On Jun 7, 6:01 pm, Albertito wrote:
On Jun 7, 1:38 pm, Albertito wrote:



Under this novel approach, we can calculate, for instance,
a escape velocity:

Kinetic energy of an orbiting test particle is

E_k = mc^2 (cosh(v/c) - 1)

And potential energy is

E_p = cosh(v/c)GMm/r

Now, we equate E_k = E_p, and it yields

mc^2 (cosh(v/c) - 1) = cosh(v/c) GMm/r,
c^2 (cosh(v/c) - 1) = cosh(v/c) GM/r,
cosh(v/c) = 1 / ( 1 - GM/rc^2),
v = c arccosh(1 / (1 - GM/rc^2))

A first-order approximation is v_e = sqrt(2GM/r),
which is the well-known Newtonian escape velocity.

Let's compute, for example, the complete escape
velocity (all higher order considered), for the mean
position of the Earth in solar system:

GM = 132,712,440,018 Km^3/s^2, is Sun's gravitational parameter,
R_e = 149597870.691 Km, is mean Earth-Sun distance, and
c = 299792.458 km/s, is the speed of light in vacuum,

Then

v = c arccosh(1/(1 - GM/R_ec^2)),
v = 42.1219 Km/s

Vis viva equation (orbital velocity):
Under this novel approach, let m be the mass of the
test body. Total energy of the test body in its orbit is the
sum of its kinetic and potential energies,

E = mc^2 (cosh(v/c) - 1) - cosh(v/c) GMm/r

For orbits that are circular or elliptical, the total energy is
also given by

E = - cosh(v/c) GMm/2a
where
a is semi-major orbital axis

Now, equating

- cosh(v/c) GMm/2a = mc^2 (cosh(v/c) - 1) - cosh(v/c) GMm/r,

and solving for v, it yields

v = c arccosh(1 - GM/rc^2 + GM/2ac^2).

For a circular orbit, it is a = r, so

v = c arccosh(1 - GM/2rc^2 ),
is the velocity of orbiting body

Let's compute, for example, the orbital velocity
(all higher orders considered), for the mean circular
orbit of the Earth in solar system:

GM = 132712440018 Km^3/s^2, is Sun's gravitational parameter,
R_e = 149597870.691 Km, is mean Earth-Sun distance, and
c = 299792.458 km/s, is the speed of light in vacuum,

v = c arccosh(1 - GM/2R_ec^2 ),
v = 29.7847 Km/s

Sorry, there is a typo, I meant equating

- cosh(v/c) GMm/2a = mc^2 (cosh(v/c) - 1) - cosh(v/c) GMm/r,

and solving for v, it yields

v = c arccosh(1/(1 - GM/rc^2 + GM/2ac^2)).

For a circular orbit, it is a = r, so

v = c arccosh(1/(1 - GM/2rc^2)),
is the velocity of orbiting body

Let's compute, for example, the orbital velocity
(all higher orders considered), for the mean circular
orbit of the Earth in solar system:

GM = 132712440018 Km^3/s^2, is Sun's gravitational parameter,
R_e = 149597870.691 Km, is mean Earth-Sun distance, and
c = 299792.458 km/s, is the speed of light in vacuum,

v = c arccosh(1/(1 - GM/2E_ec^2)),
v = 29.7847 Km/s