B = L/(4πd2)
Where, B= Brightness of star measured using magnitude scale
L= Luminosity or energy output of star per unit time also known as
intrinsic brightness of the star measured in Suns
d= Distance from the star
Also, apparent magnitude m and absolute magnitude M can be linked as
M = m + 5 - 5logd
Where, d= Distance from star
Apparent magnitude m= Brightness as observed through telescope without atmospheric interference.
Absolute magnitude M= Apparent brightness of star at 10 parsec from
observer.
A star’s luminosity in Suns can be linked to its absolute magnitude M as
logL = -0.4M + 1.884
In Astronomy, Brightness is measured using a calibrated magnitude scale first created by Greek Astronomer Hipparchus in around 130-120 BC. SI Unit for apparent brightness is W/m2 and is defined as measure of amount of energy coming from a star per unit area per unit time to Earth. Magnitude of two stars and their brightness b1 and b2 can be linked as
b1/b2 = 10(2/5)(m2-m1)
When she plotted luminosity against period of each variable, the pattern was noticeable. Period of these variable stars was directly proportional to their maximum luminosity. She analyzed 1777 of these variables to come to her conclusion and published her result in ‘Annals of the Astronomical Observatory of Harvard College’ in 1908 titled ‘1777 Variables in the Magellanic Clouds’. After further study she gave confirmation to her initial conclusion in 1912. As Leavitt put it- “A straight line can readily be drawn among each of the two series of points corresponding to maxima and minima, thus showing that there is a simple relation between the brightness of the variables and their periods”. Leavitt’s plot is known as ‘period luminosity relationship’ or ‘Leavitt’s law’ and is used by Astronomers to determine absolute brightness/luminosity/magnitude of Cepheid, having obtained its period through telescopic observation.
Tanp = r/d
From this,
d = r/(Tanp)
Distance of star from Earth will be
D = √(r2 + d2)
Here, p = parallax
r = 149 million km.
Parallax is a very small angle and is usually measured in seconds of arc. A star with parallax of 1 second of arc as observed from Earth is said to be at 1 parsec from Sun. 1 Second of arc or 1 arcsecond is 1/3600 of a degree. A parsec is equal to 3.0857×1013 km or about 19 trillion miles. Astronomical unit is distance between Earth and Sun and equals to about 149,597,870,700 meters. Yet another unit is a lightyear which is the distance light travels in a year and equals to 9.4607×1012 km or about 6 million million miles. A parsec is about 3.26 lightyears in length. Appropriate unit is used in case of small and large distances.
Hipparchus classified stars he could see in night sky according to their visual brightness. He assigned a magnitude of 1 to the brightest stars and magnitude 6 to least bright ones. Ptolemy refined Hipparchus system in 140AD. This magnitude scale is still in use having modernized and improved from time to time. Galileo using his telescope introduced seventh magnitude star. In 1856, Oxford Astronomer Norman Robert Pogson (Mar 23, 1829-Jun 23, 1891) set first magnitude star to be 100 times as bright as sixth magnitude and established the logarithmic magnitude scale. In Pogson’s scheme, difference of one magnitude is same as brightness difference of (100)1/5 which is 2.512- known as Pogson’s ratio. Pogson’s scale assigned Polaris a magnitude of 2. It was used as zero point of the scale. Later, on discovering that Polaris is slightly variable, Vega was used as standard reference with an assigned magnitude of 0, a standard still in use. Stars brighter than Vega are assigned negative magnitudes. Hubble space telescope could see stars up to a magnitude of 31. In late 19th century, after the introduction of photography in stellar photometry, Astronomers found that some stars appearing similar in brightness to eye were differing on photographic plates. It was happening because photographic emulsions used on photographic plates were more sensitive to blue light than red. This led to the creation of photographic magnitudes denoted as mp whereas visual magnitudes are denoted as mv. Henrietta Swan Leavitt analyzed 299 plates from 13 telescopes to construct her logarithmic scale, spanning 17 magnitudes. Difference between star’s photographic and visual magnitudes is called ‘color index’ which is a measure of star’s color. Color index of blue stars came to be of negative value while that of yellow, orange and red stars is increasingly positive. Now days, magnitude is obtained using standard photoelectric photometer through standard color filters, most common of which is UBV. U stands for near ultraviolet filter, B stands for Blue filter and V stands for visual band filter. Color index is obtained by subtracting V magnitude from B magnitude. This way the color index of yellow sun came to be 0.63. UBV system was extended to red and near infrared filters, becoming UBVRI system. System was extended beyond infrared to J, K, L, N, Q bands. An object’s real brightness is known as its bolometric magnitude mbol which is a measure of total radiation the object is emitting. Modern Photometry makes use of CCD sensors. CCD stands for charge coupled devices. CCD Sensors can receive more than 95% of incoming light.
Spectrum is obtained when stellar light is shown through a prism. Prism or a diffraction grating splits light into its constituents. Continuous spectrum is obtained when there is no intervening matter between the light source and prism or diffraction grating. When dust or gas cloud is heated by a proto star or active galactic nucleus, Electrons of their atoms absorb specific amount of energy and emit it in specific quantas. Such re emitted radiation when passes through prism gives of a particular type of spectrum known as emission spectrum, having bright lines corresponding to specific wavelengths depending upon type of atom that emitted the radiation. By comparing such stellar spectrum with spectrum of known elements, composition of stellar dust or gas is known. When radiation of a star is absorbed by electrons in the atoms of surrounding colder dust or gas, it gets re radiated with somewhat reduced intensity and in random direction. When such re radiated light passes through prism it gives of absorption spectrum which is a continuous spectrum except for dark lines at wavelengths where elements in dust or gas would have their bright emission lines. Therefore such spectra can also be used to obtain elemental composition of intervening dust or gas. Helium line was discovered in 1868 in solar spectra independently by Norman Lockyer and Pierre Janssen and was found on Earth in 1895. Modern Spectroscopy makes use of Holographic gratings and gratings made using Lithographic techniques. Reactive Ion Etching is another advanced method in use now for making gratings.
By the year 1913, Edwin Powell Hubble (Nov 20, 1889 – Sep 28, 1953) was sure that he wanted to get into Astronomy. He gave up law practice and went back to his alma mater- the University of Chicago, to get doctorate in Astronomy. While finishing his doctoral work in early 1917, he was invited by George Ellery Hale to work with recently finished 100 inch telescope at Mount Wilson observatory, Pasadena, California. Hubble accepted the commission as an army captain instead. After war, he joined Mount Wilson Observatory in summer of 1919. Hubble was well aware of spectroscopic work done by Vesto Melvin Slipher (Nov 11, 1875 – Nov 8, 1969) at Lowell Observatory in Arizona and his 1912 discovery of shifts in spectral lines toward red band indicating that most of the nebulae are moving away from us. Also, Slipher was first to observe the rotation of spiral Galaxies in 1914. General consensus among Physicists of those days was that the Universe is static. Even Albert Einstein (Mar 14, 1879-Apr 18, 1955) believed that Universe is unmoving which led him to introduce cosmological constant in his field equation of General Relativity to exactly balance out the crunching effect of gravity which he thought would cause the Universe to collapse on itself. General relativity showed how stress-energy causes spacetime to curve. His equations in their original form indicated an expanding or shrinking Universe which Einstein couldn’t believe.
Nebulae, which later became known as Galaxies were the great mystery of those days. They appeared as fuzzy patches in those old telescopes and were subject of great debate between Harlow Shapley and Heber Curtis in 1920. Shapley believed that the Milky Way is 300,000 lightyears wide and is our entire Universe, Sun is not at center and Nebulae are part of Milky Way system. Curtis argued that Milky Way is only 30,000 lightyears across and Nebulae are Island Universes, separate Galaxies beyond Milky Way. Hubble was interested in studying these Nebulae and so he started taking photographic plates of these objects. On the night of Oct 5, 1923, he observed 3 Novae close to Andromeda Nebula- M31. On comparing this plate with earlier plates, he noticed that one of the Novae is actually a variable star. Further observations confirmed that the variable star matches characteristic of a Cepheid variable. Using the period luminosity graph for Cepheids, Hubble was able to calculate distance to the variable star and got a value of about 900,000 lightyears for distance of Andromeda Nebula. The actual value is close to 2.2 million lightyears. Earlier, Harlow Shapley had found a value close to 300,000 lightyears for the diameter of Milky Way, current value being 100,000-120,000 lightyears. This conveniently put M31 well beyond the boundaries of Milky Way and therefore Heber Curtis’s point of view was partially confirmed that M31 and other such Nebulae are separate star systems comparable to Milky Way. Hubble published his discovery first in Nov 23, 1924 issue of New York Times and then in front of American Astronomical Society on Jan1, 1925.
Hubble continued his work on Nebulae, calculating distance to 22 Nebulae. He also calculated their velocities using shifts in their spectral lines, 4 of which were determined by his assistant Milton Lasalle Humason. He further calculated distance to 22 more Nebulae from their radial velocities, assisted by Humason. Observing the photographic plates, Hubble could see that small appearing Nebulae have greater radial velocities than bigger ones. Assuming that all Nebulae are more or less the same size, he concluded that more distant Nebulae are moving at much greater velocities than the nearer ones. Together with 2 earlier estimates by Harlow Shapley, he had data for total 46 Nebulae. He plotted the radial velocities corrected for solar motion against distance calculated using luminosities for 24 Nebulae and also for the 22 Nebulae whose distance could not be calculated individually. He found an almost linear variation in radial velocity with distance. This indicated that with increasing distance the radial velocity of Nebulae also increases by a common factor, a term which later became known as Hubble Constant. Hubble concluded that Nebulae are going away from each other and the Universe is expanding. He communicated these results in his Jan 17, 1929 paper titled ‘A Relation Between Distance and Radial Velocity Among Extra Galactic Nebulae’. The relation can be mathematically expressed as,
v = H0×d
This is known as Hubble’s law, where,
v = recessional velocity of Galaxy
d = Distance to the Galaxy
H0 = Hubble’s constant
Hubble got a value of about 500 km/s/Mpc for his proportionality constant, current value being 67.8±0.77 km/s/Mpc which means that expansion of Universe increases by about 67.8 km/s for every 3.26 million lightyears in any direction.
In 1917, Albert Einstein introduced cosmological constant to curvature side of his field equation of Gravitation to make the equations predict a static Universe as a dynamic Universe was considered absurd at that time. Russian Physicist Alexander Friedmann (Jun 16, 1888-Sep 16, 1925), worked on Einstein’s field equations without presumptions and showed that Universe might be expanding at a rate which can be calculated using the equations. Friedmann’s equation can be obtained by putting metric for homogeneous and isotropic Universe in Einstein field equations. He presented his equations in 1922. Georges Lemaître (Jul 17, 1894-Jun 20, 1966), a Belgian priest and astronomer, found something similar from Einstein’s field equations in 1927. Lemaître made the first empirical determination of Hubble constant H0 using these equations. He also suggested that if Universe is expanding then it must have been unimaginably small back in time, a state he called ‘cosmic egg’. He met Einstein and showed his results to him. Einstein held on to the popular belief of a static Universe and disapproved Lemaître’s result. After Hubble published his findings in 1929, Einstein visited him. He saw the data and was convinced that the Universe is expanding. He then dropped cosmological constant, restoring the field equations to their original form.
After Hubble, almost everyone accepted an expanding Universe. Astronomers and Physicists believed that the expansion must be slowing down due to gravitational pull of matter-energy density. Rate of slowdown was named deceleration parameter with symbol q0. Allan Rex Sandage (Jun 18, 1926 – Nov 13, 2010), a student of Walter Baade and Hubble, published a paper in 1961 titled ‘The Ability of the 200 inch Telescope to Discriminate Between Selected World Models’. In this paper he advocated that cosmology is search for two parameters- Hubble constant H0 and deceleration parameter q0. Earlier in 1958 he had measured Hubble constant to be 75 km/s/Mpc, which is its first good estimate.
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References:
1)https://www.famousscientists.org/henrietta-swan-leavitt/
2)https://apod.nasa.gov/debate/debate20.html
3)http://www.amnh.org/explore/resource-collections/cosmic-horizons/profile-georges-lemaitre-father-of-the-big-bang/
4)http://skyserver.sdss.org/dr1/en/astro/universe/universe.asp
5)http://w.astro.berkeley.edu/~mwhite/darkmatter/hubble.html
Image credits goes to respective sources.
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