- Gravity Probe B was an experiment launched in to orbit in 2004. The satellite carried a series of four precision gyroscopes designed to prove Einstein’s theory that space and time were warped around immense gravity. In this case the gravity was the Earth. If the gyroscopes pointed in the same direction over time then Newton’s equations held. If the gyroscopes precessed at a specific angle as they orbited the Earth then Einstein’s equations held.
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- The precession that is causing the axes of rotation to point in a circle is called “ frame dragging” The frame is space-time and Earth’s gravity is the drag. Einstein’s Theory of Relativity published in 1915 states that gravity curves space-time. “ Matter tells space-time how to curve and curved space-time tells matter how to move. According to Einstein’s equation the frame dragging rate orbiting Earth would be 0.0392 arcs seconds per year. This is a very small angle and it takes very precise measurements to confirm it is correct. This tiny angle is equivalent to the angular width of Pluto as seen from Earth.
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- The equation for the pointing angle of the axis that changes in degrees / second is:
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-------------------------- angle = G * L * 360 // 4 * pi * c^2 * a^3 * ( 1-e^2) ^3/2
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------------------------where:
------------------------ angle = degrees / second
------------------------ c is the speed of light = 3 * 10^8 meters / second
------------------------ c^2 C m^2 / sec^2
------------------------ L = angular momentum of the spinning Earth
----------------------- G = Gravitational Constant = 6.67*10^-11 m^3 / ( kg*sec^2)
------------------------ a = semi-major axis of orbit = 7.02*10^6 meters
------------------------ a^3 = 3.46 *10^20 m^3
------------------------ e = eccentricity of orbit, for a perfect circle = 0
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- The satellite carrying the gyroscopes was orbiting at 7020 kilometers distance for the Earth’s center. It was a circular orbit making eccentricity equal to zero, ( e = 0 ).
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--------------------------- angle = (6.67*10^-11 ) * L * 360 // 4 * pi * ( 6.67*10^-11) *( 3.46 *10^20 )* ( 1) ^3/2
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-------------------------- angle = 0.614 * 10^-46 * ( L )
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- To solve this equation we need to know the angular momentum of the spinning Earth. The angular momentum of a mass in orbit is the product of mass * velocity * radius. The largest mass traveling the fastest with the furthest orbit has the greatest momentum. This formula is for the angular momentum of the Earth orbiting the Sun.
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----------------------- L = m * v * r
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- The average radius of the Earth’s orbit is: r = 1.496*10^11 meters ( 93 million miles)
- The mass of the Earth is: m = 5.9742*10^24 kilograms.
- The velocity of orbit is: v = 29,800 metes per second, 66,663 miles per hour
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- L = 2.66*10^40 kg*m^2/sec the angular momentum of the Earth circling the Sun. However, we need the angular momentum of a spinning Earth. The angular momentum of a spinning body is the rotational inertia * angular velocity.
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--------------------- L = I * v
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- The rotational inertia changes for different configurations of the mass that is spinning. A perfectly uniform, solid sphere: I = 2/5 * m * r^2. A hollow shell sphere: I = 2/3 * m* r^2. We will assume Earth is a solid, homogeneous sphere. Jupiter being a gas planet would have a different rotational inertia.
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------------------- L = 2/5 * v * m * r
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--------------------- mass of the Earth = m = 5.9742*10^24 kilograms
-------------------- radius of the Earth = r = 6.378*10^6 ,meters.
-------------------- velocity of rotation = v = 2*pi*r / period
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------------------------ v = 2* pi *( 6.378*10^6 ) / 24 hours * 3600 seconds / hour
------------------------ v = 464 meters / second = 1,038 miles per hour.
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------------------------ L = 2/5 * ( 4.64^10^2 ) * ( 5.9742*10^24 ) * ( 6.378*10^6 )
------------------------ L = 7.022*10^33 kg * m^2 / sec
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----------------------- angle = 0.614 * 10^-46 * ( L )
----------------------- angle = 0.614 * 10^-46 * ( 7.022*10^33 )
---------------------- angle = 4.34 * 10^-13 degrees per second
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- angle / year = 4.34 * 10^-13 deg / sec * 3600 arc sec / deg * 3.1*10^7 sec / year
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-------------------- angle / year = 0.048 arc seconds / year.
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- The Gravity Probe B took data for 50 weeks and got a frame-dragging rate of 0.037 + or - 0.0072 arc seconds per year, between 0.0302 and 0.0442 arc seconds per year.
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- The refined calculation for the angular momentum of the spinning Earth is 5.861*10^33 kg*m^2/sec. This changes the precession angle to 0.040 arc seconds / year. The results of the 50 week experiment should be 0.038 arc seconds, or well within the margin of error for the experiment, an agreement to within 1%.
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--------------------------- angle = 0.614 * 10^-46 * ( L )
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---------------------------- angle = 0.614 * 10^-46 * ( 5.861*10^33 )
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--------------- angle / 50 weeks = 0.038 arc seconds
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- Therefore Einstein is right. There is no attracting force reaching out through space time tugging on the planets. The planetary trajectories around the Sun is very nearly straight in space-time, but not in space where it is circular. Earth’s orbit is a stretched-out spiral whose width is space is one Astronomical Unit , but , whose length in time is measured in lightyears. Nearly all our weight experienced on the surface of the Earth where speeds are small and the gravitational force is weak arises die to the warping of time , rather than space. In practice this means the acceleration of gravity makes clocks run slower on the surface than those in outer space
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- Gravity feels strongest where space-time is most curved and it vanishes where space-time is flat. “ Matter tell space-time how to curve, and curved space-time tells matter how to move.”.
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- Place a massive gravity bowling ball on to a stretched rubber sheet of space-time. Place a marble onto the rubber sheet and it rolls toward the bowling ball, or put the marble in orbit around the bowling ball. The marble is traveling along a surface that has been deformed by the presence of a larger mass.. Gravity is a feature of space-time itself . Your weight on Earth is due to the fact that our body is traveling through warped space-time.
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-ORBITAL ANGULAR MOMENTUM
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----------orbital radius ------ orbital period ------ mass ------------ L
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----------- ( 10^6 km ) --------( days ) -------- 10^22 *kg ) ----- ( kg*m^2/sec )
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Mercury -----58 ------------ 87.97------------------ 33------------- 9.1e38
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Venus -------108 ------------ 224.70---------------- 487------------ 1.8e40
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Earth -------150 ------------ 365.26 ----------------- 597 ------------ 2.7e40
Mars -------228 ------------ 686.98 ------------------- 64.2 ------------ 3.5e39
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Jupiter -----778------------ 4,332.71 -------------- 190,000------------ 1.9e43
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Saturn -----1429---------- 10,759.5--------------- 56,800------------ 7.8e42
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Uranus----- 2871----------- 30,685------------------ 8,680------------ 1.7e42
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Neptune ----4504 ---------- 60,190-------------- 10,200 ------------ 2.5e42
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Pluto -------5914------------ 90800 --------------- 1.27-------------- 3.6e38
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Moon -----0.384 ------------ 27.32 --------------- 7.35-------------- 2.9e34
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ROTATIONAL ANGULAR MOMENTUM:
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-------------- radius ---------- period --------- mass ----------------L
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----------------( km ) ---------- (days ) ---- (10*24 * kg ) -------- kg*m^2 / sec
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Sun--------- 69,5000 ----------- 24.6 ----------- 1,990,000----------- 1.1e42
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Earth -------- 6,378----------- 0.99 -------------- 5.97 -------------- 7.1e33
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Jupiter----- 71,492----------- 0.41----------- 1,900---------------- 6.9e38
--------------------------------------------
(1) See http: //einstein.stanford.edu/
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(2) Note that Earth’s angular momentum circling the Sun is 10 million tims greater than the angular momentum of the Earth spinning on its axis.
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RSVP, please reply with a number to rate this review: #1- learned something new. #2 - Didn’t read it. #3- very interesting. #4- Send another review #___ from the index. #5- Keep em coming. #6- I forwarded copies to some friends. #7- Don‘t send me these anymore! #8- I am forwarding you some questions? Index is available with email and with requested reviews at http://jdetrick.blogspot.com/ Please send feedback, corrections, or recommended improvements to: jamesdetrick@comcast.net. ---- “Jim Detrick” -- www.facebook.com, -- www.twitter.com, -- 707-536-3272 Sunday, December 23, 2012
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