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--- courses:ast201:2 [2023/06/07 05:09] – [4.3 Spectrum analysis] asad
+++ courses:ast201:2 [2023/10/04 00:39] (current) – [1. Probes in astronomy] asad
@@ Line 3: / Line 3: @@
 ===== - Probes in astronomy =====
+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.
+{{https://upload.wikimedia.org/wikipedia/commons/thumb/0/00/Standard_Model_of_Elementary_Particles.svg/803px-Standard_Model_of_Elementary_Particles.svg.png?nolink&500}}
+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 ([[wp>fermions]], named after [[wp>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 ([[wp>bosons]], named after [[wp>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.
 ==== - Massive particles ====
@@ Line 79: / Line 88: @@
 ==== - As a particle ====
+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.
 ==== - As a ray ====
 Photometry
@@ Line 129: / Line 144: @@
 Now if we can resolve the source, the solid angle subtended by a spherical source of radius $a$ at a distance $r$ (when $a\ll r$)
-$$ \Omega \approx \frac{\pi a^2}{r^2}. $$
+$$ \Omega \approx \frac{\pi a^2}{r^2} $$
-Then the surface brightness
+whose is angle is [[un:steradian]]. Then the surface brightness
 $$ S = \frac{F_o}{\Omega} = \frac{F_s}{\pi} $$
@@ Line 199: / Line 214: @@
 "...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.
-: Newton's prism. 1802: Wollaston's dark lines.
+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.
-Fraunhofer, 1812: sun's lines (350 precise, 225 less precise for fainter lines). 1823: bright stars and planets.
+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.
 {{https://upload.wikimedia.org/wikipedia/commons/a/a4/DBP_1987_1313_Joseph_von_Fraunhofer%2C_Sonnenspektrum.jpg?nolink&500}}
@@ Line 233: / Line 248: @@
 Blackbodies are never actually black, their color depends on one thing only: T.
-Berlin 1900, Planck function:
+In a Berlin conference in 1900, Max Planck proposed the following function for the surface brightness $S$ of a blackbody.
-$$ B(\nu,T) = \frac{2h\nu^3}{c^2} \frac{1}{e^{h\nu/(kT)}-1} $$
+$$ S_\nu = \frac{2h\nu^3}{c^2} \frac{1}{e^{h\nu/(kT)}-1} $$
-$$ B(\lambda,T) = \frac{2hc^2}{\lambda^5} \frac{1}{e^{hc/(\lambda kT)}-1} $$
+which is shown in the units of nW sr$^{-1}$ m$^{-2}$ Hz$^{-1}$.
-[{{:courses:ast201:planck-function.png?nolink&550|Surface brightness $S$ of a blackbody as a function of wavelength created using [[https://colab.research.google.com/drive/1vhrIuzj8mA5RnksjLuWGsPMko6J993gi?usp=sharing|Google Colab]].}}]
+[[https://colab.research.google.com/drive/1vhrIuzj8mA5RnksjLuWGsPMko6J993gi?usp=sharing|{{:courses:ast201:blackbody-nw-thz.png?nolink&570}}]]
-Why total power emitted by a blackbody at all angles is $\pi B$?
+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.
+[[https://colab.research.google.com/drive/1vhrIuzj8mA5RnksjLuWGsPMko6J993gi?usp=sharing|{{:courses:ast201:blackbody-kW-nm.png?nolink&570}}]]
+\\
+Why total power emitted by a blackbody at all angles is $\pi S$?
 At long wavelengths
-$$ B(\lambda,T) = \frac{2ckT}{\lambda^4} $$
+$$ S_\lambda = \frac{2ckT}{\lambda^4} $$
-$$ B(\nu,T) = 2kT\frac{\nu^2}{c^2} $$
+$$ S_\nu = 2kT\frac{\nu^2}{c^2} $$
 and we can define a brightness temperature
-$$ T = \frac{B\lambda^4}{2ck}. $$
+$$ T = \frac{S_\lambda\lambda^4}{2ck}. $$
 ==== - Stellar spectra ====
@@ Line 296: / Line 319: @@
 For a star, we could write either $m_B=5.67$ or $B=5.67$ for the magnitude at B band.
+[{{:courses:ast201:apparent-magnitude-scale.jpg?nolink&700|Credit: Pasachoff & Filippenko.}}]
 ==== - Absolute ====
+$$ m-M = 5\log(r)-5 $$
+where $M$ is the absolute magnitude and $r$ the distance in [[un: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 $M_{bol}=4.75$.
+==== - Brightness from image ====
+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.
+  - Differential photometry: two stars on the same image.
+  - All-sky photometry: two stars on different images.
-==== - Brightness from magnitude ====