Astrophysics Lecture Notes 2024/25 PDF

Astrophysics I Lecture 1: Introduction, coordinate systems used in astronomy, annual motion of the Sun Andrzej Pigulski Astrophysics: contents of the course of Astrophysics I & II Astrophysics I: 1. Introduction, basic definitions, stellar parameters (4 h) – Andrzej Pigul...

Astrophysics I Lecture 1: Introduction, coordinate systems used in astronomy, annual motion of the Sun Andrzej Pigulski Astrophysics: contents of the course of Astrophysics I & II Astrophysics I: 1. Introduction, basic definitions, stellar parameters (4 h) – Andrzej Pigulski 2. Tools of astrophysics (6 h) – Andrzej Pigulski 3. The Sun (2 h) - Urszula Bąk-Stęślicka 4. Stellar atmospheres (8 h) – Ewa Niemczura 5. Stellar structure and evolution (10 h) – Przemysław Walczak Astrophysics II: 1. Final stages of stellar evolution (8 h) – Armen Sedrakjan 2. Close binary stars (6 h) – Grzegorz Kondrat 3. Solar system, extrasolar planets (4 h) – Grzegorz Kondrat 4. Galaxies (6 h) – Grzegorz Kondrat 5. Cosmology (6 h) – Jerzy Kowalski-Glikman Astrophysics I: times and dates Lecture (30 h): Wednesday, 4 p.m. EET/CET Classes (15 h): Monday, 2 p.m. EET/CET (Kamil Bicz), Winter semester, academic year 2024/25: 1 October 2024 – 21 February 2025 Astrophysics: contents of the lecture 1. Introduction, basic definitions, stellar parameters (4 h). Astronomy as an observational science, astrophysics. Celestial sphere, coordinate systems used in astronomy. Annual motion of the Sun, precession, rises and settings. Astronomical time systems, calendar. Blackbody radiation. Stellar spectra, flux, effective temperature. Magnitude scale, bolometric magnitude, luminosity, colour index. Stellar parallax, cosmic distance scale, distance ladder. Stellar parameters. Determination of stellar masses and radii. Hertzsprung-Russell diagram. Astrophysics: contents of the lecture 2. Tools of astrophysics (6 h). Electromagnetic spectrum, observing windows. Ground-based and space observatories. Telescopes. Detectors. Infrared, ultraviolet, X-ray and 𝛾-ray astronomy. Observing techniques: imaging. Observing techniques: photometry. Observing techniques: spectroscopy. Observing techniques: optical and radio interferometry. Observing techniques: astrometry. Observing techniques: polarimetry. Astrophysics: contents of the lecture Textbooks: H. Karttunen et al., Fundamental Astronomy B. Basu et al., An Introduction to Astrophysics M. Harwit, Astrophysical Concepts B.W. Carroll, D.A. Ostlie, An Introduction to Modern Astrophysics Celestial sphere: elements of spherical astronomy Celestial sphere: elements of spherical astronomy Celestial sphere: elements of spherical astronomy Celestial sphere: elements of spherical astronomy Celestial sphere, spherical coordinates This is right-handed system of coordinates. In left-handed system, the x and y axes are exchanged. Celestial sphere Celestial sphere Plane of reference (PoR) – divides sphere into two hemispheres Celestial sphere Celestial sphere Plane of reference (PoR) – divides sphere into two hemispheres Circle of reference (CoR) Initial semicircle Celestial sphere Celestial sphere Plane of reference (PoR) – divides sphere into two hemispheres Circle of reference (CoR) Initial semicircle Two coordinates Celestial sphere Celestial sphere Plane of reference (PoR) – divides sphere into two hemispheres O Circle of reference (CoR) Initial semicircle (ISC) Two coordinates are needed Great circle Small circle Centre of the system (O) https://www.esa.int/Applications/Observing_the_Earth/Earth_from_Space_Earth_Day An example: geographical coordinates An example: geographical coordinates North pole (N) Plane of reference: plane of Earth’s equator Circle of reference: equator Initial semicircle: Greenwich (prime) meridian 𝞅 Coordinates: 𝝺 longitude 𝝺 : (0º, 360º), traditionally 0º – 180º E and 0º – 180º W latitude 𝞅 : (−90º, +90º), traditionally 0º – 90º N and 0º – 90º S Wrocław: 51º 6′ N, 17º 2′ E South pole (S) Spherical triangle Spherical excess 𝜀 : Spherical triangle Sine formula: Cosine formula for angles: Cosine formula for central angles: Systems of coordinates: location of the centre Topocentric: the centre of the coordinate system (O) located at the surface of Earth Geocentric: O in the centre of the Earth. Selenocentric: O in the centre of the Moon. Planetocentric: O in the centre of a planet. Heliocentric: O in the centre of the Sun. Barocentric: O in the centre of the Solar System. Galactocentric: O in the centre of the Galaxy. Horizontal system of coordinates / frame PoR: plane tangent to the Earth surface passing through the observer (O) CoR: horizon vertical: great circle passing through zenith Z: zenith Nd: nadir Horizontal system of coordinates / frame PoR: plane tangent to the Earth surface passing through the observer (O) CoR: horizon vertical: great circle passing through zenith Z: zenith Nd: nadir vertical N – Z – S = meridian ISC: local meridian passing through S S – south, N – north, E – east, W – west, (four cardinal points/directions) Z – E – Nd – W: first vertical P – celestial North pole, P’ – celestial South pole P – Z – S – P’ – Nd – N: astron. meridian Horizontal system of coordinates / frame almucantar(at) 𝜙 – observer’s latitude A – azimuth Angle between vertical passing through S and vertical passing through object G, measured along the horizon in the direction of W. Range: 0º – 360º. Warning: in geodesy and cartography azimuth is measured from N towards E, AGEO = AAST − 180º h – altitude / elevation Angle between the plane of horizon and the direction towards the object G. Range: from −90º (below the horizon) to +90º (above the horizon). z – zenith distance z = 90º – h (0º – 180º) Horizontal system of coordinates / frame φ = 90º ☆ φ = 52º rotation axis φ = 30º equator φ = 0º φ = –30º Equatorial system of coordinates / frame (I) PoR: plane of celestial equator CoR: celestial equator P: celestial north pole P’: celestial south pole ISC: local meridian passing through Z Equatorial system of coordinates / frame (I) t – hour angle Angle between vertical passing through Z and vertical passing through object G, measured along the celestial equator in the direction of the daily motion of the celestial sphere. Range: 0º – 360º or 0h – 24h. 1h = 15º. 𝝳 – declination Angle between the plane of celestial equator and the direction towards object G. Range: from −90º (southern hemisphere) to +90º (northern hemisphere). Dependencies between coordinate frames In general, transformation between coordinates in two 3D coordinate frames can be written as: For example, for a right-handed coordinate system and rotation around axis x by angle 𝛃: Frame HF EF Axis x OS OS’ Transformation from HF to EF: rotation by angle Axis y OW OW 90º - φ around y axis Axis z OZ OP Dependencies between coordinate frames For such a transformation we have: which means that: Dependencies between coordinate frames Writing out this matrix equation we get: Dividing Eq. (2) by Eq. (1) we get formula for tan t : Hour angle t can be derived unambiguously taking sign of numerator and denominator in this expression. Declination δ can be derived unambiguously from Eq. (3). Dependencies between coordinate frames Similarly, a reverse transformation can be obtained: Dividing Eq. (6) by Eq. (5) we get formula for tan A : Azimuth A can be derived unambiguously taking sign of numerator and denominator in this expression. Altitude h can be derived unambiguously from Eq. (7). Equatorial system of coordinates / frame (II) PoR: plane of celestial equator CoR: celestial equator P: celestial north pole P’: celestial south pole Equatorial system of coordinates / frame (II) PoR: plane of celestial equator CoR: celestial equator P: celestial north pole P’: celestial south pole ISC: hour semicircle passing through the vernal  equinox (in Aries = the ram,  ) = first point of Aries 𝛆 – angle between the plane of celestial equator and the plane of ecliptic  (axial tilt or obliquity) 𝛆 = 23º 44’ Equatorial system of coordinates / frame (II) 𝛂 – right ascension Angle between hour circle passing through vernal equinox and hour angle passing through object G, measured along the celestial equator in the direction. Range: 0h – 24h. 𝝳 – declination Angle between the plane of celestial equator and the direction towards object G. Range: from −90º (southern hemisphere) to +90º (northern hemisphere).  Since (𝛂, 𝛅) are (almost) independent of the position of the observer and the motions of Earth, they are used in catalogs and maps. https://en.wikipedia.org/wiki/Ursa_Minor#/media/File:Ursa_Minor_IAU.svg Stellar maps: northern and southern poles: UMi & Oct https://en.wikipedia.org/wiki/Octans#/media/File:Octans_IAU.svg Other frequently used coordinate systems PoR: plane of ecliptic CoR: ecliptic Ecliptic : ISC: great circle passing through vernal equinox and ecliptic poles (used for planets and Solar System bodies) Coordinates: - ecliptic longitude 𝛌 (0º – 360º) - ecliptic latitude 𝛃 (−90º – +90º) Galactic : PoR: plane of Galaxy CoR: intersection of the plane of Galaxy with celestial IAU (1958): sphere North Galactic pole: (𝜶,𝜹) = (12h49m, +27°24′) ISC: great circle passing through Galactic poles and the centre of the Galaxy Galactic centre: l = 0°. Coordinates: The Galactic equator is inclined at about - galactic longitude l (0º – 360º) 62°36′ to the celestial equator - galactic latitude b (−90º – +90º) Rises and settings Never setting objects φ = 52º (circumpolar) δ ≥ 90º − φ (for northern hemisphere) δ ≤ −90º − φ (for southern hemisphere) Rising and setting objects 90º − φ ≥ δ ≥ φ − 90º (for northern hemisphere) −90º − φ ≤ δ ≤ 90º + φ (for southern hemisphere) Never rising objects (unseen) δ ≤ φ − 90º (for northern hemisphere) δ ≥ 90º + φ (for southern hemisphere) Rises and settings: close to northern pole φ = 85º Never setting objects (circumpolar) δ ≥ 90º − φ δ ≥ +5º Rising and setting objects 90º − φ ≥ δ ≥ φ − 90º +5º ≥ δ ≥ −5º Never rising objects (unseen) δ ≤ φ − 90º δ ≤ −5º Rises and settings: close to equator φ = 11º Never setting objects (circumpolar) δ ≥ 90º − φ δ ≥ +79º Rising and setting objects 90º − φ ≥ δ ≥ φ − 90º +79º ≥ δ ≥ −79º Never rising objects (unseen) δ ≤ φ − 90º δ ≤ −79º https://www.miguelclaro.com/wp/portfolio/a-colorful-universe-revealed-in-the-earth-motion/ Star trails: northern hemisphere Portugal (φ = +38°) A.Santerne, https://www.eso.org/public/images/271109-cc/ Star trails: southern hemisphere ESO, La Silla, Chile (φ = −29°) Rises and settings: upper and lower culminations φ = 52º For upper and lower culminations: A = 0º or 180º Altitude of upper culminations: northern (north of Z) hUCN = φ + 90º − δ southern (south of Z) hUCS = 90º − φ + δ Altitude of lower culminations: northern (north of Nd) hLCN = φ − 90º + δ southern (south of Nd) hLCS = −90º − φ − δ Rises and settings: azimuth and hour angle φ = 52º For risings (R) and settings (S): h = 0º At the time of rising and setting: tR,S = arccos(− tan δ tan φ) AR,S = arccos(− sin δ / cos φ) Situation when |− tan δ tan φ| > 1 or |− sin δ / cos φ| > 1 means that rising or setting does not occur - an object is either circumpolar or invisible Earth’s orbit around the Sun Earth’s orbit around the Sun is an ellipse a – major semi-axis b b – minor semi-axis a = 149 597 871,5 km aphelion S a peryhelion e – eccentricity, e = 0.0167 4 July 2 January (b/a)2 = 1 – e2 dmin = 147.1 mln km (perihelion) dmax = 152.1 mln km (aphelion) Annual motion of the Sun As a result of the Earth's orbital motion, the Sun moves against the stars. It moves along the ecliptic, not along the celestial equator.  Due to the eccentricity of the Earth's orbit, motion along the ecliptic is not uniform. (According to Kepler's 2nd law, the field velocity is constant, not the angular velocity)  The coordinates (𝝰,𝝳) of the Sun change. d𝝰/dt ≠ const Annual motion of the Sun Cardinal points of ecliptic: Aries (), Cancer (), Libra (), and  Capricornus () Vernal equinox (), 20 or 21 March  (𝝰,𝝳) = (0h, 0º) Summer solstice (), 21 June (𝝰,𝝳) = (6h, +𝝴)  Autumnal equinox (), 21 or 22 September (𝝰,𝝳) = (12h, 0º) Winter solstice (), 22 December  (𝝰,𝝳) = (18h, –𝝴)  Annual motion of the Sun Sky & Telescope, April 2023 Changes of the declination of the Sun: -ε ≤ δ ≤ +ε Negative declinations: between autumnal and vernal equinoxes (Sep – Mar) Positive declinations: between vernal and autumnal equinoxes (Mar – Sep) Ecliptic system of coordinates 𝛌 – ecliptic longitude Angle between plane passing through vernal equinox and ecliptic poles (E, E’) and plane passing through object G and ecliptic poles, measured along the ecliptic in the direction. Range: 0º – 360º. 𝛃 – ecliptic latitude Angle between the plane of ecliptic and the direction towards object G. Range: between −90º (southern ecliptic hemisphere) and +90º (northern ecliptic hemisphere). For the Sun, 𝛃 = 0º, 𝛌 changes. Constellations according to the IAU (88) https://en.wikipedia.org/wiki/IAU_designated_constellations Constellations according to the IAU: examples https://en.wikipedia.org/wiki/IAU_designated_constellations Lunisolar precession of Earth’s axis https://www.britannica.com/science/precession-of-the-equinoxes https://en.wikipedia.org/wiki/Axial_precession Lunisolar precession is due to the gravitational forces of the Moon and the Sun acting on non-spherical Earth (non-zero torque). Period of the precession of the Earth’s rotation axis: 25 772 years Annual motion of the Sun 12 zodiacal constellations: Spring: Aries (), Taurus (), Gemini () Summer: Cancer (), Leo (), Virgo () Autumn: Libra (), Scorpion (), Sagittarius () Winter: Capricornus (), Aquarius (), Pisces () When we say that the Sun is located in a given zodiacal sign, it does not mean it is indeed in the constellation corresponding to this sign. Reasons: - Precession of the Earth’s rotation axis (~one sign shift). - Division of the sky into 88 constellations of different size and coverage. Cardinal points of ecliptic: ~one sign shift (𝝰,𝝳) = (0h, 0º) (𝝰,𝝳) = (12h, 0º) https://en.wikipedia.org/wiki/IAU_designated_constellations https://en.wikipedia.org/wiki/IAU_designated_constellations Cardinal points of ecliptic (𝝰,𝝳) = (6h, +𝝴) (𝝰,𝝳) = (18h, –𝝴) https://en.wikipedia.org/wiki/IAU_designated_constellations Ophiuchus, not a Zodiacal sign Precession: the movement of celestial poles in the sky Sky & Telescope, March 2008  Precession and nutation Precession changes the position of the cardinal points of ecliptic with respect to the background stars. They travel along the https://en.wikipedia.org/wiki/Nutation#/media/File:Praezession.svg equator at a rate of 360°/25800 years ≈ 50.2"/year. Precession is the most important factor causing the change of coordinates (α,δ) of distant objects. Therefore, (α,δ) are given for an epoch (most recently the epoch 2000.0). The term nutation is used to describe small (in comparison with precession) changes of position in space of the Earth’s rotational axis caused mainly by the influence of the Moon. The most important changes resulting from nutation have a period of 18.6 years, which is the same as the precession period of the line of nodes of the Moon's orbit (the Moon's orbit is inclined to the ecliptic at an angle of 5.9°). On the time scale of precession (tens of thousands of years), the effects of changing position of stars due to their proper motions also become important – see lecture on astrometry. Rises and settings of the Sun, Wrocław δ = +ε (21.VI) δ = 0° (20.III i 22.IX) δ = -ε (21.XII) During equinox the day is in fact slightly longer than the night. The reason is twofold: (i) a finite diameter of the disk of the Sun, (ii) atmospheric refraction φ = 52º M. Zapiór, http://solarigrafia.pl/articles/83/read.html Solarigraphy https://www.realphotographycompany.co.uk/_files/ugd/d8e46b_761478f379a84 281be24b47e620f1bb2.pdf Rises and settings of the Sun, close to equator δ = 0° (20.III i 22.IX) δ = +ε (21.VI) δ = -ε (21.XII) Close to the equator, the differences between the length of day and night become small. At the equator, day and night are of (almost) the same length and this situation is the same the whole year. φ = 11º Rises and settings of the Sun, close to north pole δ = +ε (21.VI): polar day δ = 0° (20.III & 22.IX) δ = -ε (21.XII): polar night φ = 85º Twilights and dawns Due to the scattering of sunlight in the atmosphere, the transition from night to day and vice versa is gradual: we have such phenomena as twilight and dawn. Twilight begins at sunset, its end is defined arbitrarily. A distinction is traditionally made between three types of twilight/dawn: 1. civil twilight/dawn: –6° < h < 0°. 2. nautical twilight/dawn: –12° < h < 0°. 3. astronomical twilight/dawn: –18° < h < 0°. https://www.timeanddate.com/astronomy/civil-twilight.html Digression: Comet C/2023 A3 (Tsuchinshan-ATLAS) Courtesy Dawid Moździerski Digression: Comet C/2023 A3 (Tsuchinshan-ATLAS) Courtesy Dawid Moździerski Digression: Comet C/2023 A3 (Tsuchinshan-ATLAS) Perihelion: 0.39 AU (2024 Sep 27) Closest to the Earth: 0.47 AU (2024 Oct 12) https://soho.nascom.nasa.gov/data/realtime/c3/1024/latest.html Digression: Comet C/2023 A3 (Tsuchinshan-ATLAS) https://skyandtelescope.org/astronomy-news/get-ready-for-comet-tsuchinshan-atlas-the-best-is-yet-to-come/  Refraction Refraction is the phenomenon of the apparent change in the altitude of an object and is due to the presence of the Earth's atmosphere. r = h – h0 where h is the observed altitude and h0 the actual altitude. As a first approximation for h > 20°, the formula r = (n – 1) tan z can be used, where n is the refractive index of the air. Thus, for T = +10°C and a pressure of 1013.25 hPa, we have r = 58'' tan z. For smaller h special tables have to be used. At the horizon r ≈ 35’. https://www.timeanddate.com/astronomy/refraction.html Atmospheric effects due to refraction Deformation of the https://en.m.wikipedia.org/wiki/File:Sunset_ sequence07-23-06.jpg shape of the solar disc https://atoptics.co.uk/blog/opod-flattened- sun/ https://www.fleetscience.org/activities- resources/green-flash-appears-horizon Flattening of the disc Green flash Other effects affecting apparent position: Parallax Parallax is a difference between apparent positions of a given object along two different lines of sight. The effect affects also the apparent positions of objects in the sky. In astronomy, we use trigonometric parallaxes to measure distances. Heliocentric parallax uses orbital motion of Earth. β = 45° If π = 1″, then (by definition) D = 1 pc (parsec). β = 90° β = 20° π [″] = 1 / D [pc] D [pc] = 1 / π″ β = 0° 1 pc = 3.26 light year = D = 206264.8 AU = = 3.086 · 1013 km Parallax of the nearest star (not counting the Sun) amounts to 0.772″ (Proxima Centauri = 𝛂 Cen C) Other effects affecting apparent position: Aberration Aberration causes a shift in the apparent (observed) position of a star from its mean position on the celestial sphere. It is a consequence of the Earth's orbital motion and the finite speed of light. Aberration was discovered in 1729 r. by James Bradley, when he tried to measure stellar parallaxes. Since V ~ 30 km/s, 𝛂max = 20.4958″ Aberration Aberration is the largest on the great circle perpendicular to the apex-antiapex line. β = 45° Apex moves along the ecliptic, and λapex ≈ λ − 90° β = 20° Locations of an object in the β = 0° aberration and parallax ellipses are shifted with respect to each other by about 90°. For nearby objects shifts due to aberration are the same, due to parallax – no !!! There is also diurnal aberration due to the rotation of the Earth. Aberration versus parallax Parallax (ellipses not in scale) Aberration nearby star nearby star quasar quasar