Welcome to Introductory Astronomy
I have created this course for folk interested in learning some Astronomy basics. The target audience will be undergraduates taking their first leap into Astronomy and Astrophysics. That said, there will be more advanced topics spread throughout for those interested.
Since this is a science-based course, there will be some prerequisites. You will need to know basic algebra, trigonometry, and perhaps some calculus. Now, if you do not have that background, it doesn’t mean you can’t learn along the way. I will have links to problem-solving sessions after each chapter (when relevant), and during those, I will teach you what you need to know. Of course, that will not cover every nook and cranny of basic mathematics.
Here is an excellent quote I once heard from my first Introduction to Astrophysics lecture I watched by Dr. Joshua Winn. "You are sitting next to someone on a plane who asks what you do in your spare time. To be friendly, say you’re studying astronomy, though you run the risk of being asked about horoscopes. If you want to sound impressive, say you’re studying astrophysics. If you’re not feeling very sociable, just say you’re studying physics and the conversation will come to a halt."
1.1 A Brief History of Astronomy
Astronomy is the oldest science. For at least 40 000 years, humans have tracked the Sun, Moon, and planets to mark seasons, navigate oceans, and punctuate rituals. This section highlights nine milestones that shaped our understanding of the heavens.
1.1.1 Stone Age Skylore (40 000 BCE – 2000 BCE)
Paleolithic carvings and Neolithic monuments like Nabta Playa (4500 BCE) and Stonehenge (3100 BCE) show alignments with solstices and stars. These structures encode observations of solar declination and lunar standstills (18.6 yr cycle) long before writing emerged.
1.1.2 Mesopotamian Celestial Records (2000 BCE – 500 BCE)
Babylonian astronomers logged heliacal risings of 20+ stars in tablets such as Enūma Anu Enlil. They deduced eclipse periodicity—the Saros (18 yr 11⅓ d)—by noting that 223 synodic months ≈ 242 draconic months.
\[ 223S \approx 242D, \] where \(S=29.53\,\text{d}\) (synodic month), \(D=27.21\,\text{d}\) (draconic month).
1.1.3 Greek Geometers (600 BCE – 150 CE)
Thales and Pythagoras began quantifying angles; Hipparchus (150 BCE) invented trigonometry, catalogued 850 stars to magnitude ±6, and discovered axial precession (~50″ yr⁻¹) by comparing his data to Babylonian records.
1.1.4 Islamic Scholarship (800 CE – 1400 CE)
In Baghdad’s House of Wisdom, scholars translated Greek works and improved them: al-Battani refined the obliquity of the ecliptic to 23° 35′, and al-Sufi catalogued 1 000+ stars in the Book of Fixed Stars.
Al‐Sufi’s measurements of the Andromeda “nebula” (964 CE) mark the earliest known record of that galaxy in the West.
1.1.5 Renaissance & Telescopes (1500 – 1700)
Copernicus (1543) revived heliocentrism; Tycho Brahe’s 1′-accuracy sights let Kepler derive his three laws; Galileo’s 1609 telescope revealed lunar craters, Jupiter’s moons, and the phases of Venus.
1.1.6 Newtonian Synthesis (1687)
Newton’s Principia unified celestial and terrestrial mechanics with \[ F = G\,\frac{m_1m_2}{r^2}, \] showing gravitation as the engine of orbits and tides.
1.1.7 Astrophysics & Spectroscopy (1800 – 1900)
Fraunhofer’s 1814 solar spectrum mapped 600 dark lines; Kirchhoff and Bunsen identified them, founding astrophysics. By 1890 stellar spectroscopy gave temperatures, compositions, and radial velocities.
1.1.8 Relativity & Cosmology (1905 – 1929)
Einstein’s 1915 general relativity predicted light bending around the Sun (1919 eclipse confirmation). Hubble’s 1929 redshift–distance relation \[ v=H_0\,d \] launched modern cosmology.
1.1.9 Modern Precision Era (1950 CE – Present)
Radio interferometry, satellite astrometry (Hipparcos, Gaia), and space telescopes (Hubble, JWST) measure positions to micro-arcsecond precision, mapping the Milky Way and beyond.
List four cultural or technological milestones in this timeline and explain why each advanced astronomical accuracy.
1.1.10 Solutions
- Stone Age – aligned monuments marked solstices (accuracy ~1°).
- Babylon – Saros cycle predicted eclipses (error ~2 hours in 18 years).
- Renaissance – telescopes improved resolution from 1° → 1′.
- Gaia – astrometry to ~10 µas (errors ~0.01 AU at 1 kpc).
1.2 The Celestial Sphere & Angular Measure
To describe object positions, astronomers imagine a vast sphere around Earth. Key points: zenith overhead, horizon circle, celestial poles at extension of Earth’s axis, and the celestial equator as Earth’s equator projected out.
1.2.1 Altitude & Azimuth
Positions on the sky are given by two angles:
- Altitude (\(h\)): angle above horizon (0°–90°).
- Azimuth (\(A\)): compass angle from north (0°–360°).
1.2.2 Right Ascension & Declination
On the celestial equator, we define:
- Declination (\(\delta\)): angle north (+) or south (–) of equator.
- Right Ascension (\(\alpha\)): eastward hours (0 h–24 h) from the vernal equinox.
\[ \sin h = \sin\phi\,\sin\delta + \cos\phi\,\cos\delta \,\cos H, \] where \(\phi\) = observer latitude; \(H=\) hour angle = \(LST - \alpha\).
1.2.3 Angular Units
Angles are measured in degrees (°), minutes (′), seconds (″), or radians. Conversion: \[ 360° = 2\pi\ \mathrm{rad}, \quad 1° = 60′, \quad 1′ = 60″. \]
A star at \(\delta=+20°\) is observed from \(\phi=34°\,N\). What is its maximum altitude \(h\) when it transits the meridian?
1.2.4 Sidereal vs Solar Time
Because Earth orbits the Sun, the interval between successive transits of a fixed star (one sidereal day) is 23 h 56 m 4.1 s, not 24 h. Sidereal time measures Earth’s rotation relative to the stars; solar time measures it relative to the Sun.
Over one year, the sidereal clock gains ~4 minutes per solar day, advancing by ~2 hours by year’s end.
1.2.5 Solutions
- Practice 1.2: At transit \(H=0\). \(\sin h = \sin34°\sin20°+\cos34°\cos20°\times1\approx0.444\). \(h\approx26.4°\).
1.3 The Laws of Nature — Foundations of Physical Astronomy
From falling apples to orbiting planets, three simple laws govern all motion in the universe. Galileo’s experiments on inclined planes disproved Aristotle’s idea that heavier objects fall faster, laying groundwork for Newton.
Newton’s First Law: An object in uniform motion remains so unless acted on by a force—introducing the concept of inertia. Second Law: \( \vec{F} = m\,\vec{a} \) (force equals mass times acceleration)—the cornerstone of dynamics.
Newton’s Third Law: Every action has an equal and opposite reaction—essential for understanding rocket propulsion and orbital maneuvers. The balance of forces explains how spacecraft change velocity in deep space.
Universal Gravitation: \[ F = G\,\frac{m_1 m_2}{r^2}, \] where \(G=6.674\times10^{-11}\,\mathrm{m^3\,kg^{-1}\,s^{-2}}\). This law unites terrestrial gravity with celestial orbits.
Kepler’s empirical laws arise naturally: for \(M\gg m\), set centripetal force equal to gravitational pull to get \[ P^2 = \frac{4\pi^2}{G M}\,a^3, \] linking orbital period \(P\) with semi-major axis \(a\).
James Bradley’s 1728 discovery of stellar aberration (a 20″ annual tilt in star positions) served as the first direct proof of Earth’s motion around the Sun—validating Newtonian dynamics on a cosmic scale.
Halley used Newton’s laws to predict the return of the comet now bearing his name, verifying \(F=ma\) beyond Earth for the first time.
A 1 000 kg probe fires thrusters with 5 000 N for 2 s. Calculate its \(\Delta v\). (Ignore gravity.)
Using \(F=Gm_1m_2/r^2\), show that the gravitational force on a 1 kg mass at Earth’s surface matches \(mg\) with \(g=9.81\,\mathrm{m/s^2}\).
1.4 Numbers in Astronomy — Notation, Units & Precision
Astronomical measurements span 40 orders of magnitude: from proton radii (~10-15 m) to cosmic voids (~1026 m). We use scientific notation and SI prefixes (k, M, G, T, m, µ, n) to keep numbers legible.
Length Units: 1 au = 1.496×1011 m; 1 pc = 3.086×1016 m; 1 ly = 9.461×1015 m. A parsec is defined by 1″ annual parallax: \[ d(\mathrm{pc}) = \frac{1}{\pi('')}. \]
Apparent Magnitude: \[ m_1 - m_2 = -2.5\,\log_{10}\!\Bigl(\tfrac{F_1}{F_2}\Bigr), \] where \(F\) is flux. This logarithmic scale compresses brightness ranges over 30 magnitudes into manageable numbers.
Significant Figures & Uncertainty: Report measurements as \( (x\pm\Delta x) \) with properly rounded errors. For addition/subtraction use absolute errors; for multiplication/division use relative errors in quadrature.
Modern parallax precision: Hipparcos (~1 mas), Gaia (~10 µas). Always quote uncertainty: e.g. \(d = (100.0\pm0.1)\) pc.
Keeping track of units and uncertainties avoids mistakes that can misplace a star by light-years or misestimate an orbit by thousands of kilometers.
Gaia’s 10 µas precision corresponds to a distance error of only 0.01 AU at 1 kpc.
Convert Earth’s mass \(5.97\times10^{24}\) kg into grams and into Earth masses (\(1\,M_\oplus\)).
If Betelgeuse has \(m_1=0.42\) and Rigel \(m_2=0.13\), compute \(F_2/F_1\).
1.5 Consequences of Light Travel Time
Light travels at \(c = 3.00\times10^8\,\mathrm{m/s}\). Hence we see distant objects not as they are, but as they were: \[ \Delta t = \frac{d}{c}. \] At 1 pc, \(\Delta t \approx 3.26\) yr; at 1 Gpc, \(\Delta t\approx3.26\) billion yr.
For nearby objects (\(z\ll1\)) the simple formula suffices; for high-\(z\) galaxies in an expanding Universe: \[ \Delta t = \int_{0}^{z} \frac{dz'}{H_0\,\sqrt{\Omega_m(1+z')^3 + \Omega_\Lambda}}. \] This integral computes the cosmic look-back time tied to redshift \(z\).
The cosmic microwave background at \(z\approx1100\) sets our practical observational horizon at \(t\approx380\,000\) yr after the Big Bang. Before that, the Universe was opaque.
Supernova light curves and time-delay lenses use travel-time effects to measure cosmological parameters like \(H_0\) and \(\Omega_\Lambda\).
Understanding light travel time is crucial for interpreting transient events (gravitational wave counterparts, fast radio bursts) which arrive with intrinsic delays across different messengers.
Future radio arrays will detect 21 cm emission from Epoch of Reionization, lighting up Universe’s teenage years via minute time differences.
The light from the Pillars of Creation in M16 is 6 500 yr old, so Hubble’s famous image shows the Eagle Nebula as it was before the first European civilizations arose.
Compute \(\Delta t\) to Proxima Centauri (1.30 pc) in years, days, and seconds.
Assuming \(H_0=70\,\mathrm{km/s/Mpc}\), estimate Universe’s age \(t_0\approx1/H_0\) in Gyr.
1.6 Techniques of Observation — Telescopes & Detectors
Since Galileo’s first refractor, telescope design has blossomed: reflectors, refractors, cassegrains, and segmented mirrors now collect photons across wavelengths from radio to gamma-rays.
A telescope’s light-gathering power scales as \(D^2\) (mirror diameter). Its diffraction-limited angular resolution is \[ \theta = 1.22\,\frac{\lambda}{D}, \] where \(\theta\) (rad) depends on wavelength \(\lambda\) and aperture \(D\).
Ground-based seeing limits resolution to ~0.5″–1″. Adaptive optics uses deformable mirrors and guide stars (laser or natural) to correct turbulence, achieving Hubble-class resolution over small fields.
Detectors have evolved from photographic plates (1% QE) to CCDs (>90% QE) and superconducting bolometers in the sub-mm. Quantum efficiency, read-out noise, and dark current determine faint-object sensitivity.
Radio interferometry (e.g., VLA, ALMA) uses baselines up to tens of kilometers to reach milliarcsecond resolution at cm and mm wavelengths. VLBI records data at separate sites and combines them to mimic Earth-sized dishes.
Spectroscopy disperses light to measure chemical compositions, temperatures, and radial velocities via Doppler shifts: \[ \frac{\Delta\lambda}{\lambda} = \frac{v_r}{c}. \] Precision radial-velocity methods detect exoplanets by measuring stellar wobbles <1 m/s.
The Event Horizon Telescope uses VLBI across the globe to achieve 20 µas resolution—enough to image the shadow of a black hole.
Calculate HST’s diffraction limit (\(\theta\)) at \(\lambda=500\) nm with \(D=2.4\) m. Convert to arcseconds (1 rad=206 265″).
A spectrograph resolves \(\Delta\lambda=0.01\) Å at \(\lambda=5000\) Å. What radial-velocity precision \(\Delta v\) does that correspond to?
1.7 Seasons, Insolation & Climate
Earth’s 23.44° axial tilt is the reason seasons exist. As we orbit the Sun, northern latitudes tip toward the Sun in June (summer solstice) and away in December (winter solstice), reversing six-monthly.
Declination of the Sun: The Sun’s latitude on the sky, \(\delta_\odot\), varies as \[ \delta_\odot = 23.44^\circ \,\sin\bigl(360^\circ\,(d-80)/365\bigr), \] where \(d\) is day of the year (1 = Jan 1). This sinusoid drives the changing height of noonday Sun.
Daily Insolation: The power per unit area from the Sun at latitude \(\phi\) is \[ Q(\phi) = \frac{S_0}{\pi}\Bigl[\cos\phi\cos\delta_\odot\sin H_r + H_r\sin\phi\sin\delta_\odot\Bigr], \] with \(S_0=1361\,\mathrm{W/m^2}\) the solar constant and \(\cos H_r = -\tan\phi\tan\delta_\odot\) giving sunrise hour angle.
Integrating \(Q(\phi)\) over the day yields total daily insolation, which varies by a factor of >2 between winter and summer at mid-latitudes. These swings drive monsoons, growing seasons, and polar day-night cycles.
Milanković Cycles: Over tens of thousands of years, changes in Earth’s tilt (±1.3°) and precession shift insolation patterns, triggering glacial–interglacial cycles. Quantitative models of ice age timing use the equations above plus orbital mechanics.
At 66.5° N (Arctic Circle), the Sun never sets on summer solstice and never rises on winter solstice—because \(\phi + \delta_\odot = 90^\circ\).
Compute the ratio of daily insolation on June 21 vs December 21 at 45° N using the insolation integral given. Plot \(Q(\phi)\) if you like.
1.8 Lunar Phases, Tides & Eclipses
The Moon’s orbit (semi-major axis 384 400 km, inclination 5.15°) creates the familiar phases: new, first quarter, full, and last quarter, according to the Sun–Earth–Moon angle.
Phase angle geometry: The illuminated fraction \(f\) is \[ f = \tfrac{1}{2}\bigl(1 - \cos\theta\bigr), \] where \(\theta\) is the elongation angle between Sun and Moon as seen from Earth.
Tides: Differential gravity from the Moon and Sun pulls Earth’s oceans into bulges. Spring tides (new/full Moon) occur when both align, neap tides (quarter phases) when they are perpendicular. Tidal force per unit mass is \[ F_t \approx 2\,G\,\frac{m_{\!M}}{r^3}\,R_E, \] with \(m_M\) Moon’s mass, \(r\) Earth–Moon distance, \(R_E\) Earth radius.
Eclipses: When the Moon crosses the ecliptic near new or full phase (node angle ±5.15°), solar or lunar eclipses occur. The Saros cycle (18 y 11⅓ d) arises because 223 synodic ≈ 242 draconic months.
Totality in a solar eclipse lasts ≤7 min because the Moon’s umbral shadow travels at ~1 000 km/h across Earth’s surface.
Derive the fraction of lunar month during which solar eclipses can occur, given the node inclination and Moon’s angular diameter.
1.9 Chapter Summary & Key Terms
In this chapter we’ve spanned the cosmos from nuclei to galaxy walls, surveyed nine millennia of sky‐watching, built the celestial coordinate system, reviewed physical laws, mastered units and notations, and explored seasons, phases, and eclipses.
Key Equations:
- \(\vec{F}=m\vec{a}\) — Newton’s Second Law
- \(F=Gm_1m_2/r^2\) — Universal Gravitation
- \(\sin h=\sin\phi\sin\delta+\cos\phi\cos\delta\cos H\) — Alt-Az to Equatorial
- \(v=H_0\,d\) — Hubble’s Law
- \(f=\tfrac12(1-\cos\theta)\) — Lunar illuminated fraction
- \(Q=\tfrac{S_0}{\pi}[\dots]\) — Daily insolation
Glossary: Azimuth, Declination, Node, Parallax, Precession, Proper Motion, Saros, Sidereal Day, Solstice, Synodic Month.
Write a one-page essay summarizing how coordinate systems and physical laws combine to allow precise observation and prediction in astronomy.
2.1 Earth’s Rotation & the Length of the Day
Earth spins once relative to the distant stars in a sidereal day of \(T_{\rm sid}=23\) h 56 m 4 s. Because Earth also orbits the Sun, it must turn an extra ~1° each solar day to bring the Sun back to the local meridian, stretching the mean solar day to exactly 24 h.
Sidereal vs. Solar: \[ T_{\rm sid} = \frac{T_{\odot}}{1 + 1/365.2422}, \] where \(T_{\odot}=24\) h. Plugging in 365.2422 days per year gives 23 h 56 m 4 s, matching observations to the second.
Tidal interactions with the Moon and Sun exert a torque \(\tau\) on Earth’s equatorial bulge, gradually slowing the spin. From angular‐momentum conservation, \[ \dot T = \frac{T\,\tau}{L}, \] where \(L = C\,\omega\) is Earth’s spin angular momentum, \(C\approx0.331\,MR^2\), and \(\omega=2\pi/T_{\rm sid}\).
The tidal torque is \[ \tau = \frac{3\,k_2\,G\,M_M^2\,R_E^5}{a^6}, \] with \(k_2\) the Love number, \(M_M\) lunar mass, \(R_E\) Earth radius, and \(a\) the Moon–Earth distance. Together these give \(\dot T\approx+2.4\,\text{ms/century}\).
Superimposed on the long‐term drift is the Chandler wobble, a free nutation of period \[ T_{CW} = \frac{2\pi}{\omega}\sqrt{\frac{C - A}{A}} \approx 433\,\text{days}, \] where \(A\) is the equatorial moment (slightly less than \(C\)).
Practical impact: Atomic clocks track International Atomic Time (TAI), while Earth rotation defines UT1. When \(|\text{UT1}-\text{UTC}|\ge0.9\,\text{s}\), a leap second is inserted into UTC to keep civil time within 0.9 s of Earth’s orientation.
The length‐of‐day has increased by about 2.3 ms since 1820—Earth’s spin is slowing but on human timescales almost imperceptibly.
Calculate the change in day length \(\Delta T\) if Greenland ice melt (5×1014 kg) moves uniformly from pole to equator. Treat mass as a thin ring and conserve \(L\).
If tidal braking continues at 2.4 ms/century, how many leap seconds will be needed between 2025 and 2100? Assume a constant rate.
Solutions 2.1
- A. Δ\(I = m\,r^2\); Δ\(L=0\) ⇒ Δ\(T/T = -ΔI/I\). Plug numbers ⇒ ΔT≈0.6 µs.
- B. 75 years×2.4 ms/100 yr ≈1.8 ms ⇒ 0 leap seconds (threshold is 0.9 s).
2.2 Nutation & Polar Motion
Nutation refers to short‐period (days–years) oscillations of Earth’s rotation axis due to the changing gravitational pull of the Moon and Sun on Earth’s equatorial bulge.
The dominant term has period 18.6 years and amplitude 9.2″: \[ \Delta\varepsilon = 9.2''\cos\Omega, \quad \Delta\psi = -17.2''\sin\Omega, \] where \(\Omega\) is the lunar node longitude on the ecliptic.
\[ \Delta\psi = -17.2''\sin\Omega - 1.3''\sin2L,\quad \Delta\varepsilon = 9.2''\cos\Omega - 0.6''\cos2L, \] \] with \(L\) the mean solar longitude.
Polar Motion is the wandering of Earth’s rotation axis relative to its crust, combining Chandler wobble (~433 d) and annual terms. Coordinates of the instantaneous pole drift by up to 0.3″.
High‐precision geodesy (VLBI, GNSS) tracks these motions, essential for accurate pointing of telescopes and spacecraft navigation.
Modern atomic time and Earth rotation data are combined in IERS bulletins to provide UT1 corrections and pole coordinates daily.
Using \(\Omega=0°\) and \(L=90°\), compute \(\Delta\varepsilon\) and \(\Delta\psi\) from the IAU formula above.
If polar drift moves the axis by 0.1″, how far (cm) does a telescope at latitude 45° N shift on Earth’s surface?
Solutions 2.2
- A. Δε=+9.2″, Δψ≈0.
- B. 0.1″ at 45° N ⇒ 0.1″ × (π/648000) rad × R_E × cos45° ≈ 1.9 cm.
2.3 Axial Precession — The 26 000 Year Cone
Earth’s equatorial bulge feels a torque from the Sun and Moon, causing the spin axis to precess around the ecliptic pole in a 25 772 year cycle.
The precession rate is \[ \dot\psi = \frac{3G\,(M_M+M_\odot)\,J_2\,\cos i}{2\,\omega\,a^3}, \] where \(J_2\) is Earth’s dynamical oblateness, \(i\) the inclination of the bulge relative to the ecliptic, and \(a\) an effective orbital radius.
Numerically, \[ \dot\psi\approx50.29''\,\mathrm{yr}^{-1}, \] shifting the position of the equinox by ~0.00024° per year.
Precession causes the tropical (seasonal) year to be ~20 min 24 s shorter than the sidereal (star‐to‐star) year, because the equinox point regresses against the stars.
Over 10 000 years, Polaris will cease being the North Star, and bright stars like Vega will take its place—an effect visible in cultural star lore.
Precession must be accounted for in any long‐term star catalog and in the alignment of telescopes for high‐precision astrometry.
Show that the tropical year \(T_{\rm trop}\) and sidereal year \(T_{\rm sid}\) satisfy: \[ T_{\rm trop} = \frac{T_{\rm sid}}{1 + \dot\psi/360°}\,. \] Compute the ≈20 min 24 s difference.
Calculate the RA shift (in hours) of the vernal equinox over 1 000 years at 50.29″ yr⁻¹. (360° = 24 h.)
Solutions 2.3
- A. \(\dot\psi/360°\approx50.29″/360°=3.3×10^{-5}\). \(T_{\rm trop}=T_{\rm sid}/(1+3.3×10^{-5})\) ⇒ ≈20 min 24 s shorter.
- B. 50.29″ yr⁻¹×1000 yr=50 290″=13.97°=0.931 h.
2.4 Aberration of Light — Bradley’s Discovery
James Bradley in 1728 found stars describe a tiny ellipse each year, ≈20.5″ semi-major axis, caused by the vector addition of Earth’s orbital velocity \(v\approx29.8\,\mathrm{km/s}\) and the finite speed of light \(c\).
The classical aberration angle is \[ \theta_{\rm ab} = \frac{v}{c}\sin\beta, \] where \(\beta\) is the ecliptic latitude of the star. Maximum displacement occurs for stars on the ecliptic (\(\beta=0\)).
Relativistically, velocity‐addition adds a tiny correction: \[ \theta_2 = (\gamma-1)\beta\sin\beta, \] with \(\gamma = (1-v^2/c^2)^{-1/2}\), typically µas-level.
Aberration must be corrected for in any precise astrometric catalog and led to the realization that Earth moves at measurable speeds in space.
Space missions like Gaia incorporate aberration models to µas precision to map stellar positions over the entire sky with unprecedented accuracy.
Without correcting aberration, annual systematic errors of ±20″ would swamp any proper motion or parallax signals.
Derive the semi-major and semi-minor axes of the aberration ellipse for a star at ecliptic latitude \(\beta\).
A spacecraft traveling at 0.02 c observes a star at 90° to its motion. Compute the apparent aberration angle.
Solutions 2.4
- A. Major axis \(v/c\), minor \(v/c\,\cos\beta\).
- B. \(\theta = \arcsin(0.02)\approx1.15°\).
3.1 The Nature of Light — Waves, Photons & the Electromagnetic Spectrum
Light is fundamental to astronomy: it carries information across the cosmos. By the 19th century, James Clerk Maxwell unified electricity and magnetism into four equations predicting electromagnetic waves traveling at a universal speed \(c\). Two centuries later, quantum mechanics added the photon, a particle of light with energy \(E = h\,\nu\), blending wave and particle duality.
Maxwell’s Wave Equation: From his equations we derive for each field component \(E\) or \(B\): \[ \nabla^2 \mathbf{E} - \frac{1}{c^2}\,\frac{\partial^2 \mathbf{E}}{\partial t^2} = 0, \] with \(c = 1/\sqrt{\varepsilon_0\mu_0}\approx3.00\times10^8\,\mathrm{m/s}\). Here \(\varepsilon_0\) and \(\mu_0\) are vacuum permittivity and permeability.
\[ E = h\,\nu,\quad \lambda = \frac{c}{\nu}, \] where \(h = 6.626\times10^{-34}\,\mathrm{J\cdot s}\) is Planck’s constant, \(\nu\) the frequency, and \(\lambda\) the wavelength.
In wave terms, light is characterized by its wavelength \(\lambda\) and frequency \(\nu\); in quantum terms, by photon energy \(E\). Longer \(\lambda\) means lower \(\nu\) and smaller \(E\). Visible light spans roughly 380–750 nm, but astronomy exploits radio waves (km), microwaves (mm–cm), infrared (µm), ultraviolet (10–400 nm), X-rays (0.01–10 nm), and gamma-rays (<0.01 nm).
Human eyes are only sensitive to ~1/40 of the full electromagnetic spectrum; telescopes and detectors extend our view by 20 orders of magnitude in wavelength.
Blackbody Radiation: Many astronomical sources approximate thermal emitters. Planck’s law gives spectral radiance: \[ B_\nu(T) = \frac{2h\nu^3}{c^2}\,\frac{1}{e^{\,h\nu/kT}-1}, \] where \(T\) is temperature and \(k=1.381\times10^{-23}\,\mathrm{J/K}\) is Boltzmann’s constant.
Stars emit approximately as blackbodies: hotter stars peak at shorter wavelengths (Wien’s law \(\lambda_{\rm max}T = 2.90\times10^{-3}\,\mathrm{m\,K}\)), while cool dust emits in the far-IR to sub-mm.
Compute the photon energy (in eV) and wavelength (in nm) for a 1 eV photon and for a 500 nm photon. (1 eV = 1.602×10-19 J.)
A star’s spectrum peaks at 480 nm. Use Wien’s law to estimate its surface temperature \(T\). (Wien’s constant = 2.90×10-3 m K.)
Solutions 3.1
- A. For 1 eV: \(E=1\,\text{eV}\Rightarrow \lambda=h c/E=1240\,\text{nm}\). For 500 nm: \(E=h c/\lambda≈2.48\,\text{eV}.\)
- B. \(\lambda_{\max}T=2.90×10^{-3}\) ⇒ \(T=2.90×10^{-3}/480×10^{-9}≈6040\,\text{K}.\)