Re: graviton emissions

[John Mallinckrodt <ajmallinckro@CSUPomona.Edu>]

On Mon, 31 Mar 1997, Mark Sylvester wrote:

> Question: (i) How long would a Rutherford-type electron in a Rutherford-type
> H atom (i.e. a classical orbit) take to lose its energy by em radiation?
>           (ii) Ignoring the em radiation, how long would the same system
> take to radiate its energy gravitationally?
>           (iii) Perhaps instead of (ii) - How long would the earth take to
> fall into the sun due to its gravitational radiation?

For nonrelativistic gravitational oscillators one can get an order of
magnitude estimate for the radiated power as

P ~ (G/c^5)(mass)^2(amplitude)^4(frequency)^6

Whereas for nonrelativistic electromagnetic oscillators the corresponding
formula is

P ~ (k/c^3)(charge)^2(amplitude)^2(frequency)^4      

where k = 1/(4 pi epsilon_0).  The reason for the different dependences on
amplitude and frequency is the quadrupole vs. dipole natures of the
fundamental radiation modes.  Now, rather than calculate the collapse time
for an orbiting system, let's just settle for an estimate of how long it
would take for the radiated energy to equal the initial kinetic energy,
i.e., tau ~ K/P so that 

tau_grav ~ (c/G)(radius)^2/[(mass)(v/c)^4]

tau_em ~ (c/k)(radius)^2(mass)/[(charge)^2(v/c)^2]

For the hydrogen atom tau_em ~ 10^(-10) s and tau_grav ~ 10^(37) s ~ 10^19
ages of the universe.  For the earth-sun system, tau_grav ~ 10^(32) s ~
10^14 ages of the universe.  The moral is, don't sell your stocks yet.

John
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