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Orbiting Atomic Clock Rate Equations GPS Satellite Clocks Clock Rates Adjusted for Inertial Motion and Gravitational Motion By James Carter In the 1980s, it was discovered that atomic clocks run at different rates in different orbits. Clocks run slowest in low space shuttle orbits and then run faster and faster in higher and higher orbits until, at an orbit of about 1.5 times Earth’s radii, they surpass the rates of sea level clocks. Physicists were able to use the equations of Einstein’s special and general theories relativity to calculate the correct rates for orbiting clocks but these equations were based on unmeasured metaphysical assumptions of relative motion. It is shown here that the same accurate calcula- tions can be made using physical principles of measurements of absolute motion. It is obvious and simple physics that clocks must change the length of their measured inter- vals as the mass of their component parts change. A clock’s time interval is based on its conser- vation of angular momentum. Consider the 24 hour clock of the rotating Earth. If Earth could be accelerated to a higher velocity, its mass would increase and it would slow its rotation in order to conserve its angular momentum. This is true for any rotational clock regardless of the design of its mechanism. The beauty of satellite clock measurements is that they combine the satellite clock’s inertial orbital velocity and its gravitational escape surface velocity to form the single vector of absolute motion for the Lorentz transformation of the clock’s mass and time. Thus, there is no need for metaphysical assumptions and undetectable parameters such as the potentials of a gravitational field or the invention of a four-dimensional spacetime continuum. Changes in satellite clock rates are the result of the Lorentz transformations in mass. As a clock’s mass and momentum is increased, it rate slows due to the conservation of angular momentum. Orbiting Atomic Clock Rate Equations 2 2 Orbital time dilation results not V tdV = esV + oV es V from a combination of td Time dilation velocity (tdV) of an orbit is gravitational potential and orbital equal to the square root of the sum of the motion. Rather, it is caused by escape velocity squared (esV2) and the the combined velocity vector orbital velocity squared (oV2). (tdV) of two velocities at right T angles to one another. The T = 0 Lorentz mass transformation at k es 2 + o 2 the combined vector of orbital = Velocity Escape 1- V V Graviational Space 2 velocity (oV) and escape velocity Time DilationIntrinsic Velocity Motion =Through C es ( V) causes the time dilation of A clock’s kinetic time interval (Tk) is equal 11 12 1 Orbital Velocity = oV orbiting clocks, 10 2 9 3 to its rest time interval (T0) divided by the 8 4 7 6 5 square root of one minus the escape velocity squared (esV2) plus the orbital velocity Orbit squared (oV2) divided by the speed of light squared (C2). © 2017 by James Carter 1 Orbiting Clocks Geo-synchronous Comsat Orbit = 6.615 Radii td Escape Velocity = 4,351 m/s V = 5329 m/s Orbital Velocity = 3,076 m/s Time Dilation Velocity = 5829 m/s Clock interval = 1.000000000158 oV = 3076 m/s 11 12 1 10 2 V = 4351 m/s 9 3 8 4 es 7 6 5 11 12 1 10 2 9 3 8 4 Comsat 7Orbit6 5 Anti-Comsat3,0676 m/s Orbit 3,076 m/s V = 6,700 m/s GPS Orbit = 4.175 R = 5,471 m/s V td es 1.000000001045 Escape Velocity =5,471 m/s 11 12 1 10 2 oV = 3,868 m/s 9 3 1.0000000001665 8 4 Orbital Velocity = 3,868 m/s 7 6 5 1.0000000000832 1.0000000006976 Sea Level Interval Time Dilation Velocity = 6700 m/s 1.0000000002497 GPS Interval GPS clock interval .0000000004479 GPS Clock Adjustment 1.0000000002497 GPS clocks must be slowed down by 4.479 parts in 10 billion Clock synchronous Orbit 1.498 R to be synchronous with Escape Velocity = 9,143 m/s td V = clocks at sea level. Orbital Velocity = 6,465 m/s Time dilation Velocity = 11,198 m/s Clock interval at this orbit is the same 11,198 as at sea level. 1.000000006976 oV = 6465 m/s m/s V = 9,143 m/s 11 12 1 10 2 es 9 Space Shuttle Orbit = 1.046 R 3 Borneo 8 4 7 6 5 Escape Velocity = 10,940 m/s Sea Level 11 12 1 448 m/s 10 2 Orbital Velocity = 7,736 m/s 9 3 Escape Velocity = 11,189 m/s 8 4 7 5 Time dilation Velocity =13,399 m/s Equtorial Velocity =448 m/s 6 Clock interval at space shuttle Time dilation Velocity =11,198 m/s orbit = 1.0000000009988 Clock interval at sea level 1.0000000006976 11 12 1 10 2 esV = 10,940 m/s 9 3 8 4 esV = 11,189 m/s 7 6 5 o V = 7,736 m/s V o td V = 448 m/s V V = 13,399 m/s 1.0000000009988 tdV = 11,198 m/s 1.0000000006976 11 12 1 10 2 9 3 8 4 7 6 5 Ecuador 448 m/s Clock interval in deep space and at rest relative to earth = 1.000000000000000000 GPS Time Dilation Equations Newton Absolute Motion Special Relativity General Relativity Absolute Motion Escape Velocity Time Dilation Time Dilation Time Dilation Gravitational Time Dilation T0 T0 T0 Tk = Tk 2 Tk = = T0 V 2 Tk = es 2GM 1 V 2GM V = - 2 1- 1- es 2 + o 2 R C 2 RC 2 1- V V C C2 Escape velocity (esV) is equal to the square root of two times the gravitational constant (G) times the mass of the earth (M) divided by the distance to the earth’s center (R). The duration of an interval of Kinetic Time (Tk) measured by a clock experiencing gravitational acceleration is equal to the duration of an interval of Inertial Time (T0) measured by a clock at rest in deep space divided by the square root of one (√1) minus the escape velocity squared (esV2) plus the orbital velocity squared (oV2) divided by the speed of light squared (C2). © 2017 by James Carter 2 Orbital & Escape Orbital velocity Orbital Acceleration Mass-Length Velocity meters/sec Energy of Gravity sea level = 1 Escape Velocity R Angular meters/sec 2,192.4 13 1/13 g=.0574 momemtum 1/13 1/169 ML=1/13 3,100.5 2,281.9 12 R 1/12 1/12 g=.0674 IW=13 1/144 ML=1/12 3,227.1 2,383.4 R 11 1/11 g=.0802 1/11 1/121 IW=12 ML=1/11 3,370.6 2,499.7 R 10 g=.0970 1/10 1/10 1/100 IW=11 3,535.1 2,634.9 9 R ML=1/10 1/9 1/9 g=.1198 1/81 IW=10 3,762.3 R 2,794.7 8 ML=1/9 1/8 1/8 g=.1516 1/64 3,952.4 7 R 2,987.7 1/7 1/7 g=.1980 IW= 9 ML=1/8 3,073.3m/s 6.6154 R 1/49 4,225.3 .2241 m/s 3,227.1 6 R 1/6 1/6 2 4,346.4 m/s ML=1/7 IW=8 Comsat orbit 3,535.1 g=.2696 4563.8 5 R 1/5 1/5 1/36 3,952.4 4,999.4 4 R IW=7 1/4 1/4 g=.3882 1/25 ML=1/6 4,563.8 IW=6.6 5,589.5 3 R 1/3 1/3 g=.6065 1/16 ML=1/5 5,589.5 2 R 6,454.2 IW=6 1/2 1/2 g=1.078 R 7,904.7 1 1/9 ML=1/4 E=1 g=2.426 IW=5 7,904.7 1 1/4 ML=1/3 g=9.807 IW=4 Sea level g = 1 Escape Velocity= 2gR ML=1/2 esV@ R=11,179 m/s=1 ML=1 IW=1 IW=3 IW=2 Earth Orbits to 13 Radii © 2017 by James Carter 3.
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