Astrophysics Lecture 2 - 2024 PDF

Astrophysics I Lecture 2: Time in astronomy, blackbody radiation, effective temperatures. Andrzej Pigulski Astronomical time systems Astronomical time systems have their origin in generally observable astronomical phenomena. The most commonly used units of time...

Astrophysics I Lecture 2: Time in astronomy, blackbody radiation, effective temperatures. Andrzej Pigulski Astronomical time systems Astronomical time systems have their origin in generally observable astronomical phenomena. The most commonly used units of time are: Day: related to the period of the Earth's rotation around its axis. Month: associated with the orbital period of the Moon around the Earth. Year: associated with the Earth's orbital period. Unfortunately, these phenomena are independent of each other, so the aforementioned periods are not commensurate. For practical reasons we try to make them commensurate, which poses problems when creating calendars. Local times are usually defined as the hour angle of a specific object. They are therefore closely related to the rotation of the Earth around its axis. Locations that share the some astronomical meridian also share the same geographical meridian (the same longitude). So also the same local time. Astronomical time systems: definition of a second The basic modern unit of time measurement (also in the SI) is 1 second. 1 second (1 s) is defined as the time equal to 9 192 631 770 periods of radiation corresponding to the transition between two levels of the hyperfine structure of the ground state of caesium 133Cs atom. (the above definition refers to caesium atom at rest, at 0 K). Previous definitions (up to 1967) linked a second to the rotation of the Earth: it was defined as 1/86400 of an average solar day or 1/31 556 925.9747 part of the tropical year 1900 (tropical year: from equinox to equinox). According to Special Theory of Relativity, time is not an absolute concept, it depends on velocity. Clocks moving relative to the observer go slower (time dilation, 2nd order Doppler effect), e.g. if V = 100 km/h, the difference is 2.6 × 10−13 s per minute. According to the General Theory of Relativity, time runs slower for an observer in a stronger gravitational field. For example, an alarm clock located on the floor (1 m lower) will be 3.1 ps late in 8 hours of sleep relative to our watch (gravitational redshift). Sidereal time Sidereal time is defined as the hour angle of the (mean) vernal equinox: T ∗ = t However, the first point of Aries is a mathematical point of the celestial sphere, not an object that can be observed. It is clear that T∗ = t = t∗ + 𝛂∗ This allows to measure time with an observable object. Instead of measuring the hour angle of an object, it is sufficient to note that for upper culmination t∗ = 0h, which means that T∗ = t∗ + 𝛂∗ = 𝛂object at upper culmination Sidereal day Sidereal day is defined as the time interval between two successive culminations of the vernal equinox. Sidereal day divides into 24 sidereal hours and 86400 sidereal seconds. If there were no precessional motion of the vernal equinox, the length of the sidereal day would be equal to the period of the Earth's rotation around its axis. Because of this motion, the sidereal day is shorter than the period of the Earth's rotation by 0.0084 s. Sidereal day = 23h56m04.0914s Rotation period of the Earth = 23h56m04.0998s Δ = 0.0084 s  True solar time The object for which we measure hour angle can be, for example, the Sun. For practical reasons (date change at midnight, not at noon) the true solar time is defined as: T = t + 12h The true solar time is indicated by sundials, provided they are correctly oriented. https://pl.wikipedia.org/wiki/Zegar_słoneczny#/me True solar day: interval between two consecutive lower culminations of the Sun. dia/Plik:Sundial_Warsaw.jpg Unfortunately, as a result of the two effects (the eccentricity of the Earth's orbit and variability of Δ𝛂/Δλ), the true solar days are not of the same length. Mean solar time In order to establish uniformly flowing time, the concept of the mean Sun and associated mean solar time were introduced. The mean Sun is a mathematical point that moves along the equator with a constant rate (d𝛂/dt = const). True Sun Mean Sun Moves along ecliptic equator d𝛂/dt changes is constant In consequence, we can define now (mean) solar time as: TM = tM + 12h (Mean) solar day is time interval between two consecutive lower culminations of the mean Sun. Solar day is 3m56s longer than sidereal day.  Equation of time https://www.researchgate.net/publication/350481425_From_time_ frames_to_temperature_bias_in_temperature_series/figures?lo=1 Equation of time [min] T∗ = t∗ + 𝛂∗= tM + 𝛂M T – TM = t – tM = 𝛂 – 𝛂M = ET ET = equation of time |ET| does not exceed 16 min The equation of time contains two factors: The first, with a period of half a year, resulting from the fact that the true Sun moves along the ecliptic and the mean Sun moves along the equator (Amplitude = 9.87 min). The second, with a period of one year, resulting from the eccentricity of the Earth's orbit around the Sun (Amplitude = 7.68 min). https://epod.usra.edu/blog/2023/06/analemma-captured-from-lake-varese- italy.html Analemma Lake (Italy) Varese 16:00 UT Local time and geographical longitude For any two locations at longitudes λ1 and λ2, the relationship between the local times at these locations, t1 and t2, are expressed by the formula: t2 – t 1 = λ 2 – λ 1 or t – λ = const These relationships are valid for all three defined previously local times. Local time and geographical longitude Mean solar time at Greenwich meridian is called Universal Time (UT). UT = TM,Greenwich From the practical point of view, zone times have been introduced, which most often differ from the UT by an integer number of hours (less often by a half number of hours). For example: Central European Time (CET), our winter time: CET = UT(C) + 1h Eastern European Time (EET), our summer time: EET = UT(C) + 2h https://en.wikipedia.org/wiki/Time_zone#/media/File:World_Time_Zones_Ma p.png Time zones  International Date Line In the vicinity of the 180° meridian, an arbitrary international date line has been defined. Crossing this line, the date must be changed by one day forward (when travelling from east to west) or one day back (when travelling from west to east), see Jules Verne’s `Around the World in Eighty Days’ + 1 day – 1 day WIkipedia Calendars Many calendars were (and still are) in use. They can be divided into: lunar solar combined solar-lunar The basic units of time that a calendar must take into account are: - day, 1d = 86400 s, mean solar day = 86400.0009 s - synodic month (from new Moon to new Moon) = 29.530589 d - tropical year (from equinox to equinox, average period of the recurrence of seasons) = 365.24219879 d – 0.00000614 T, (T – centuries counted from 1900). currently = 365.242190 d Gregorian calendar. Mean length of a year: 365.2425 d Units of time: a summary DAY (Earth): 1d= 86 400 s mean solar 86 400.0009 s mean sidereal 86 164.0914 s mean rotation period of Earth 86 164.0998 s MONTH (Moon): draconic (node to node) 27.212 221 d tropical (vernal equinox to vernal equinox) 27.321 582 d sidereal (fixed star to fixed star) 27.321 662 d anomalistic (perigee to perigee) 27.554 550 d synodic (new Moon to new Moon) 29.530 589 d YEAR (Sun): tropical/solar (vernal equinox to vernal equinox) 365.242 190 d average Gregorian 365.242 5 d (leap years) average Julian 365.25 d sidereal (fixed star to fixed star) 365.256 363 d anomalistic (perihelion to perihelion) 365.259 635 d Time scales LMST = Local Mean Sidereal Time, T∗ (local) GMST = Greenwich Mean Sidereal Time, T∗ (LMST for Greenwich) LMT = Local Mean (Solar) Time, TM (local) UT = Universal Time, TM, Greenwich All these mean time scales are based on Earth’s rotation. They are as uniform as that rotation. The Earth's rotation becomes slower and slower with time. A continuous time scale cannot therefore be related to Earth's rotation. Accelerations of Earth rotation: secular slowing due to tidal friction – the day becomes 1 s longer about every 40 000 years random accelerations (both positive and negative), can be an order of magnitude higher than secular short-term periodic due to lunar-induced tides, meteorological phenomena https://en.wikipedia.org/wiki/Earth%27s_rotation Deviation of day length from SI-based day Uniform time scales Uniform dynamical time scale known as Ephemeris Time (ET), was in use in the years 1952 – 1984. The ephemeris second, was defined in 1960 as a certain fraction of the tropical year 1900. In 1972 International Atomic Time (TAI) was introduced (based fully on atomic clocks). Set to agree with UT1 on 1958 Jan. 1. In 1976 Barycentric Dynamic Time (BDT) was introduced to account for relativistic effects. In 1984 ET was replaced with terrestrial dynamical time, TDT, the proper time of an observer moving with the Earth. The time scale is affected by the relativistic time dilation due to the orbital speed of the Earth, but not rotation. In 1991, a new standard, Terrestrial Time (TT) was adopted. It is practically equivalent to TDT. TT = TAI + 32.184 s Uniform time scales We would prefer to use an Earth rotation-based time. That’s why in practice we use universal time that accounts for the lengthening of the rotation period of the Earth (UTC). UT – Universal Time is a time standard based on Earth's rotation. Equivalent to mean solar time at 0° longitude (Greenwich), precise measurements of the Sun are difficult. UT1 – accounts for polar motion UTC – Coordinated Universal Time, the primary time standard by which the world regulates clocks and time. 𝚫T = TT – UT1 https://en.wikipedia.org/wiki/ΔT_%28timekeeping%29 https://ascom-standards.org/Help/Developer/html/732148ca-fc8a-4f0d-bc1d- 8f5fe2907744.htm Time scales https://ascom-standards.org/Help/Developer/html/732148ca-fc8a-4f0d-bc1d- 8f5fe2907744.htm Time scales https://www.cambridge.org/core/books/time-from-earth-rotation-to-atomic- physics/coordinated-universal-time- utc/7FAC66BFB86567A4471E476665834696 Time scales  DUT = UTC – UT1 https://en.wikipedia.org/wiki/Coordinated_Universal_Time The last leap second was introduced on 2016 December 31  Black body radiation Black body – an idealized physical body that absorbs all incident electromagnetic radiation, regardless of frequency. The radiation emitted by a black body in thermal equilibrium with its environment is called black-body radiation. https://en.wikipedia.org/wiki/File:Wiens_law.svg Planck function  Black body radiation https://www.astronomynotes.com/cosmolgy/s5.htm John C. Mather George F. Smoot (NASA) (UCB) Nobel Prize in physics 2006  Terms related to radiation field Specific intensity I𝛎 [W m−2 rad−2 Hz−1] https://link.springer.com/chapter/10.1007/978-981-15-4174- Flux: a measure of net energy flow across the area dS over a time dt in a spectral range d𝛎. F𝛎 [W m−2 Hz−1] 2_20 Stefan-Bolzman law, effective temperature If I𝛎 does not depend on 𝜙 and 𝜃 we get: Stefan-Boltzmann law: F [W m−2] Stefan- Boltzmann constant  Effective temperature Stars do not radiate as black bodies !!! https://en.wikipedia.org/wiki/Color_temperature https://en.wikipedia.org/wiki/Color_temperature Effective temperature  Effective temperature: a local quantity Zhao et al. (2009, ApJ 701, 209) Effective temperature: the Sun Teff, = 5780 K Teff, = 5780 K https://www.iau.org/public/images/detail/iau1508d/ Fröhlich & Lean (2004, The Astron. Astrophys. Rev. 12, 273) Effective temperature From the law of the conservation of energy: D – distance, f – total flux measured at Earth, R – stellar radius 𝜃 – angular diameter, 𝜃 = 2R / D Fundamental method to derive Teff (by measuring f and 𝜃) Effective temperature: fundamental method to derive Code et al., 1976, ApJ 203, 417 http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/difopa.html#c1 Angular diameters: diffraction at opaque barrier http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/bardif.html https://skyandtelescope.org/astronomy-news/observing-news/the-april-1st- pleiades-occultation/ Lunar occultation Michael Richmond, http://spiff.rit.edu/richmond/occult/bessel/bessel.html Angular diameters: diffraction at opaque barrier Michael Richmond, http://spiff.rit.edu/richmond/occult/bessel/bessel.html Angular diameters: diffraction at opaque barrier Angular diameters: diffraction at opaque barrier Michael Richmond, http://spiff.rit.edu/richmond/occult/bessel/bessel.html This dependece shows that θ can be derived from lunar occultations Lunar occultation https://skyandtelescope.org/astronomy-news/observing-news/the-april-1st- Occultation of Pleiades by the Moon February 21, 1991 Białków, 60-cm telescope + photomultiplier time step = 0.6 ms pleiades-occultation/