Light and color

“Astronomers are storytellers… Astronomy is a lot like paleontology than it is like physics.” — Frederick R. Chromey.

Astronomy deals with particles or waves of matter or energy coming from outer space. Many of the fundamental particles illustrated in the standard model of particle physics below can be found in space.

Quarks are not found in isolation, but in packets called protons or neutrons. We detect mostly protons from space using specialized detectors. Among the leptons, electrons and neutrinos are the most common particles found streaming through space. Quarks and leptons are particles of matter (fermions, named after Enrico Fermi) and they are commonly called cosmic rays in astronomy; cosmic-ray astronomy is a large community.

The particles of energy or force carriers (bosons, named after Satyendranath Bose) constitute two more communities within astronomy. The photons or electromagnetic waves create the largest community of astronomers and they are the ones who use telescopes. The gravitational waves are detected by specialized detectors creating the most recent community of astronomy and astrophysics. Note that only the photon collectors and detectors are called telescopes, the detectors of cosmic rays or gravitational waves are simply called detectors.

Fermions are massive, bosons are massless. The two are described below in the context of observational astronomy.

Cosmic rays are high-energy particles entering the atmosphere of earth from outer space.

Primary cosmic rays consist of mainly nuclei of hydrogen (84%) and helium (14%). The remaining 2% are heavier nuclei or free electrons and positrons.

Some primary particles come from solar flares, but most are extrasolar. On each square meter of the upper atmosphere of earth, 6000 cosmic rays strike every second.

The speed of the particles is close to the speed of light. Their high energy is measured in electron-volts.

$$ 1 \text{ eV} = 1.602\times 10^{-19} \text{ J}. $$

The energy of primary cosmic rays can be 1 MeV to $10^{20}$ eV. The higher the energy the rarer the rays, the mean being around 10 GeV. The relation between their mass and energy is given as

$$ E = \frac{mc^2}{\sqrt{1-v^2/c^2}} $$

where $m$ is the rest mass of a particle, $v$ its speed and $c$ the speed of light. It is standard to express the mass in terms of energy as $mc^2$.

What is the rest mass of proton in eV?

The speeds of cosmic ray particles are much higher than anything possible in our laboratories. What is the highest energy attained at LHC so far?

Secondary cosmic rays are created when the primaries strike the atmosphere and create more particles that shower down to the surface of earth. This collision happens around 50 km above ground.

The secondaries give information about the primaries because total energy is conserved.

Cosmic ray collisions can create nuclei, individual nucleons, electrons, and new particles like positrons and gamma rays, and exotic things like kaons. Pions, muons and positrons were first discovered in secondary cosmic rays.

Primary cosmic rays can only be detected from space. But secondary cosmic rays were discovered by Victor Hess in 1912. He took an electrometer to an altitude of 5 km during a solar eclipse and found that the ionization levels still increase with altitude. As this could not be because of the sun, the only explanation was radiation coming from outer space.

Neutrinos are created in nuclear reactions involving weak nuclear force. Their rest mass is very small, around 0.05 eV, but they could be the most numerous particles in the universe. Many of them were created immediately after the big bang and they are routinely produced inside stellar cores. They can penetrate right through the earth or the sun.

So they are very difficult to detect. Super-Kamiokande (Super-Kamioka Neutrino Detection Experiment) or SK is made of a 50-kilo-ton tank of water 1 km underground, 125 km west of Tokyo. Photodetectors on the walls of the tank detect light coming from the reaction of neutrino with the water.

So far we have detected neutrinos from only two celestial objects: Sun and SN 1987A.

Meteorites can be considered as another probe in astronomy. They come mainly from the asteroid belt of the solar system. But a trace amount has come from the surface of the Moon and Mars as well.

The age of the solar system (4.56 Gy¹⁾) has been found by calculating the abundance of radionuclides in meteorites.

The presence of Al-26 (half-life around 1 My²⁾) in a meteorite suggested that part of the material in the molecular cloud that gave birth to our system came from a nearby supernova.

Gravitons are particles associated with gravitational field and waves. They are bosons; they do not have mass.

In 2015, gravitational waves were directly detected by LIGO from the merger of two black holes as shown above. LIGO detected the gravitational wave strain $h$ which is the relative change in distances due to a passing gravitational wave. It is a dimensionless number. If the wave has a strain $h\sim 10^{-21}$ for a $b=4$ km baseline, the distance along $x$-axis will change by $(1+h/2)b \approx 10^{-18}$ m or 1 exa-meter (Em). Such a small distance change has really been measured.

Photons are particles associated with electromagnetic field and waves and we will focus on only photons in this course. We study astronomical objects mainly through the light coming from them. And light is an electromagnetic wave made of photons.

“Some (not astronomers!) regard astronomy as applied physics.” — Chromey.

Huygens proposed a wave theory of light in a 1678 book, but Newton did not support the theory; Newton’s particle theory flourished in the next century.

Wave is a disturbance that can propagate in space in a coherent way. Particles also propagate. Both waves and particles reflect (change directions at the interface between two media) and refract (change velocity when transmitting medium changes). But waves have two unique properties: diffraction and interference. Diffraction is the ability of waves to bend around obstacles. Interference is the ability to combine with other waves in a determinate way.

Waves can be quantified using amplitude, wavelength, frequency, wave speed and phase. Scientists beginning from Fresnel were able to measure these for light waves.

The great achievement of the 19th century was that light was understood to be a wave, an electromagnetic wave. An accelerated charge produces waves.

https://www.compadre.org/osp/EJSS/4126/154.htm

https://ophysics.com/em3.html

$$ c = (\epsilon \mu)^{-1/2} $$

$$ c = \lambda \nu $$

From the beginning of the twentieth century, quantum mechanics claimed that there are situations where light cannot be described as a wave, but rather as a stream of particles called photons.

The energy of a photon, however, is related to the frequency of the light when it exhibits its wave property according to the equation

$$ E = h\nu = \frac{hc}{\lambda} $$

where $h=6.626\times 10^{-34}$ J s is Planck’s constant.

Photometry

Spectrometry

Astrometry

Polarimetry

Imaging

$L$: luminosity, J s$^{-1}$ or W.

$$ L_\odot \approx 10^{26} \text{ W} $$

Flux

$$ F = \frac{E}{At} $$

Unit: W m$^{-2}$

Flux is the apparent brightness.

If there is nothing between the source and the astronomer, the average brightness

$$ \langle F \rangle = \frac{L}{4\pi r^2} $$

and if the radiation is isotropic, the average flux will be the same as the flux measured at any point on the circumference, then local flux for the observer

$$ F_o = \frac{L}{4\pi r^2}. $$

Left hand side is easily known. But there are two unknowns in the right hand side: luminosity and distance. One cannot be known without the other. The greatest problem for an astronomer.

But there is one quantity that does not depend on the distance: surface brightness of an extended object as opposed to a compact object (point source).

Let us define $F_s$ as the amount of energy leaving from the surface of a spherical object per unit area:

$$ F_s = \frac{L}{4\pi a^2} $$

where $a$ is the radius of the source object. This $F_s$ has the same unit W m$^{-2}$ as $F_o$ but they are not the same at all, $F_s$ is related to the source, $F_o$ to the observer.

Now if we can resolve the source, the solid angle subtended by a spherical source of radius $a$ at a distance $r$ (when $a\ll r$)

$$ \Omega \approx \frac{\pi a^2}{r^2} $$

whose is angle is steradian. Then the surface brightness

$$ S = \frac{F_o}{\Omega} = \frac{F_s}{\pi} $$

which does not depend on the distance. We cannot measure the $S$ of stars, but surface brightness of planets, nebulae and galaxies are easily measurable.

1 steradian = 1 rad$^2$ = $4.25\times 10^{10}$ arcsec$^2$.

So 1 arcsec$^2$ is almost $10^{-10}$ steradian.

$$ f_\nu = \frac{dF}{d\nu} = \lim_{\Delta\nu\rightarrow 0} \frac{F(\nu,\nu+\Delta\nu)}{\Delta\nu} $$

Here $f_\nu$ is the spectrum, unit W m$^{-2}$ Hz$^{-1}$.

Low-resolution spectrum: $\Delta\nu$ large.

High-resolution spectrum: $\Delta\nu$ small.

$$ f_\lambda = \frac{dF}{d\lambda} = \lim_{\Delta\lambda\rightarrow 0} \frac{F(\lambda,\lambda+\Delta\lambda)}{\Delta\lambda} $$

Both $f_\nu$ and $f_\lambda$ are spectra, but they look different.

An ideal bolometer records radiation at all wavelengths within a band with perfect efficiency.

Bolometric flux

$$ F_{bol} = \int_0^\infty f_\lambda d\lambda $$

In reality,

$$ F(\lambda_1,\lambda_2) = \int_{\lambda_1}^{\lambda_2} f_\lambda d\lambda = F(\nu_1,\nu_2) = \int_{\nu_1}^{\nu_2} f_\nu d\nu $$

Introducing the efficiency function $R(\lambda)$ we get

$$ F = \int_{\lambda_1}^{\lambda^2} R(\lambda) f_\lambda d\lambda $$

where $R(\lambda)$ can come from the instrument or the atmosphere.

Earth’s atmosphere is transparent in visible and near-infrared pass-band: from 0.32 to 1 micron³⁾.

Standard bands are mostly used in visible and near-infrared, but very rare at lower or higher frequencies.

“[With regard to stars] . . . we would never know how to study by any means their chemical composition . . . In a word, our positive knowledge with respect to stars is necessarily limited solely to geometrical and mechanical phenomena…” — Auguste Comte, 1835.

“…I made some observations which disclose an unexpected explanation of the origin of Fraunhofer’s lines, and authorize conclusions therefrom respecting the material constitution of the atmosphere of the sun, and perhaps also of that of the brighter fixed stars.” — Gustav Kirchhoff, 1859.

Astronomers are fond of comparing these two sentences uttered within one generation of each other. Comte could not predict that Kirchhoff would find chemical explanation of Fraunhoffer’s lines so soon.

The spectrum of visible white sunlight was first analyzed by Newton in 1666 using glass prisms. In 1802, Wollaston’s observed dark lines in the solar spectrum. In 1812, Fraunhoffer made a detailed map of solar absorption lines using a superior spectroscope. He found precise positions of 350 lines and approximate positions of 225 fainter lines. for fainter lines). 1823: bright stars and planets.

The two absorption lines D occur at the same wavelength (589 nm, 589.6 nm) as the two emission lines produced by a candle flame. The cause is sodium.

Gases in a flame produce emission lines, but heated solids only produce continuous spectra.

Physicist Kirchhoff and chemist Bunsen come together in Heidelberg, 1859.

The two quickly found sodium, potassium, iron and calcium in the solar atmosphere but no lithium.

Almost all stars show absorption spectra.

1940s: more precise measurements, all stars made of hydrogen and helium, H being 12 times more than He.

Blackbody spectra: Kirchhoff, 1860. Reflects nothing but emit. Absorbs and emits. A hole in an oven, the hole is the blackbody.

Any dense solid, liquid or gas emits blackbody radiation. Low-density gas produces line emission.

Stefan-Boltzmann law says, the surface brightness of a blackbody

$$ s = \sigma T^4 $$

where $\sigma$ is the Stefan-Boltzmann constant.

In 1893, Wien’s displacement law said $T\lambda_m \approx 2.9\times 10^{-3}$ m K or $T/\nu_m\approx 1.7\times 10^{-11}$ Hz$^{-1}$ K.

Blackbodies are never actually black, their color depends on one thing only: T.

In a Berlin conference in 1900, Max Planck proposed the following function for the surface brightness $S$ of a blackbody.

$$ S_\nu = \frac{2h\nu^3}{c^2} \frac{1}{e^{h\nu/(kT)}-1} $$

which is shown in the units of nW sr$^{-1}$ m$^{-2}$ Hz$^{-1}$.

The same brightness can be represented as a function of wavelength as

$$ S_\lambda = \frac{2hc^2}{\lambda^5} \frac{1}{e^{hc/(\lambda kT)}-1} $$

and the corresponding plot in the units of kW sr$^{-1}$ m$^{-2}$ nm$^{-1}$ is given below.

Why total power emitted by a blackbody at all angles is $\pi S$?

At long wavelengths

$$ S_\lambda = \frac{2ckT}{\lambda^4} $$

$$ S_\nu = 2kT\frac{\nu^2}{c^2} $$

and we can define a brightness temperature

$$ T = \frac{S_\lambda\lambda^4}{2ck}. $$

Spectra mainly depends on temperature, surface gravity and chemical composition.

Spectral type indicates temperature. More precise type can also indicate surface gravity, radius, luminosity.

Spectral type has 3 designations: the letter indicates temperature class, the number the temperature subclass, and the roman numeral the luminosity (surface gravity, radius) class. Luminosity class I (supergiant) is the most luminous, III intermediate (giants) and V least luminous (dwarves). Most stars are dwarves.

L and T stars are called brown dwarf.

There are carbon stars, Wolf-Rayet stars.

Hipparchus published a catalog of 600 stars with magnitudes.

His magnitude basically measured $F_v$, the flux in the visual pass-band.

At high illumination, cones (one type of receptor cells) operate with a peak sensitivity at 555 nm (green, photpic), at low illumination rods operate with peak sensitivity at 505 nm (yellow, scotopic). Hipparchus’ system works at low light levels.

$$ m = -2.5 \log_{10} F + C $$

Set $C$ such that Vega has $m\approx 0$.

$$ \Delta m = m_1 - m_2 = -2.5 \log_{10} \frac{F_1}{F_2}. $$

$$ \frac{F_1}{F_2} = 10^{-0.4\Delta m} $$

For a star, we could write either $m_B=5.67$ or $B=5.67$ for the magnitude at B band.

$$ m-M = 5\log(r)-5 $$

where $M$ is the absolute magnitude and $r$ the distance in parsec.

($m-M$) is called the distance modulus of a source.

The absolute magnitudes of the sun are $M_B=5.48$, $M_V=4.83$ and $M_{bol}=4.75$.

The size of a star in an image only depends on the telescope and the atmosphere, not on the star. All stars have the same size in an image. The flux

$$ F = \frac{1}{tA} \sum_{x,y}S_{xy} $$

where $S_{xy}$ is the value of the pixel $(x,y)$, the exposure time is $t$ and $A$ is the area of the camera lens (if the image is taken by a lens camera).

But $S_{xy}$ has two contributions: $E_{xy}$ is the flux from a star, and $B_{xy}$ the flux from the background, from anything other than the star along the line of sight or surroundings. So

$$ E_{xy} = S_{xy} - B_{xy}. $$

Assuming $B_{xy}=B$, the average background noise in the image, the flux of the star

$$ F_\star = \frac{1}{tA} \sum_{x,y} (S_{xy}-B). $$

Now if we take two pictures, one of our target star and another of Vega as a standard star, then we can measure the magnitude of the target star with respect to Vega as

$$ m_\star-m_{Vega} = -2.5 \log_{10} \frac{F_\star}{F_{Vega}} = -2.5 \log_{10} \frac{\sum (S_{xy}-B)_\star}{\sum (S_{xy}-B)_{Vega}}. $$

Instead of Vega we can use any other standard star.

1. Differential photometry: two stars on the same image.
2. All-sky photometry: two stars on different images.