its star...
:::

::: {.fragment .mt-4}
By Newton's Third Law, the **star also wobbles** in response.

Both orbit their common center of mass.
:::

::: {.fragment .mt-4}
This tiny wobble is **detectable** with precise Doppler measurements.
:::

::: {.notes}
7–8 min. "The star wobbles, and we can measure it."
:::

---

## {background-color="black"}

{{< fig radial-velocity-method >}}

::: {.fragment .mt-2}
As the star wobbles toward/away from us, spectral lines shift back and forth.

The periodic pattern reveals an orbiting planet.
:::

::: {.notes}
8–9 min. Diagram of the RV method.
:::

---

## Example: Detecting a Hot Jupiter {background-color="#1a1a2e"}

::: {.text-sm}
**51 Pegasi b** (first exoplanet around Sun-like star, 1995):

| Measurement | Value |
|-------------|-------|
| Stellar wobble | ±56 m/s |
| Period | 4.2 days |
| Inferred planet | ~0.5 Jupiter mass |
| Orbit | Very close to star |
:::

::: {.fragment .mt-2}
This "hot Jupiter" was discovered entirely through Doppler shifts!
:::

::: {.notes}
9–10 min. "The discovery that won the 2019 Nobel Prize."
:::

---

## Quick Check: Exoplanet Detection {background-color="#1a1a2e"}

::: {.quiz}
The radial velocity method detects exoplanets by measuring:

- [ ] The planet's light directly
- [x] The star's periodic Doppler shift due to its wobble
- [ ] The planet blocking starlight during transit
- [ ] Gravitational lensing by the planet
:::

::: {.notes}
1 min. iClicker poll.
:::

---

## {.center}

::: {.r-fit-text}
Dark Matter
:::

*When orbits don't add up*

---

## Galaxy Rotation Puzzle {background-color="black"}

::: {.text-center}
1970s: Vera Rubin measured how fast stars orbit within galaxies.
:::

::: {.fragment .mt-4}
**Expected:** Outer stars orbit slower (like outer planets)

$$v_{orbit} = \sqrt{\frac{GM}{r}}$$
:::

::: {.fragment .mt-4}
**Observed:** Outer stars orbit **just as fast** as inner stars!
:::

::: {.notes}
10–11 min. "The galaxy rotation puzzle."
:::

---

## {background-color="black"}

{{< fig galaxy-rotation-curve >}}

::: {.fragment .mt-2}
The flat rotation curve means there's **more mass** than we can see.
:::

::: {.notes}
11–12 min. Galaxy rotation curve showing the dark matter problem.
:::

---

## Dark Matter {background-color="#1a1a2e"}

::: {.text-center .text-xl}
The leading explanation: **dark matter**
:::

::: {.fragment .mt-4}
- Doesn't emit, absorb, or reflect light
- **Does** have gravitational effects
- ~5× more abundant than normal matter
- Forms invisible "halos" around galaxies
:::

::: {.fragment .mt-4 .text-muted}
**Motion reveals mass** — even invisible mass.
:::

::: {.notes}
12–13 min. "Dark matter: we can't see it, but gravity reveals it."
:::

---

## Quick Check: Dark Matter Evidence {background-color="#1a1a2e"}

::: {.quiz}
Galaxy rotation curves provide evidence for dark matter because:

- [ ] Stars at galaxy edges move slower than expected
- [x] Stars at galaxy edges move faster than expected from visible matter
- [ ] Galaxies don't rotate at all
- [ ] We can see dark matter directly
:::

::: {.notes}
1 min. iClicker poll.
:::

---

## {.center}

::: {.r-fit-text}
Part 2: Telescopes
:::

*Collecting the light*

---

## Two Key Properties {background-color="black"}

::: {.text-center}
Telescopes have two main jobs:
:::

:::: {.columns .mt-4}
::: {.column width="50%"}
**1. Collect Light**

More photons → fainter objects detected

$\propto D^2$ (area)
:::

::: {.column width="50%"}
**2. Resolve Detail**

Separate closely-spaced objects

$\theta \propto \lambda/D$
:::
::::

::: {.notes}
13–14 min. "The two key telescope properties."
:::

---

## Light-Gathering Power {background-color="#0a2f5c"}

::: {.text-center .text-xl .bg-muted .p-3}
$$\text{Light collected} \propto D^2$$

**Double the diameter → 4× the light**
:::

::: {.fragment .mt-4}
| Telescope | Diameter | Relative Power |
|-----------|----------|----------------|
| Human eye | 7 mm | 1 |
| Hubble | 2.4 m | ~120,000 |
| Keck | 10 m | ~2,000,000 |
:::

::: {.notes}
14–15 min. "Bigger mirrors collect more light."
:::

---

## Quick Check: Light-Gathering Power {background-color="#1a1a2e"}

::: {.quiz}
Telescope A has twice the diameter of Telescope B. A's light-gathering power is:

- [ ] 2× greater
- [x] 4× greater
- [ ] 8× greater
- [ ] The same
:::

::: {.notes}
1 min. iClicker poll.
:::

---

## Angular Resolution {background-color="#0a2f5c"}

::: {.text-center .text-xl .bg-muted .p-3}
$$\theta = 1.22 \frac{\lambda}{D}$$

**Larger D → finer detail (smaller θ)**
:::

::: {.fragment .mt-4}
- **Larger telescope** → better resolution
- **Shorter wavelength** → better resolution
:::

::: {.fragment .mt-4 .text-muted}
This is why radio telescopes must be huge — radio waves are long.
:::

::: {.notes}
15–16 min. "Resolution depends on diameter and wavelength."
:::

---

## Quick Check: Resolution {background-color="#1a1a2e"}

::: {.quiz}
Which change would improve a telescope's angular resolution?

- [ ] Using a smaller mirror
- [ ] Observing at longer wavelengths
- [x] Using a larger mirror
- [ ] Observing from a lower altitude
:::

::: {.notes}
1 min. iClicker poll.
:::

---

## Why Space Telescopes? {background-color="black"}

::: {.text-sm}
| Wavelength | Ground? | Why Space? |
|------------|---------|------------|
| Radio | ✓ | Atmosphere transparent |
| Infrared | Partial | Water vapor absorbs |
| Visible | ✓ | But atmosphere blurs |
| UV | ✗ | Ozone absorbs |
| X-ray | ✗ | Atmosphere absorbs |
| Gamma | ✗ | Atmosphere absorbs |
:::

::: {.fragment .mt-2}
**Different wavelengths require different observatories.**
:::

::: {.notes}
16–17 min. "Atmospheric windows determine what we can observe from Earth."
:::

---

## Modern Observatories {background-color="#1a1a2e"}

**Ground-based:**

- Keck (10m), VLT (4×8m), ALMA (radio)
- ELT (39m, under construction)

**Space:**

- Hubble (optical/UV)
- JWST (infrared)
- Chandra (X-ray)
- Fermi (gamma-ray)

::: {.notes}
17–18 min. "Modern astronomy uses telescopes across the spectrum."
:::

---

## Key Takeaways {background-color="#0a2f5c"}

::: {.text-sm}

**Doppler Effect:**

1. $\Delta\lambda/\lambda_0 = v/c$ measures **radial velocity** only
2. Blueshift = approaching; Redshift = receding
3. Exoplanets detected via stellar wobble
4. Dark matter revealed by flat galaxy rotation curves

**Telescopes:**

5. Light-gathering power ∝ $D^2$
6. Resolution: $\theta \propto \lambda/D$
7. Space telescopes access blocked wavelengths

:::

::: {.notes}
18–20 min. Rapid recap.
:::

---

## Toolkit Complete! {background-color="black"}

::: {.text-sm}
| Tool | What It Reveals |
|------|-----------------|
| L5-L6: Orbits | Mass (Kepler + Newton) |
| L7-L8: Blackbody | Temperature |
| L9: Spectral lines | Composition |
| L10: Doppler | Motion/velocity |
| L10: Telescopes | How we collect light |
:::

::: {.fragment .mt-4 .text-center}
**Now we apply these tools** — solar system, exoplanets, life in the universe.
:::

::: {.notes}
20–21 min. "Your toolkit is complete."
:::

---

## Coming Up: Capstone Week {background-color="black"}

::: {.text-center .text-xl}
**Lectures 11-13:** Apply the toolkit
:::

::: {.fragment .mt-4}
- L11: Solar System Architecture
- L12: Planetary Climates & Exoplanets
- L13: Are We Alone? (Module 1 Synthesis)
:::

::: {.fragment .mt-4 .text-muted}
**Demo:** Doppler Shift Spectrometer: [../../../demos/doppler-shift-spectrometer/](../../../demos/doppler-shift-spectrometer/)

**Demo:** Telescope Resolution: [../../../demos/telescope-resolution/](../../../demos/telescope-resolution/)
:::

::: {.notes}
21–22 min. Preview the capstone lectures.
:::

---

## {.center background-color="#1a1a2e"}

::: {.r-fit-text}
Numerical Verification
:::

*Calculating radial velocities and telescope properties*

::: {.notes}
Let's verify our equations with explicit calculations.
:::

---

## Doppler Calculation: 51 Pegasi b {background-color="#1a1a2e"}

**The discovery that won the 2019 Nobel Prize!**

**Observed:** Hα line (rest wavelength λ₀ = 656.28 nm) oscillates between:

- Maximum: λ = 656.29 nm
- Minimum: λ = 656.27 nm

::: {.fragment}
**Step 1: Calculate the shift**

$$\Delta\lambda_{\text{max}} = 656.29 - 656.28 = +0.01 \text{ nm (redshift)}$$
$$\Delta\lambda_{\text{min}} = 656.27 - 656.28 = -0.01 \text{ nm (blueshift)}$$
:::

::: {.notes}
"The star wobbles toward us and away from us as the planet orbits."
:::

---

## 51 Pegasi: Velocity Calculation {background-color="#1a1a2e"}

**Step 2: Apply Doppler formula**

$$v = c \times \frac{\Delta\lambda}{\lambda_0}$$

::: {.fragment}
$$v = (3.00 \times 10^8 \text{ m/s}) \times \frac{0.01 \text{ nm}}{656.28 \text{ nm}}$$
:::

::: {.fragment}
$$v = 3.00 \times 10^8 \times 1.52 \times 10^{-5} \text{ m/s}$$
:::

::: {.fragment}
$$v = 4570 \text{ m/s} \approx 4.6 \text{ km/s}$$
:::

::: {.fragment .bg-muted .p-2}
**Unit check:** $\frac{\text{m/s} \times \text{nm}}{\text{nm}} = \text{m/s}$ ✓

The star wobbles at ~5 km/s — walking speed for an elephant, but detectable!
:::

::: {.notes}
This example uses a deliberately large shift for arithmetic clarity. Modern instruments can measure much smaller velocity signals (we won’t need a specific number today).
:::

---

## From Stellar Wobble to Planet Mass {background-color="#1a1a2e"}

**Given:**

- Star's wobble velocity: $v_* = 56$ m/s
- Orbital period: $P = 4.23$ days
- Star's mass: $M_* = 1.06 M_\odot$

::: {.fragment}
**Step 1: Find semi-major axis from Kepler III**

First convert period: $P = 4.23$ days $\times \frac{1 \text{ yr}}{365.25 \text{ days}} = 0.0116$ yr
:::

::: {.fragment}
$$a^3 = M_* \times P^2 = 1.06 \times (0.0116)^2 = 1.43 \times 10^{-4} \text{ AU}^3$$
$$a = 0.052 \text{ AU}$$
:::

::: {.fragment .mt-2 .text-muted}
This uses Newton’s Kepler III in a convenient unit system: $a$ in AU, $P$ in years, $M_*$ in solar masses, and assumes $M_p \ll M_*$.
:::

::: {.notes}
"The planet orbits very close to its star — a 'hot Jupiter'."
:::

---

## Planet Mass Calculation {background-color="#1a1a2e"}

**Step 2: Use momentum conservation**

The star and planet orbit their common center of mass:

$$M_* v_* = M_p v_p$$

::: {.fragment}
**Planet's orbital velocity:**

$$v_p = \frac{2\pi a}{P} = \frac{2\pi \times 0.052 \text{ AU}}{4.23 \text{ days}}$$
:::

::: {.fragment}
Convert: 0.052 AU = $7.78 \times 10^9$ m; 4.23 days = $3.65 \times 10^5$ s

$$v_p = \frac{2\pi \times 7.78 \times 10^9 \text{ m}}{3.65 \times 10^5 \text{ s}} = 1.34 \times 10^5 \text{ m/s} = 134 \text{ km/s}$$
:::

::: {.notes}
"The planet moves much faster than the star wobbles."
:::

---

## Planet Mass: Final Calculation {background-color="#1a1a2e"}

$$M_p = M_* \times \frac{v_*}{v_p} = 1.06 M_\odot \times \frac{56 \text{ m/s}}{134,000 \text{ m/s}}$$

::: {.fragment}
$$M_p = 1.06 M_\odot \times 4.2 \times 10^{-4}$$
:::

::: {.fragment}
Convert to Jupiter masses ($M_J = 10^{-3} M_\odot$):

$$M_p = 1.06 \times 4.2 \times 10^{-4} M_\odot = 4.45 \times 10^{-4} M_\odot$$
$$M_p = 0.45 M_J$$
:::

::: {.fragment .bg-success-subtle .p-2}
**51 Pegasi b has about half Jupiter's mass.**
:::

::: {.notes}
"From a tiny wavelength shift, we determined a planet's mass from 50 light-years away!"
:::

---

## Telescope Resolution: JWST {background-color="#1a1a2e"}

**Given:**

- JWST mirror diameter: $D = 6.5$ m
- Observing wavelength: $\lambda = 2000$ nm (infrared)

::: {.fragment}
$$\theta = 1.22 \frac{\lambda}{D} = 1.22 \times \frac{2000 \times 10^{-9} \text{ m}}{6.5 \text{ m}}$$
:::

::: {.fragment}
$$\theta = 1.22 \times 3.08 \times 10^{-7} = 3.76 \times 10^{-7} \text{ radians}$$
:::

::: {.fragment}
**Convert to arcseconds:**

$$\theta = 3.76 \times 10^{-7} \text{ rad} \times \frac{206,265 \text{ arcsec}}{1 \text{ rad}} = 0.078 \text{ arcsec}$$
:::

::: {.fragment .bg-muted .p-2}
**JWST resolves ~0.08 arcsec at 2 μm** — a dime from 50 km away!
:::

::: {.notes}
"The conversion factor 206,265 arcsec/rad comes from (360° × 3600 arcsec/°)/(2π rad)."
:::

---

## Comparing Telescope Resolutions {background-color="#1a1a2e"}

**Resolution = 1.22 λ/D (in arcseconds)**

| Telescope | D | λ | θ (arcsec) | Calculation |
|-----------|---|---|------------|-------------|
| Human eye | 7 mm | 550 nm | 20" | $1.22 \times 550/7 \times 10^6 \times 206265$ |
| Hubble | 2.4 m | 550 nm | 0.05" | $1.22 \times 550/2.4 \times 10^9 \times 206265$ |
| JWST | 6.5 m | 2000 nm | 0.08" | $1.22 \times 2000/6.5 \times 10^9 \times 206265$ |
| Keck | 10 m | 2000 nm | 0.05" | $1.22 \times 2000/10 \times 10^9 \times 206265$ |

::: {.fragment .mt-2}
**Note:** JWST at IR has similar resolution to Hubble at visible because of the wavelength difference!
:::

::: {.notes}
"Longer wavelengths need larger mirrors to achieve the same resolution."
:::

---

## Light-Gathering Power: Full Calculation {background-color="#1a1a2e"}

**How much more light does Keck collect than your eye?**

::: {.fragment}
**Human eye (dark-adapted):** $D_{eye} = 7$ mm = 0.007 m
**Keck:** $D_{Keck} = 10$ m
:::

::: {.fragment}
**Ratio:**

$$\frac{\text{Power}_{Keck}}{\text{Power}_{eye}} = \left(\frac{D_{Keck}}{D_{eye}}\right)^2 = \left(\frac{10 \text{ m}}{0.007 \text{ m}}\right)^2$$
:::

::: {.fragment}
$$= (1429)^2 = 2.04 \times 10^6$$
:::

::: {.fragment .bg-success-subtle .p-2}
**Keck collects 2 million times more light than your eye!**
:::

::: {.notes}
"This is why we build big telescopes. More photons means fainter objects visible."
:::

---

## {.center background-color="#0a2f5c"}

::: {.r-fit-text}
Deeper Dive
:::

*Relativistic Doppler and interferometry*

::: {.notes}
Optional advanced content.
:::

---

## When v Approaches c: Relativistic Doppler {background-color="#0a2f5c"}

Our simple formula $\Delta\lambda/\lambda = v/c$ assumes $v \ll c$.

**For fast objects, use the relativistic formula:**

$$\frac{\lambda_{observed}}{\lambda_{rest}} = \sqrt{\frac{1 + v/c}{1 - v/c}}$$

::: {.fragment}
**Example:** A quasar receding at $v = 0.9c$

$$\frac{\lambda_{obs}}{\lambda_{rest}} = \sqrt{\frac{1.9}{0.1}} = \sqrt{19} = 4.36$$

Light is **redshifted by a factor of 4.36!**
:::

::: {.notes}
"Distant quasars and galaxies require relativistic corrections."
:::

---

## Cosmological Redshift {background-color="#0a2f5c"}

**For very distant galaxies, there's another effect:**

The **expansion of space itself** stretches light waves.

$$z = \frac{\lambda_{obs} - \lambda_{rest}}{\lambda_{rest}} = \frac{\Delta\lambda}{\lambda}$$

::: {.fragment}
**Relationship to expansion:**

$$1 + z = \frac{a_{now}}{a_{then}}$$

where $a$ is the cosmic scale factor.
:::

::: {.fragment .mt-2 .text-muted}
A galaxy at $z = 1$ has light that was emitted when the universe was **half its current size**.
:::

::: {.notes}
"Cosmological redshift isn't really a Doppler shift — it's space expanding."
:::

---

## Interferometry: Virtual Giant Telescopes {background-color="#0a2f5c"}

**Problem:** Building a 100 m mirror is impractical.

**Solution:** Combine light from multiple telescopes!

::: {.fragment}
**Resolution:**

$$\theta = 1.22 \frac{\lambda}{B}$$

where $B$ = **baseline** (distance between telescopes)
:::

::: {.fragment}
**Examples:**

- VLBA (radio): $B = 8600$ km → milliarcsecond resolution
- Event Horizon Telescope: $B$ = Earth diameter → imaged black hole shadows!
:::

::: {.notes}
"This is how we got the famous M87 black hole image."
:::

---

## EHT Black Hole Image: The Math {background-color="#0a2f5c"}

**Event Horizon Telescope parameters:**

- Baseline: $B \approx 10,000$ km = $10^7$ m
- Wavelength: $\lambda = 1.3$ mm = $1.3 \times 10^{-3}$ m

::: {.fragment}
$$\theta = 1.22 \times \frac{1.3 \times 10^{-3}}{10^7} = 1.6 \times 10^{-10} \text{ rad}$$
:::

::: {.fragment}
**Convert to microarcseconds:**

$$\theta = 1.6 \times 10^{-10} \times 206,265 \times 10^6 = 33 \text{ μas}$$
:::

::: {.fragment .bg-muted .p-2}
**EHT resolution: ~30 microarcseconds**

That's like reading a newspaper in New York from Los Angeles!
:::

::: {.notes}
"This resolution was needed to image M87's black hole shadow (about 40 μas)."
:::

---

## Adaptive Optics: Beating the Atmosphere {background-color="#0a2f5c"}

**Problem:** Earth's atmosphere blurs images ("seeing").

**Solution:** Deformable mirrors that correct in real-time.

::: {.fragment}
**How it works:**

1. Observe a bright reference star (or create one with a laser)
2. Measure how atmosphere distorts its image
3. Flex a mirror hundreds of times per second to compensate
:::

::: {.fragment .mt-2}
**Result:** Ground-based telescopes achieve near-space resolution!

Keck with AO: 0.04 arcsec (better than Hubble's 0.05 arcsec)
:::

::: {.notes}
"Adaptive optics turns 10 m ground telescopes into Hubble-beaters."
:::

---

## Questions? {background-image="../../../assets/images/common/cosmic-cliffs-jwst.png" background-opacity="0.25"}

::: {.text-center .text-xl .mt-4}
**Doppler reveals motion.**

**Telescopes collect the light.**

**Your toolkit is complete.**
:::

::: {.notes}
Remaining time for questions.
:::
