Introduction to astronomical quantities

Astro-particles physicists need to know a bit of astronomy. Especially if you want to point your telescope to some point in the sky and say something interesting!

Historical hints

Astronomy comes from the Greek words: astron, meaning stars, and nomos meaning law. Modern Astronomy was born at a very precise date:Jan. 7, 1610, as reported by Galileo in the Sidereus Nuncius: "Adì 7 di gennaio 1610 Giove si vedeva col cannone con 3 stelle fisse, delle quali senza il cannone niuna si vedeva" (Today, Jan 7, 1610, Jupiter could be seen with the telescope with 3 fixed stars of which none could be seen without).

The Sidereus Nuncius by Galielo text by Galileo

Since the 13th century, the "cannone" or the galilean telescope was used for navigation (as reported also by Robert Baccus). The first telescopes were built with a divergent lens close to the eye, the ocular lens, and one convergent light-entrance lens. In 1590 the telescope arrives in the Netherlands, taken by an anonymous venetian artisan. In 1604 the optician Sacharias Janssen copies the italian model with one tune of 50 cm, a diameter of 3-4 cm and a magnification factor of 3-4. Afterwards it was reproduced by many and commercialized in Frankfurt. Galileo had news about it from 1608 by his friend Paolo Sarpi and later by a French pupil. In 1609 Galileo understood the importance of the instrument and fabricated his own with a magnification factor of 8. The two telescopes below are his telescopes with resolving power of 10 arcsec and 20 arcsec.

Two of Galileo's cannons and the broken lens with which he observed Jupiter

Galileo and his `cannone'.

Galieleo's greatest idea was to point the telescope towards the sky in fall 1609. Following this, he observed the Moon and also drew wonderful acquerellos. It is noticeable that Galileo knew exactly what he was observing, possibly also inspired by the reading of Plutarco's "The face of the Moon", where, in the 1st century after Christ, the Greek philosopher speculates on the Moon spots as being similar to what we find on the Earth (valleys and mountains).

Acquerellos by Galileo's on his Moon observations.

Why dating the birth of Modern Astronomy to the date of observation of the satellites of Jupiter? In that occasion Galileo observed 3 little stars that the day after became 4, which were named later "i pianeti medicei" in honor of Cosimo dei Medici. These were the satellites of Jupiter, and they represented a revolution in astronomy since they would contradict the tolemaic conception that the center of any motion was the Earth. It is the start of an astronomy that sees the Earth not as an exceptional place in the universe but as part of a system governed by the laws of gravitation. Galileo's imagination went even further in imaginig the rings of Saturn, that today we see including the aurorae borealis at its Pole with Hubble. By the way, did you know that on Dec. 21, 2020 we went through the Great Conjunction of Saturn and Jupiter, when they got as close as 1 billion kilometers!

Saturn and its aurorae borealis (ESA/Hubble, NASA, A. Simon (GSFC) and the OPAL Team, J. DePasquale (STScI), L. Lamy (Observatoire de Paris).

More recently, a new Astronomy is being funded which is named 'High-Energy Multi-Messenger Astrophysics', and this is what this text is about. It focuses on the most powerful cataclismatic events and their remnants observed with high-energy photons and other particles, such as neutrinos and Ultra-High-Energy cosmic rays (UHECRs) and through gravitational waves, as well. The subjects we discuss belong to Astroparticle Physics (ApP), a field born at the interception between Particle Physics, Plasma Physics and Astronomy.

If you wonder what are the main subjects on which astronomers and particle physicists worked on recently and what they have to work on in the future, we can briefly summarize them, though the list is surely not exahustive. Astronomers and Cosmologists have:

• observed the expansion of the universe and its acceleration at very large distances;

• inferred the existence of dark matter and dark energy;

• limited the parameter space of the inflation model;

• observed the cosmic microwave background, the echo of Big Bang, which gives a picture of the universe $\sim$380’000 yrs after it;

• measured the relative abundance of light elements in the universe (H, Li, $^2H$, He) due to the nuclear reactions in the first seconds of its life;

• observed the globular clusters and certain radioactive isotopes that do not seem to have passed the age of 13-14 milliards of years.

In recent years Particle Physicists have:

• understood the composition of matter to the level of its fundamental constituents, the quarks, that compose neutrons and protons, and the leptons (electrons, muons, taus ad neutrinos);

• explained how the interactions shape matter;

• are understanding what gives mass to particles (Higgs mechanism)

• have to understand if Physics Beyond the Standard Model exists, and found first signs of it in the neutrino sector, of which they have to understand the nature as well.

Together, Astrophysicists and Particle Physicists are understanding the profound relationship between Cosmology and Particle Physics, the immense cosmo and the microscopic world.

Galaxies

A typical galaxy, as our Milky Way, contains order of $10^{11}$ stars. In a spiral galaxy the oldest (population II) are in a central spherical hub surrounded by a flat disc with younger stars (population I) moving in circular orbits. Matter is not uniformly distributed in the cosmos, but it forms clusters, superclusters and walls of galaxies with large voids between them in which very few galaxies seem to exist. The shown map, making up only ~7% of the diameter of the visible universe, shows the nearest structures close to Earth.

The mass of a supercluster is typically of $10^{15} M_\odot$ ( $M_\odot \sim 2 \times 10^{30} \rm kg$ ) with a dimension of 50 Mpc.

Taken from R. Powell, The Atlas of the Universe, http://www.atlasoftheuniverse.com/index.html

The Milky Way is surrounded by $\sim 70$ known satellite galaxies gravitationally attracted by the Milky Way and Andromeda.This is known as the Local Group. Many of the galaxies are dwarf galaxies (a galaxy with $\sim$billion stars compared to our Milky Way 200-400 billion stars). The most notable are the Large Magellanic Cloud ($\sim 45$ kpc) and the Small Magellanic Cloud (60 kpc). The LMC mass is . Next nearest full-fledged galaxy is Andromeda or M31 () at a distance of approximately 780 kpc. The group contains the Milky Way and also other galaxies, e.g. M31(Andromeda), M33 (Triangulum Galaxy). It has volume of diameter of about 3 Mpc. The Local Group in turn is part of the Virgo supercluster, extending out to about 33 Mpc and with total mass of $10^{15} \rm M_\odot$, and of the even bigger Laniakea Supercluster including other $10^5$ other nearby galaxies. This supercluster spans over 500 Mlyr, while the Local Group spans over 10 Mlyr. The number of superclusters in the observable universe is estimated to be 10 M. Below some reference extragalactic distances are summarized.

Cosmic object	Distance
Proxima Centaury (closest star)	4.3 ly = 1.3 pc
Large Magellanic Cloud (LMC)	45 kpc
Local group diameter	3.1 Mpc
Centaurus A (closest Active Galactic Nuclei)	$\sim 3$ Mpc
Mrk 421 (close-by blazar)	$\sim 136$ Mpc
3C 273	1 Gpc

Recently a detailed 3D map of the large scale structure comprising several 4 millions of galaxies was produced by eBOSS with the Sloan Digital Sky Survey data.

The ordinary matter in the universe is estimated to be $M_{universe} \sim 10^{23} M_\odot$ in the Hubble length (which is about a factor of 3 $ct_0 = c/H_0 = 4.2 \, \rm Gpc$, with $H_0 = 67.8 \pm 0.9 \, \rm km/s/Mpc$and $t_0 \sim \frac{1}{H_0} =14.45 \rm Gyr$ (Planck, 2015), corresponding to a rest energy of $E_0 = M_{universe} c^2 \sim 2 \times 10^{70} \rm J$ (see Basic notions of Cosmology).

The Milky Way

The Milky Way

Schemr of the Galaxy from [By RJHall at English Wikipedia, CC BY-SA 3.0 ].

The halo of our spiral galaxy has diameter of about 150'000 ly and the disk thickness of $\sim 300$pc \sim 1 kly. The solar system is located at $\sim 8$kpc $\sim$ 28'000 ly from the Galactic Centre. The speed of the Sun around the Galactic Centre is about 220 km/s. It is worth to note the values of: the energy density of the galactic cosmic ray and magnetic fields are comparable $\sim$ 1 eV/cm$^3$ and this value is also not far from what is found for the photon fields in the intergalactic space (Optical - Infrared and Cosmic Microwave Background).

In the Galactic Centre, in the constellation of Sagittarius there is a black hole (BH) of $4.31 × 10^6 M_\odot$ (as measured from orbits of surrounding stars) confined in a sphere of $44 \times 10^6$ km. It is shown in the figure below as seen in X-rays (a meal of the supermassive BH Sgr A*), seen by the space telescope Chandra in X-rays. Astronomers believe that a mass equivalent to the planet Mercury was devoured by the BH yr earlier, causing an X-ray outburst which then reflected off gas clouds near Sagittarius A*. While the primary X-rays from the outburst would have reached Earth about 50 years ago, before X-ray observatories were in place to see it, the reflected X-rays took a longer path and arrived in time to be recorded by Chandra.

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Collection of images of the X-ray eco produced by a tidal disruption event Courtesy of NASA/CXC/Caltech/M.Muno et al.

Flux, Intensity, Luminosity, Magnitude

A star sends to us a luminous signal characterized by a certain spectrum and intensity. The observables are the quantity of radiation and the star color, but this radiation is altered during propagation and in the atmosphere.

We define the energy flux density passing through the area dA in the interval dt. It depends on the orientation of dA and also to the frequency of the radiation: $S_\nu = \frac{dE}{dA \, dt \, d\nu}$ , and its units are $\rm W m^{-2} Hz^{-1}$ and we will see that it depends on the position from the source of the flux at which we measure it.

An element of area dA intercepting a flux of particles or radiation

Notice that every element of spherical surface of an isotropic source emits equal amount of energy in all directions. If the flux at distance $r_1$ , e.g. the surface of the star, is $S_{\nu,1}$ at any distance $r$ the following equality applies in vacuum and not absorbing medium:

$4 \pi r_1^2 S_\nu (r_1) = 4 \pi r^2 S_\nu ( r) \Rightarrow S_\nu = \frac{S_\nu r_1^2}{r^2} = \frac{\rm const}{r^2} = \frac{L_\nu}{4\pi r^2}$

For a spherical source there is a relationship between the flux density and the monochromatic luminosity : $L_\nu = 4 \pi d^2 S_\nu$, where d is the distance of the star (expressed in W/Hz). The uniformity of the emission across the spherical surface is approximately true for thermal sources, but not for non thermal accelerators where often the emission is beamed within an angle inversely related to the Lorentz factor.

The total or bolometric luminosity is obtained integrating on all frequencies (or wavelengths):

$$L = \int_0^\infty L_\nu d\nu = 4\pi r^2 \int_0^\infty S_\nu d\nu$$

which is the relation between luminoities and flux.Clearly the flux of a source depends on the distance between source and observer while the intensity or luminosity not, it is a property of the source.

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We consider the energy ${\rm d}E_\nu$ emitted by a source with a certain frequency range between$\nu$ and $\nu+{\rm d}\nu$, which flows through a surface element ${\rm d}\sigma$ in the time ${\rm d}t$ in the solid angle ${\rm d}\Omega$ along a ray which points at an angle $\theta$ from the surface normal:

$${\rm d}E_\nu = I_\nu \cos\theta {\rm d}\sigma {\rm d}\Omega {\rm d}t {\rm d}\nu$$

The area in de figure is what in the text is called dσ

Dividing this energy by the time interval we obtain the emitted power or the spectral brightness: $dW (\rm Watt) = I_\nu cos\theta d\sigma d\Omega d\nu$ in (m$^2$ sr Hz).

Notice that the spectral brightness is related to the specific intensity:

$I_\nu \equiv \frac{dW}{cos\theta d\sigma d\nu d\Omega}$ measured in W m$^{-2}$ sr $^{-1}$ Hz $^{-1}$.

The specific radiative intensity of the source is conserved in empty space (there is no absorption nor emission) $I_{\nu,1} = I_{\nu,2}$ and it is independent of the distance r from the source to the detector; it is a property of the source alone. For two surfaces along the same ray (eg. the detector surface seeing the emitting area), we have the relations describing the solid angles (explained for instance by the plot below):

$d\Omega_1 = \frac{cos\theta_2 d\sigma_2}{r^2}$ and $d\Omega_2 = \frac{cos\theta_1 d\sigma_1}{r^2}$ .

The solid angles subtending the element of areas at distance r

As a matter of fact, because the energy is conserved, also the power is:

$dW_1 = I_{\nu,1} cos\theta_1 d\sigma_1 \frac{\cos\theta_2 d\sigma_2}{r^2} d\nu$ = $I_{\nu,2} cos\theta_2 d\sigma_2 \frac{\cos\theta_1 d\sigma_1}{r^2} d\nu = dW_2$.

Hence, specific intensity, and also brightness, are constant over distance and are porperties of a source. The brightness is the same at the source or at the detector.

We can connect the flux density, the emitted energy in the frequency interval $d\nu$ per unit time and surface, with the specific intensity. In doing this, we need to consider that each surface element of the emitting source is seen under a different angle:

Emitting surface element with respect to the observer viewing it.

$S_\nu \equiv \int_{source} \frac{dW}{d\sigma d\nu} = \int_{source} I_\nu(\theta, \phi) \cos\theta d\Omega$ (W m$^{-2}$ Hz$^{-1}$)

Notice that the solid angle is inversely proportional to the square of the distance between source and detector $d$. Hence, $S_\nu \propto d^{-2}$ . Again we that, while brightness is a property of a source the flux depends on the source distance.

Let's consider an optically thick source (see figure) of which we can only see its spherical surface and the intensity is constant on its surface, then there is symmetry in the $\phi$ direction and $d\Omega = 2\pi \sin\theta d\theta$. Hence,

$$S_\nu = 2\pi I \int_{0}^{\theta_c} \sin\theta \cos\theta d\theta = 2\pi I \left[ \frac{\cos^2\theta}{2}\right]_{0}^{\theta_c} = \pi I \sin^2\theta_c = \pi I \left( \frac{R}{d} \right)^2$$

At the surface of the source, $r = R \Rightarrow S_\nu = \pi I; \rm if \, r>>R \Rightarrow S_\nu = \pi \theta_c^2 I = \Omega_c I$ .

Clearly the flux of a source depends on the distance between source and observer while the intensity or luminosity not, it is a property of the source. This inverse-square law of the distance is due to the decreasing solid angle under which the element of the emitting source is seen.

Energy density:

the radiative specific energy density per unit volume, frequency and solid angle is : $dE = u_\nu(\Omega) dV d\Omega \, d\nu = u_\nu(\Omega) c dt dA d\Omega d\nu$, where for radiation we use the volume element is $dV = c dt dA$. Because of the definition of intensity${\rm d}E_\nu = I_\nu {\rm dA} {\rm d}\Omega {\rm d}t {\rm d}\nu$(where the intercepting area dA is perpendicular to the line of sight). We find: $u_\nu(\Omega) = \frac{I_\nu}{c}$, which is the energy per unit volume per unit solid angle and frequency range. Integrating over the solid angle we obtain the specific energy density: $u_\nu = \int u_\nu(\Omega) d\Omega = \frac{1}{c}\int I(\nu) d\Omega \equiv \frac{4\pi}{c} J_\nu$ , where we have introduced the mean intensity $J_\nu$. This is equal to the intensity for an isotropic radiation field. The total radiation energy density $\rm [erg \, cm^{-3}]$ is: $u = \int u_{\nu} d\nu = \frac{4\pi}{c}\int J_\nu d\nu$.

Fluxes at Earth

Effectively the flux observed at Earth from a star will depend on other factors. The absorption in the interstellar medium is represented by the term $A_\nu$, which depends on the frequency. The absorption in the Earth atmosphere can be taken into account by a factor $D_\nu (\theta)$, which depends on the direction at which the radiation penetrates the atmosphere (see image). For instance, only optical and radio telescopes detect directly the radiation from ground; gamma-rays are absorbed and gamma-ray astronomy from ground is an inderect technique where gamma-rays convert into electromagnetic showers in the atmosphere.

The detector itself is characterized by a sensitivity curve which depend on the adopted photosensors and filters. A common photometric system is the Johnson & Morgan UBV, based on 2 filters, whose sensitivity curves correspond to the visual one (V) and the photographic one (B), centered between $\lambda = \frac{c}{\nu}$ = 400 and 450 nm, and a third one (U) centered at 365 nm in the near UV. Here you can find a extensive table of photometric filters. The most known detectors are the eyes, which are not at all bad detectors The visual curve is expecially sensitive to the region between 550 nm and 650 nm (if you want to know more of color perception read Light and the Eye in Color Vision by Bruce MacEvoy, 2009). It is not a chance that our closest star, the Sun has a black body spectrum which peaks at 550 nm as the one shown in the section discuss the thermal radiation. Notice the the eyes are not bad detectors at all. Astronomer's estimates indicate that there ar about 170 billion galaxies in the observable universe, stretching out over a radius of some 45.7 billion* light-years. The Milky Way has some 400 billion stars. When you gaze the sky you observe order of 2500 stars. We compare to the high-energy accelerators that we are mostly interested to in this course. You can derive by yourself the sources observed in the X-rays and gamma rays: the Kifune's plot ! You can see the ramp up of the number of sources visible by telescopes in the high energy domain of X-rays (green), gamma-rays from space (blue) and from ground (red). This is the range of energy most interesting for this course.

The Kifune's plot

The sources of extinction in the Earth's atmosphere, that must be considered when dealing with ground-based astronomical photometry, are: molecular absorption, Rayleigh scattering by molecules, Mie or aereosol non-molecular scattering and non selective scattering. Rayleigh scattering occurs when radiation interacts with molecules and particles in the atmosphere that are smaller in diameter than the wavelength of the incoming radiation. Shorter wavelengths are more readily scattered that longer wavelengths. Mie scattering occurs when the wavelength of the electromagnetic radiation is similar in size to the atmospheric particles. Mie scatter generally influences radiation from the near UV through the mid- infrared parts of the spectrum. Non-selective scattering occurs when the diameter of the particles in the atmosphere is >> the wavelength of radiation and it is primarily caused by water droplets in the atmosphere. Non-selective scattering scatters all radiation evenly through out the visible and infrared portions of the spectrum - hence the term non-selective.

Atmospheric opacity due to absorption of radiation in gases. Ground astronomy is mostly optical and radio due to this, and in the gamma region ground based detectors do not directly detect gamma-rays but the atmospheric showers they induce. Source NASA

Going back to our photometer with a filter (eg V), the measured flux will then be:

$f(\nu) = S_V(\nu) D_\nu(\theta)A_\nu S_\nu$

where $S_\nu$ is defined above and $A_\nu$ and $D_\nu$ are defined above (see figure on the atmospheric opacity below for this last term, and also GSP_216 course of the Humbolt State University).

Integrating over the frequency (or wavelength) the equation above, one obtains the intensity:

$I_V = \int f(\nu) d\nu$

Emission of radiation: radiation can be emitted (eg. decay of electrons in atomic levels) adding energy to a beam $d E= j_\nu dV d\Omega dt d\nu,$ where $j_\nu \rm [erg \, cm^{-3} s^{-1} sr^{-1} Hz^{-1}]$is the coefficient of spontaneous emission = energy added per unit volume, unit solid angle, unit time and unit frequency, which depends on the direction. From this one can realize from $dE = I_\nu dA dt d\Omega d\nu$ that $dI_\nu = j_\nu dx$ , where dx is the distance along the ray dV = A dx.

Absorption of radiation: radiation of specific intensity $I_\nu \rm [W m^{-2} Hz^{-1} sr^{-1}]$ can be absorbed in a layer of thickness dx of a medium of density of particles n $\rm [cm^{-3}]$:

• Each absorber has an effective area of $\sigma_\nu \rm [cm^2]$;

• The number of target absorbers in the area dA is: n dA dx;

• Each of the target absorbers has an effective absorbing area, called the cross section: $\sigma_\nu$;

• The total absorbing area $n \sigma_\nu dA dx$;

• The energy absorbed out of the beam is : $dE = I_\nu (n\sigma_\nu dx) dA d\Omega \, dt \, d\nu$ ;

• The intensity variation is: $dI_\nu = - I_\nu n \sigma_\nu dx = - \alpha_\nu I_\nu dx$, where we introduced the absorption coefficient $\alpha_\nu = n\sigma_\nu$ $[cm^{-1}]$.

• A beam of initial intensity $I_0$ after a distance $x$ will be attenuated and will have $I(x) = I_0 e^{-\alpha x}$ or more generally $I_\nu (x) = I_\nu (0) exp[\int_0^x \alpha_\nu(x)' dx' ]$.

• We can define the optical depth: $\tau_\nu (x) = \int_0^x \alpha_\nu (x')dx' = \int_0^x n(x') \sigma_\nu dx'$ . Hence, $I_\nu (x) = I_\nu (0) e^{-\tau_\nu}$ . If $\tau_\nu > 1 (<1)$ the medium is told 'optically thick (thin)'.

• The probability that a photon travels at least an optical depth is: $e^{-\tau_\nu}$ , hence the mean optical depth is $<\tau_\nu > = \int_0^\infty e^{-\tau_\nu} \tau_\nu d\tau_\nu = -e^{-\tau_\nu}\tau_\nu|_0^\infty + \int_0^\infty e^{-\tau_\nu}d\tau_\nu = -e^{-\tau_\nu}|_0^\infty = 1$

• The average distance that a photon can travel without being absorbed is $<\ell_\nu>= \frac{1}{n\sigma_\nu}$ (see also Particle Physics elements).

Example: The Sun has radius $R_\odot = 7 \times 10^{10} \rm cm$ and mass $M_\odot = 2 \times 10^{33} \rm g$. Hence, assuming it as uniform and spherical its average density is $\rho_\odot = \frac{M_\odot}{\frac{4}{3}\pi R_\odot^3} = 1.4 \rm g cm^{-3}.$ Let’s consider that the Sun is made up predominately of ionized hydrogen (1 proton of mass $m_p = 1.7 \times 10^{-24}$ g ), hence the mean number of protons (equal also to the number of electrons) is: $n = \frac{\rho_\odot}{m_p} = \frac{1.4}{1.7 \times 10^{-24}} = 10^{24} \rm particles/cm^3.$ The cross section of a classical process of scattering of a photon on free electrons is the Thomson cross-section: $\sigma_{Th} = \frac{8\pi}{3} r_e^2 = 6.65 \times 10^{-25} \rm cm^2,$ where $r_e = \frac{e^2}{4\pi\epsilon_0 m_e c^2}$ is the classical electron radius. Thus the mean free path of a photon traveling in the Sun is$<\ell > = \frac{1}{n \sigma_{Th}} = \frac{1}{10^{24} \times 6.65 \times 10^-{25}} \sim 1 cm.$

Phase space occupation number of photons

If $-\vec{n}$ is a unit vector in the direction specified by the polar angles $(\theta, \phi)$ (the sign follows the convention that the telescope points along $\vec{n}$ ) then the momentum of photons traveling along directions in the solid angle $d\Omega = d\cos\theta d\phi$ across an area perpendicular to their direction base $dA_\perp = dA \cos\theta$is given by: . Hence, considering all space coordinates and differentiating: $d^3\vec{p} = \left( \frac{h}{c}\right)^3 \nu^2 d\nu d\cos\theta d\phi$, where the momentum of a photon is $\vec{p}= -\frac{h\nu}{c} \vec{n}$ . The spatial element volume occupied by these photons is a cylinder of height $cdt$ and base $dA_\perp$ = : $d^3x = c \, dt dA_\perp$. The number of quantum mechanical states in the phase space volume $d^3\vec{p} d^3\vec{x}$ (`modes') is:

$$n_{\rm modes} =2 \frac{d^3\vec{p} d^3\vec{x}}{h^3} = 2 \left(\frac{\nu}{c}\right)^2 d\nu \, d\cos\theta d\phi dA_\perp dt$$

where we account for the 2 possible polarisations of radiation (photons have spin $\pm 1$ or left/right-handed circular polarization of light). We compare with the definition of specific intensity: $dE = I_\nu d\nu dA_\perp d\Omega$, and obtain: $n_{\rm modes} = 2 \left( \frac{\nu}{c}\right)^2 \frac{dE}{I_\nu}$. Hence, the energy per mode is given by: $\frac{dE}{n_{\rm modes}} = \frac{c^2 I_\nu}{2 \nu^2}$. Because each photon carries an energy of $h\nu$, the number of photons per quantum state or `occupation number' is:

$$\rho= \frac{c^2 I_{\nu}}{2\nu^2 h\nu} = \frac{c^2 I_\nu}{2 h \nu^3}$$

Black body and thermal radiation

In this part we mostly address the 'colour' of stars. Thermal energy consists of the kinetic energy of random movements of atoms and molecules in matter at T>0 °K, i.e. charged protons and electrons which emit radiation due to charge-acceleration and dipole oscillation. The thermal spectrum of the Sun is shown in the figure below. While in this course we are not interested in emissions related to the temperature of objects , nor to the emission due to transitions of electrons between atomic/molecular energy levels, we remind here some fundamental aspects.

Solar radiation spectrum for direct light at the top of the atmosphere (yellow) and at sea level (red). The Sun produces light with a similar spectral emission of a black body at 5778 °K, which corresponds approximately to the Sun surface temperature. From Wikipedia: Nick84 [https://en.wikipedia.org/wiki/Sunlight#/media/File:Solar_spectrum_en.svg]

Ideally, a black body in equilibrium at constant temperature T emits a power per unit surface, solid angle and for a given frequency $\nu$ (it is a specific intensity!) given by the Planck function:

$B_\nu(\nu,T) = \frac{2 h\nu^3}{c^2}\frac{1}{e^{\frac{h\nu}{k_B T}}-1}$

where $k_B = 1.38 \times 10^{-16} erg/^\circ K$ is the Boltzmann constant and $h = 6.63 \times 10^{-27} \rm erg \, s$ is the Planck constant (see Basic Concepts and Notations). We recognise the relation between the specific intensity and the photon occupation number derived above: $I_\nu = \frac{2 h \nu^3}{c^2 \rho}$ and this last is given by the Bose-Einstein distribution for thermodinamical equilibrium (chemical potential $\mu = 0$): $\rho = f(E) = \frac{1}{exp[(E-\mu)/kT] - 1}$. Notice, by the way, that the photon occupation number is a relativistic invarant, and from this it comes out that $\frac{I_\nu}{\nu^3} \propto \rho$ is a relativistic invariant (see Introduction to Special Relativity). Two regimes of the Planck function are relevant:

1) The Rayleigh-Jeans approximation $h\nu << k_B T \Rightarrow \nu << 2 \times 10^{10} T/^\circ K.$ At these frequencies the specific intensity $B_\nu \sim \frac{2k_B T \nu^2}{c^2}$ . In the radio the brightness temperature provided by this formula is often used as a measurement of the intensity;

2) At $h\nu >> k_B T \Rightarrow B_\nu \sim \frac{2h\nu^3}{c^2} e^{-\frac{h\nu}{k_BT}}$.

Moreover we can determine the frequency corresponding t the maximum of the Planck function by differentiating it and equatng to zero: $\frac{\partial B_\nu}{\partial \nu} = 0 \Rightarrow x = 3 (1-e^{-x}),$ where $x = \frac{h\nu}{k_B T}$. We can solve this equation numerically to find $x \sim 2.9 \Rightarrow h\nu_{max} = k_B T.$ Hence, the frequency corresponding to the maximum of the black body spectrum is: $\nu_{max} \sim \frac{c}{\lambda_{max}} = 10^{11}$ ($T$/°K) Hz. Translating this into wavelength using $\lambda = \frac{c}{\nu}$, we obrain the Wien's displacement law, connecting the maximum of the black body spectrum to its effective temperature: $\lambda_{max} T = 0.29 \rm cm ^\circ K = 2.9 \times 10^6 \rm nm ^\circ K$.

As defined above, the specific energy density is: $u_\nu = \int u_\nu(\Omega) d\Omega = \frac{1}{c}\int I(\nu) d\Omega \equiv \frac{4\pi}{c} J_\nu$from which we get thee total radiation energy density $\rm [erg \, cm^{-3}]$: $u = \int u_{\nu} d\nu = \frac{4\pi}{c}\int J_\nu d\nu$. Hence, the total energy density of the black body is obtained by integrating its isotropic emission over the solid angle and by integrating over its frequency spectrum: $u_{BB}(T) = \frac{4\pi}{c} \int_0^\infty B_\nu d\nu \propto T^4$ and so we obtain that the total flux emitted by the black body out of its surface is $\Phi = \frac{c}{4}u_{BB} = \sigma_{SB} T^4$ , which is the Stephan - Boltzmann law with $\sigma_{SB} = 5.67 \times 10^{-5} \rm erg \, cm^{-2} s^{-1} K^{-4}$ .

At low frequency the Rayleigh-Jeans function inversely proportional to the square of the frequency approximates the Planck function, while at high frequency the Wien's law approximates it (Wikipedia CC BY-SA 3.0).

Integrating over the frequency (or the wavelength for the similar function $B_\lambda$) we obtain the Stephan-Boltzmann law: $W = \sigma T^4$ where $\sigma = 5.67 \times 10^{-8}$ W m$^{-2}$ K$^{-4}$.

The luminosity integrated over all wavelengths of a source is related to its temperature: $L = 4\pi R^2 \sigma T_{eff}^4$[units of Watt], where R is the star radius.

For the Sun $L_{\odot} = 4 \pi R_\odot^2 \sigma T^4\sim 3.86 \times 10^{26}$W, $\rm R = 6.95 \times 10^{8}$ m and d = distance Earth-Sun= $1.49 \times 10^{11}$ m.

Example 1: calculate the effective temperature of the Sun (corresponding to $\lambda_{max} \sim 500$ nm). From the Wien's displacement law: $T_{eff} = \frac{2.9 \times 10^6}{500} \sim 5800$°K.

Example 2:

Example 3: calculate the effective temperature of the Earth. At the Earth the energy is passing through a spere of radius d. We can define the solar constant = the received power per unit area at the Earth or irradiance = $E_{\oplus} = \frac{L_\odot}{4\pi d^2} = 1384 W/m^2$= $\sigma T^4$. The Earth has a radius of 6400 km, hence the solar power absorbed by the Earth is $\Phi_{abs} = E_\oplus \pi R_\oplus^2$and $4 \pi R_\oplus^2 \sigma T_{\oplus}^4 = E_\oplus \pi R_\oplus^2$ = $\frac{4 \pi R_\odot^2 \sigma T_\odot^4}{4 \pi d^2} \times \pi R_\oplus^2$$\Rightarrow T_{\oplus}^4 = \frac{R_\odot^2 T_{\oplus}^4}{4d^2}$ so the resulting effective temperature of the Earth is $\sim 279$ °K

$T_{eff} \sim 5777$ °K. The Human body has $T_{eff} \sim 310$°K and 9 $\mu$m; molecular clouds have typically $T_{eff} \sim 15$°K and $\lambda_{max}\sim 200 \, \mu$m.

Magnitude

At this point it is useful to introduce the definition of the apparent visible magnitude of a star (Pogson definition from 1856): $\rm m_V = -2.5 log I_V + const$. The magnitude follows a logarithmic scale of luminosity where smallest values correspond to brightest objects. The Greeks defined stars visible by eye in 6 classes, for which each steps in magnitude is separated by about a factor of 2.5 from the other. This definition follows our eyes since we can distinguish the brightness of 2 objects if their magnitude varies by this factor. Hence, a star with $\rm m_V = 1$ (first magnitude star) is about 2.5 times brighter than a second star with $m_V =2.5$, $2.5^2$ than a third magnitude star, $2.5^3$than a fourth magnitude star and so on. In other words: $\frac{I_{V,1}}{I_{V,2}} = 10^{(m_2 - m_1)/2.5} = 10^{\Delta m/2.5}$ and for $\Delta m = 1$ this means that $\frac{I_{V,1}}{I_{V,2}} = 10^{1/2.5} \sim 2.5$. The definition above requires fixing an absolute zero: for instance we can take Vega for which $\rm m_{V,0} \sim 0 = -2.5 log I_{V,0} + const$and use $m_V = -2.5 log\frac{I_{V}}{I_{V,0}}$. The naked eye is sensitive to about $m_V = +6$.

The absolute magnitude is the apparent magnitude that the object would have if it would be seen from a distance of 10 pc (32.6 ly). Because the intrinsic brightness of a source $I_V \propto \frac{1}{d^2}$, with $d$ distance of the source, we can define the absolute magnitude M through:

$\frac{I_V}{I_{V,0}} = 10^{\frac{\Delta m}{2.5}} \sim 2.5^{(M-m)} = \left( \frac{10 \rm pc}{d_{pc}}\right)^2$$\Rightarrow (M-m) log_{10}2.5 = 2 log_{10}\left(\frac{10 \rm pc}{d_{pc}}\right)$

Hence, $M = m - 5(log_{10}(d_{pc})-1)$.

The bolometrci absolute magnitude is defined as:

$M = 2.5 log_{10} \left(\frac{L}{L_\odot} \right)+ 4.74$, where the Sun magnitude is 4.74 and its luminosity inetgra tedover the full is $L_{\odot} = 3.85 \times 10^{33}$erg/s.

Visual magnitude of celestial objects. Many of the objects we will encounter in this Course are on the right region (eg black holes in galaxies are between 20-25 depending on there activity state)

Resolving power

When light crosses the objective of a telescope diffraction occurs which limits the capability to distinguish two objects into individual images. Considering diffraction through a circular aperture, this translates into: $\theta = 1.22 \frac{\lambda}{D}$ , where D is the diameter of the objective. Below the resolving power is expressed in arcsec where 1 arcsec = 1/3600 degree.

The historical evolution of the resolving power of telescopes.

Measuring distances in astronomy

Various methods are used depending on the distance of the sources. For near stars the common method adopted is the parallax, which is the apparent displacement that a star undergoes on the celestial sphere with respect to a reference direction, due to the real variation of the observer's position. The parallax can be daily or annual, using the rotation of the Earth on its axis (12 hours maximum) or the rotation of hte Earth around the Sun (6 months maximum - see the image below and notice that 1 AU = $1.495978 \times 10^{11}$ m.) . The nearest star is proxima centauri, which exhibits a parallax of 0.762 arcsec, and therefore is 1.31 parsecs away. The maximum measurable distance of 100 pc (326 light years) has been achieved by the Hipparcos satellite.

By R Nave, Hyperphysics [http://hyperphysics.phy-astr.gsu.edu/hbase/Astro/para.html#c1].

A summary of methods is here. The Cepheid Variables (visible to about 20 million light years with the brightest of them having $m \sim -6$ ) is a valuable method based on the relation between their apparent luminosity and their period. Othe rstandard candles are supernovae, whic hcan reach peak magnitudes of $m \sim -19$, giving access to distances larger than 300 Mly.

Coordinate Systems

The first thing you need to know is how to locate sources, like galaxies or gamma-ray bursts, in the sky. And also how to unequivocaly say the time of an astronomical event, something that is more tricky than what you might think. Coordinate systems are used to identify directions of events in an experiment (local coordinate system) and to map objects positions in the sky (celestial coordinate systems). If you want to know more on how to transform the local coordinate system in the celestial reference frames or how to define time of an event visit the SLALIB library site or the course of Prof. Majewski.

The Local Coordinate System

In each experiment we need to define the local reference frame (also named horizontal) by positioning the x-axis at a certain angle from the North. Then two angles determine directions: For example in the horizontal coordinate system zenith angle (defined from the z axis at 0° to 180°) and the azimuth angle (defined between 0°-360°). Also the elevation/altitude angle can be used, the complementary angle of the zenith from -90° to + 90°. These coordinates are not useful to inequivocally identify the position of an object in the sky due to the Earth rotation, nutation and precession of the rotation axis. We will need the latitude of the experiment and the time of detection of an event to transform its coordinates from the local to the frame of the stars.

Celestial coordinate systems

Those systems are independent of the observer's local position. Two of the mostly used coordinates systems in astroparticle are:

• Equatorial coordinates: It's defined by an origin at the center of the Earth, a fundamental plane consisting of the projection of the Earth's equator onto the celestial sphere (celestial equator). The position of a star is defined by α = right ascension and δ = declination. The NCP is the North Celestial Pole, on the mean direction of the rotation axis of the Earth and at declination δ= 90°. The celestial equator is at δ = 0 an the SCP at δ = -90°. At the epoch 2000 the Earth rotation axis was tilted 23° 26’ 21.448’’ from the direction perpendicular to the plane of ecliptic (= plane of Earth’s orbit about the Sun -indicated in red) meaning that this is the angle between the celestial equator and the plane of the ecliptic. α and δ are referred to epoch 2000. The Earth’s rotation axis points about to the same direction when the Earth moves around the Sun (hence seasonal changes), but for precessional changes due to tidal torques acting on the aspherical Earth, which require corrections from the epoch 2000 to the epoch of observation. Another effect is nutation which slightly tilts the rotation axis during precession due to Sun and Moon tidal forces.

Equatorial coordinate system

Image of nutation and precession of the rotation axis. Image from Wikipedia (Designed by Dr. H. Sulzer - Original)

• Galactic coordinates: The galactic longitude, $\ell$ is the angular distance Eastward (counter-clockwise looking down on the Galaxy from the GNP) from the Galactic Center where $\ell = 0$and the galactic latitude, $b$, is the angular distance outside of the plane of the Galaxy, positive up, negative down. The Galactic equator is chosen to be that great circle on the sky approximately aligned with the Milky Way mid-plane.

  This plane is inclined by 62° 36' to the celestial equator.The NCP is at 27.5° in the 1950 epoch.

You can find more information in the Slalib document and in the lectures of ASTR 5610, Prof. Majewski.

Time Scales and Standards

There are several time standards or ways to express the measurement of time. A time standard specifies the rate at which time passes or points in time, or both. Before defining a standard we need first agree on what is a day.

Solar Day vs Siderial Day

The definition of what is a day might not be as straight forward as one can think. Indeed there are two different ways to define a day. The time it takes the Earth to complete a full rotation, what is called siderial day and the time it takes to see the Sun rising from the same point at a given location, what is called a solar day. For the given orbit of the Earth, it happens that the siderial day is shorter than the solar day, typically about 4 minutes.

Differences between a solar day and a siderial day

Time scales and time standards

Once we have define the basic unit of a day, we could derive some time standards:

• Mean solar time There are two solar times, the apparent one (also called true one) which depends on latitude and the year and the Mean solar time which the time of mean sun, the difference between the two is called the equation of time. The length of the mean solar day is slowly increasing due to the tidal acceleration of the Moon by the Earth and the corresponding slowing of Earth's rotation by the Moon.

• Universal Time (UT0, UT1) Is a time scale based on the mean solar day, defined to be as uniform as possible despite variations in Earth's rotation

• International Atomic Time Is the primary international time standard from which other time standards, including UTC, are calculated. TAI is kept by the BIPM (International Bureau of Weights and Measures), and is based on the combined input of many atomic clocks around the world.

• Coordinated Universal Time (UTC) is an atomic time scale designed to approximate Universal Time. UTC differs from TAI by an integral number of seconds. UTC is kept within 0.9 second of UT1 by the introduction of one-second steps to UTC. The difference with UT1 is known as DUT1.

Time representations: JD and MJD

These are not technically standards (or scales), they are just representations (formats) of the aforementioned standards typically used in astronomy:

• Julian Date Is the count of days elapsed since Greenwich mean noon on 1 January 4713 B.C., Julian proleptic calendar. Note that this day count conforms with the astronomical convention starting the day at noon, in contrast with the civil practice where the day starts with midnight (in popular use the belief is widespread that the day ends with midnight, but this is not the proper scientific use).

• Modified Julian Date Is defined as MJD = JD - 2400000.5. The half day is subtracted so that the day starts at midnight in conformance with civil time reckoning. There is a good reason for this modification and it has to do with how much precision one can represent in a double (IEEE 754) variable. Julian dates can be expressed in UT, TAI, TDT, etc. and so for precise applications the timescale should be specified, e.g. MJD 49135.3824 TAI.

Time in experiments

Practically speaking, in experiments time comes from one or more of the following sources:

• Atomic clocks (Cs, Rb) -They use the microwave signal that electrons in atoms emit when they change energy levels. These have very good short term performance but a Rb clock left by itself will wander by several ns per day. Cs clocks are perhaps better by a factor of 100x.

• GPS - The GPS gives precision timing too. The system consists of the space segment of O(30) satellites each equipped with Caesium atomic clocks and each constantly getting corrections from the central control facility. GPS broadcasts navigation and time messages synchronized to this ultraprecise time from which the user segment can extract time and space coordinates accurate to O(10) ns and meters, respectively. GPS time is based on the 86400 second day. It indirectly accounts for leap years. There are no leap seconds in GPS time.

• Many of the large arrays in Astroparticle need to synchronize the various elements which compose them. To do this with sub-nanosecond accuracy,White Rabbit can be used.