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------------------------------------ 2340 - How to Weigh a Galaxy
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- Weight is how much force gravity pulls on a given mass. When we say weigh a galaxy we mean we want to determine how much mass is their creating how much gravity. Let’s measure the mass of the Milky Way Galaxy. We can not measure its mass directly. We can measure its size directly:
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- The Milky Way Galaxy is 2,000 parsecs thick and 40,000 parsecs across. From Earth the galaxy center is 8,500 parsecs away. Parsecs is a distance measurement used by astronomers. A parsec is a very long distance, it is longer than a lightyear. A lightyear is the distance a beam of light will travel in one year traveling at 670,000,000 miles per hour.
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- That distance is 5,880,000 million miles and the parsec equates to a distance of 3.26 lightyears. A parsec is the distance one astronomical unit looks like when it is so far away it subtends an angle of one arcsecond.
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- An arcsecond is a very small angle. A full circle has 360 degrees. Each degree is divided into 60 arcminutes. There are 21,600 arcminutes in a circle. Each arcminute is divided into 60 arcseconds. There are 1,296,000 arcseconds in a circle.
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- An astronomical unit is the distance between the Earth and the Sun. The parsec is defined as the distance one astronomical unit would appear if it were the base of a triangle and the vertex was an angle of one arcsecond.
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-------------- A parsec = 3.1 * 10^16 meters = 3.26 lightyears
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--------------- A lightyear = .95 * 10^16 meters = 1 lightyear
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-------------- An AU = 1.5 * 10^11 meters. There are 633,757 AU’s in a lightyear.
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Therefore, the Milky Way Galaxy is 6,400 light years thick and 128,000 light years in diameter, with the center 27,200 light years from Earth. It takes 128,000 years for light to travel across our galaxy.
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- That is how big the Milky Way Galaxy is. We need to us Kepler’s laws of orbiting bodies to calculate the mass of the Milky Way:
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- The Sun is traveling 220,000 meters / second (492,130 miles / hour) in a near circular orbit around the center of the Milky Way with a radius of 8,500 parsecs. It takes 230 million years for the Sun to complete one orbit, that is, the last time we were at this point in our orbit 96% all species on Earth died out and the Dinosaurs began their rule of Pangaea. Pangaea was the super-continent in the Jurassic period 230 million years ago.
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- Johan Kepler in 1619 discovered the laws that will allow us to calculate the mass from the data we can measure. He determined that the square of the period of revolution of a planet is proportional to the cube of its distance from the Sun. He was working with planets but the same laws apply stars, and orbiting mass in general.
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-------------- Period ^2 in years = distance of semi-major axis in astronomical units ^3
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- Period is the time it takes for one full orbit. Kepler's Third Law was originally applied to an ellipse, thus a semi-major axis rather than a radius was used. It turns out the Sun’s orbit is a very flattened ellipse but it definitely was not a perfect circle.
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- Using Newton's laws, Kepler's Third Law can be converted to:
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--------------------- Period^2 = 4 * pi^2 * axis^3 / G ( m+M )
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- M + m are the masses of the two bodies. The Milky Way mass is so much larger than the Sun’s mass we will assume (M+m) = Mass. And, we will assume that the orbit is a perfect circle of Radius. We can calculate the rotational velocity of the orbiting body:
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-------------------- Since velocity = distance / time = circumference / period:
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-------------------- distance^2 / velocity^2 = 4 * pi^2 * Radius^3 / G * Mass
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------------------- The distance is the circumference of the circle = 2 * pi * radius
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-------------- 4 * pi^2 * Radius^2 / velocity^2 = 4 * pi^2 * Radius^3 / G * Mass
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-------------- G is the gravitational constant = 6.67 * 10^-11 m^3 / ( kg/sec^2)
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- “G” is a natural constant for the force of gravity and its value depends on the units used in the measurement.
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----------------------- 1 / velocity^2 = Radius / G * Mass
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- Solving for the mass:
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--------------------- Mass = velocity^2 * Radius / G
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----- Mass = (220,000 m/s)^2 * 8,500parsecs / 6.67 * 10^-11 m^3 / ( kg/sec^2)
---------------------- Mass = 1.9 * 10^41 kg
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- The Mass of the Milky Way inside the Sun's orbit = 1.9 * 10^41 kilograms
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- The mass of the Sun is 1.99 * 10^30 kilograms
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- So, the mass of the Milky Way inside the Sun's orbit is about 10 ^ 11 or 100 billion solar masses.
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- When Johan Kepler was doing this math in 1609, he was using data on the planets orbiting the Sun. Kepler was a German astronomer, born in 1571, the son of a professional soldier. He had smallpox as a child that crippled his hands and eyes so his parents put him into a religious school because they thought the work would be less strenuous for him as a minister.
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- He got his masters degree in 1591, abandoned ministry and taught mathematics in the University of Graz, Austria. Because of the Thirty Years' War Graz was too dangerous and he moved to Prague were he worked with astronomer, Tycho Brahe. Tycho died in 1601 and left Johan all his data on the orbiting planets.
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- With his religious background, and influence of Plato and the Greeks, Johan tried for years to fit the data on orbiting planets to circles and spheres. The circle was recognized as the perfect geometry and ,of course, the heavens would not be ruled by anything less than perfect.
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- Johan could not make the math work. He nearly gave up before he tried making Mar's orbital data fit an ellipse. It matched up with a high degree of accuracy. Valhalla !
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- Once Johan was working with ellipses he discovered these three laws: Planets orbit in ellipses, with the Sun at one focus; the line connecting the planet and the Sun will sweep over equal areas in equal times as the planet moves about its orbit ( also stated as the product of distance from the focus and transverse velocity is a constant); and, the square of the period of the revolution of a planet is proportional to the cube of its distance from the Sun.
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----------------------------- Radius ------- Period^2 -------Radius^3 ----- Velocity
----------------------------- AU ------- years^2 ----------- AU^3 ----- Km/sec
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------- Mercury -------- .39 ----------- .06 ---------------.06 --------- 47.9
------- Venus ---------- .72 ----------- .37 ---------------.37 --------- 35.0
------- Earth ------------ 1.00 -----------1.00 -------------1.00 -------- 29.8
------- Mars ----------- 1.52 ----------- 3.53 ------------3.51 --------- 24.1
------- Jupiter ----------- 5.20 -------- 140.7 ----------140.6 ----------- 13.1
------- Saturn ---------- 9.54 --------- 867.9 ----------868.3 ----------- 9.6
------- Uranus --------- 19.19 -------- 7,058 ----------7,067 ------------ 6.8
------- Neptune -------- 30.06 ------- 27,160 ----------27,160 ------------5.4
------- Pluto ------------ 39.53 ------ 61,770 ----------61,770 ------------- 4.7
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- Note in the above table that for each of the 9 planets (Pluto included) the square of the period of the orbit is equal to the cube of the radius from the Sun. Also note that as you go further out from the Sun the rotational velocity of the planet decreases. Let’s see what happens when we do similar calculations for the Milky Way?
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- If we go to the very edge of the Milky Way, at a distance of 16,000 parsecs , the orbiting speed of the stars is 230,000 meters / second.
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- Again substituting in to Kepler's formula:
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--------------------- Mass = velocity^2 * Radius / G
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-- Mass = * (230,000 m/s)^2 * 16,000 parsecs / 6.67 * 10^-11 m^3 / (kg/sec^2)
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- The mass of the entire Milky Way is = 3.9 * 10 ^ 41 Kg. Or, 2* 10^11 solar masses.
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- Conclusion, about half the mass of the Milky is inside the Sun's orbit and half is outside. This fact has proven to be a real problem for astronomers. The inside orbit is much, much denser with visible light than the outside orbits. The mass of the outside orbits is larger than we can visible see.
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- This fact has caused astronomers to invent another object in the Universe, Dark Matter. This invention assumes that each galaxy is surrounded by a giant halo of invisible mass that accounts for the constant rotational velocity. (See #718“ Dark Matter, What Could it Be“?)
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- Johan and Galileo were pen pals although they never met and Galileo sent Johan one of his first constructed telescopes. Johan used it to study the moons of Jupiter. Johan must have had some time away from the telescope because he had thirteen children in two marriages. He died in 1631 due to a fever and an over enthusiastic doctor that was practicing medical bleeding as a remedy.
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- He taught us how to weigh a Galaxy. His laws are used universally in Astronomy today.
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- (1) See " Time Comes to Us in Particles" to learn more about G
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- (2) See Review 33 to learn more about Kepler’s laws of physics.
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- (3) See Review 718 to learn more about Dark Matter
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- (4) See Review 2283 “Galaxies that make up the Universe” that also lists 14 more Reviews about galaxies.
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- April 24, 2019 13
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-------------------------- Wednesday, April 24, 2019 --------------------------
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