Chapter 1 — History, Coordinates & Foundations

1.1 The Long History of Astronomy

Astronomy is humanity’s oldest science. Ancient peoples tracked cycles of Sun, Moon, and stars to regulate agriculture and ritual. Over time, those records evolved into predictive mathematical astronomy.

Babylonians discovered the Saros eclipse cycle, Greeks applied geometry to stellar positions, and Islamic scholars refined observational accuracy. With Galileo’s telescope, the cosmos revealed new structures. Newton’s synthesis in 1687 unified all motion with a single law of gravitation.

Accuracy leaps
Stonehenge (~1°), Hipparchus (~20′), Tycho Brahe (~1′), telescopes (~1″), Gaia (~10 μas). Each leap in precision birthed new physical theory.

1.2 The Celestial Sphere

To describe star positions, we use a model of a sphere surrounding Earth. Two common coordinate systems:

  • Horizontal (Alt–Az): altitude \(h\), azimuth \(A\).
  • Equatorial (RA–Dec): declination \(\delta\), right ascension \(\alpha\).
Transform
\[ \sin h = \sin\phi\sin\delta + \cos\phi\cos\delta\cos H, \] with \(H=LST-\alpha\).
Deeper Math: Derivation via Spherical Trig
Consider the spherical triangle formed by celestial pole, object, and zenith. Apply the spherical law of cosines: \[ \cos z = \sin\phi\sin\delta + \cos\phi\cos\delta\cos H, \] where \(z=90^\circ-h\). Substituting gives the altitude formula.

1.3 Newtonian Laws

Newton’s laws of motion + universal gravitation unify sky and Earth. Example: derive Kepler’s 3rd law.

Kepler’s 3rd Law
\[ P^2=\frac{4\pi^2}{GM}a^3. \]
Deeper Math: Energy & Angular Momentum
From energy: \(\epsilon=v^2/2-\mu/r\). For bound orbit, \(\epsilon=-\mu/2a\). Angular momentum \(h=\sqrt{\mu a(1-e^2)}\). Together → ellipse parameters, periapsis/apapsis speeds, and Kepler’s laws.

1.4 Units, Parallax & Magnitudes

Distance is measured via parallax:

\[ d(\text{pc})=\frac{1}{\pi('')} \]

Brightness is measured logarithmically:

\[ m_1-m_2=-2.5\log_{10}(F_1/F_2). \]

Chapter 2 — Earth’s Rotation & Timekeeping

2.1 Sidereal vs Solar Day

Earth spins relative to stars in 23h 56m 4s. Because Earth also orbits, the Sun returns to meridian after ~24h. The difference accumulates into leap years and calendars.

Relation
\[ T_{sid}=\frac{T_\odot}{1+1/365.2422}. \]

2.2 Tidal Braking & Nutation

Lunar tides slow Earth’s rotation (~2 ms/century). Nutation adds short-term wobbles (9.2″ amplitude, 18.6 yr).

Deeper Math: Torque & Angular Momentum
Torque from tidal bulge: \(\tau\sim \frac{3k_2GM_M^2R_E^5}{a^6}\). Rate of change in rotation: \(\dot T=T\tau/L\), with \(L=C\omega\).

2.3 Precession

Earth’s axis precesses ~25,772 yr. Caused by torque on equatorial bulge.

\[ \dot\psi \approx 50.29''/yr. \]

2.4 Aberration of Light

Aberration: apparent displacement of stars due to Earth’s velocity.

\[ \theta_{ab}=\frac{v}{c}\sin\beta. \]

Chapter 3 — The Nature of Light

3.1 Waves & Photons

Maxwell’s equations predict EM waves. Quantum mechanics adds the photon: \[ E=h\nu=\frac{hc}{\lambda}. \] Duality explains diffraction (wave) and photoelectric effect (particle).

3.2 Blackbody Radiation

Stars approximate blackbodies. Their spectrum follows Planck’s law.

\[ B_\nu(T)=\frac{2h\nu^3}{c^2}\frac{1}{e^{h\nu/kT}-1}. \]
\[ \lambda_{max}T=2.90\times10^{-3}\,mK. \]
\[ F=\sigma T^4. \]
Deeper Math: Planck Derivation Sketch
Energy density of photons in cavity: \(\rho(\nu)d\nu=\frac{8\pi\nu^2}{c^3}\frac{h\nu}{e^{h\nu/kT}-1}d\nu.\) Multiplying by \(c/4\) gives radiative flux per frequency, yielding Planck’s law.

3.3 Spectral Lines

Spectral lines arise from quantum transitions. Line width sources:

  • Natural (uncertainty principle)
  • Doppler (thermal motions)
  • Pressure (collisions)
\[ \frac{\Delta\lambda}{\lambda}=\sqrt{\frac{2kT}{mc^2}}. \]

Chapter 4 — Orbital Mechanics & Spaceflight

Skeleton content (to be expanded later).

\[ v^2=\mu\left(\frac{2}{r}-\frac{1}{a}\right). \]

Chapter 5 — Photometry, Magnitudes & Signal-to-Noise

Skeleton content (to be expanded later).

\[ m-M=5\log_{10}(d/10\,pc). \]