Relativistic Orbital Velocity. Copyright 2009 Joseph A. Rybczyk

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1 Relativistic Orbital Velocity Copyright 2009 Joseph A. Rybczyk Abstract The relativistic orbital velocity formula for circular orbits is derived through direct application of the Newtonian orbital velocity to escape velocity relationship using the corresponding relativistic forms of orbital velocity and escape velocity. 1

2 Relativistic Orbital Velocity Copyright 2009 Joseph A. Rybczyk 1. Introduction The fundamental relationship between the Newtonian orbital velocity formula for circular orbits and the corresponding Newtonian escape velocity formula is applied to the previously derived relativistic formula for escape velocity 1 to derive the relativistic form of the orbital velocity formula for circular orbits. 2. Newtonian Orbital Velocity for Circular Orbits In accordance with the principles of Newtonian physics the acceleration of an object in circular uniform motion about the center of gravity of a primary body is given by 1 where a is the rate of acceleration directed toward the center of gravity of the primary body, v is the constant rate of speed along the circumference of the circle, and R is the radius of the circle or radial distance to the center of gravity of the primary body. This form of acceleration is referred to as centripetal acceleration to distinguish it from the straight line acceleration associated with an object in free fall toward a primary gravitating body where there is an increase in speed with no change in direction of the accelerating object. The force associated with the centripetal acceleration given by equation (1) is defined by 2 where F is the centripetal force, and m is the mass of the accelerating object. Since Newton s law of universal gravitation gives the gravitational force above the surface of the primary body as 3 where F is the gravitational force, G is the universal gravitational constant, and M is the mass of the primary body, we can by way of substitution with equations (1) and (2) arrive at 4 2

3 or simply 5 that when solved for v gives 6 where the subscript o is added to show that v o is the orbital velocity for a circular orbit. Given the gravitational potential Φ as defined by 7 equation (6) can be rewritten as 8 where, for purposes of simplicity, v o is the orbital velocity for circular orbits expressed in terms of gravitational potential Φ. 3. Newtonian Escape Velocity To obtain the Newtonian escape velocity formula we simply equate the Newtonian formula for gravitational potential energy Φ e 9 to the Newtonian formula for kinetic energy K in the manner

4 that gives the minimum energy required for mass m to escape the gravity of a primary body of mass M. Solving this equation for v then gives 2 12 or by use of equation (7) 2 13 where in each case the subscript e is added to show that v e is the escape velocity. 4. Newtonian Orbital Velocity to Escape Velocity Relationship By re-expressing equation (13) for Newtonian escape velocities as 2 14 and by way of substitution with the left side of equation (8) into the right side of equation (14) we obtain 2 15 or conversely 2 16 giving the relationship of the Newtonian orbital velocity v o to the Newtonian escape velocity v e. By squaring both sides of equations (15) and (16) the relationship between the Newtonian orbital velocity and escape velocity can alternately be expressed as 2 17 and

5 respectively for improved understanding of the relationship between the two kinds of velocities. 5. Relativistic Orbital Velocity to Escape Velocity Relationship Given the orbital to escape velocity relationships of equations (15) through (18) in combination with the earlier mentioned relativistic escape velocity formula derived in previous works 1 by this author a solution toward the derivation of the relativistic version of the orbital velocity formula for circular orbits is strongly suggested. That is, where 2 19 is the relativistic escape velocity formula, it would appear that the relativistic form of the orbital velocity formula for circular orbits is given by the relationship which is simply a modified form of equation (16) using the right side of equation (19) in place of v e that previously represented the Newtonian escape velocity and not the relativistic escape velocity. Simplification of equation (20) then gives and therefore 2 22 or 2 23 where v o now represents the relativistic orbital velocity for circular orbits. 5

6 6. Validating the New Relativistic Orbital Velocity Formula Validating the new relativistic orbital velocity formula through an independent derivation process is a far more complicated procedure than that just conducted and thus far has led to a similar though not entirely identical result. Using equation (22) from the author s previous work involving kinetic energy 2 as a guide, we modify equation (1) of the present work to derive 24 for the relativistic form of the centripetal acceleration formula. The factor 25 operating on variable v in equation (24) is simply another form of the Lorentz transformation factor 3 often referred to as gamma and is the form used throughout the author s own work to distinguish it from Einstein s theory. Simplification of equation (24) gives 26 where a is the relativistic rate of centripetal acceleration. Using this same procedure to modify equation (2) to its relativistic form gives 27 that simplifies to 28 where F is the relativistic form of centripetal force acting on mass m of the accelerating object. If we now equate the relativistic forms of centripetal acceleration and force given in equations (26) and (28) to the universal gravitation equation (3) as done previously with equations (1) and (2) to obtain equation (4) given earlier, we obtain 29 6

7 for the relativistic relationship of gravitational acceleration. From this we obtain 30 that simplifies to 31 and upon substitution with the left side of equation (7) gives 32 for the relativistic relationship of velocity v to gravitational potential Φ. Solving for v then gives ending with 7

8 39 as the final form of the alternative relativistic orbital velocity formula for circular orbits. Similar to the previous derivations, the subscripts oa were added to the variable v to show it is the alternate orbital velocity and not the original orbital velocity nor the escape velocity. 7. Comparison of the Two Orbital Velocity Formulas Extensive analysis conducted by the author strongly supports the original derivation for orbital velocity formula over the alternative derivation just completed. The primary reasoning behind that conclusion involves the fact that the original formula for relativistic orbital velocity as given in equations (22) and (23) maintains its expected relationship to the relativistic escape velocity formula (19) throughout the latter formula s entire range of applicability. This conclusion appears to be valid in spite of the fact that the predicted behavior for both velocities quite unexpectedly reverses itself as the gravitational potential continues to increase. More specifically, both velocities increase asymptotically toward peak values of for orbital velocity, and 41 for escape velocity as the gravitational potential increases toward the value of c 2, then, as the gravitational potential continues to increase toward the value of 2c 2, both velocities decrease toward 0. Any further increase in gravitational potential results in an imaginary result for both velocities. Although the alternative orbital velocity also rises asymptotically, similar to but not identical to the accepted orbital velocity, and intersects the same peak value point as given by equation (40), it continues to increase asymptotically with the continued increase in gravitational potential. This behavior is not supported by any of the preceding research and analyses that led to the original discovery of the relativistic escape velocity formula and the resulting accepted orbital velocity formula (22) and (23) introduced in this present work. And since all of the preceding research and analyses are based on the accepted principles of relativistic theory, it would seem that the alternative formula (39) cannot be entirely correct. To facilitate understanding of the relationships just discussed, they are given in chart form in Figure 1 that follows: 8

9 1.2 Relativistic Orbital Velocity vs Escape Velocity 1 Speed in Terms of c Ve Vo Voa At point 10, Φ = c^2, at point 20, Φ = 2c^2 FIGURE 1 Relativistic Orbital Velocity vs. Escape Velocity 8. Conclusion The findings presented in this work are based upon extensive research and analyses involving every reasonable approach the author could conceive to derive the relativistic form of the orbital velocity formula for circular orbits through the direct application of appropriate relativistic principles. After exhaustive analyses and comparison of the many different results, the alternative formula (39) for the relativistic orbital velocity given in this work was the only reasonable alternative to the accepted formula given as equations (22) and (23). This is not to say that some future breakthrough might not give additional insights needed to verify the accepted version of the relativistic orbital velocity formula for circular orbits through an independent derivation process that produces the exact same formula. But in the author s opinion there is nothing is to be gain in the meantime by not advancing forward with the results already obtained. The future will speak for itself. REFERENCES 1 Joseph A. Rybczyk, Relativistic Escape Velocity, (2009); Relativistic Escape Velocity using Relativistic Forms of Potential and Kinetic Energy, (2009), 9

10 2 Joseph A. Rybczyk, Time and Energy, Inertia and Gravity, (2001), equation, The mathematical factor devised by Dutch physicist H. A. Lorentz in 1895 to explain the null results of the Michelson and Morley interferometer experiments of The use of the factor was expanded upon by Albert Einstein to include time dilation and other effects in his original work, On the Electrodynamics of Moving Bodies, (1905), now referred to as the Special Theory of Relativity. This same factor took on a different form in its use in the Millennium Theory of Relativity, (2001) by the present author. Relativistic Orbital Velocity Copyright 2009 Joseph A. Rybczyk All rights reserved including the right of reproduction in whole or in part in any form without permission. Note: If this document was accessed directly during a search, you can visit the Millennium Relativity web site by clicking on the Home link below: Home 10