LOG#115. Bohr’s legacy (III).

Dedicated to Niels Bohr

and his atomic model


3rd part:

From gravatoms to dark matter



Imagine a proton an an electron were bound together in a hydrogen atom by gravitational forces and not by electric forces. We have two interesting problems to solve here:

1st. Find the formula for the spectrum (energy levels) of such a gravitational atom (or gravatom), and the radius of the ground state for the lowest level in this gravitational Bohr atom/gravatom.

2nd. Find the numerical value of the Bohr radius for the gravitational atom, the “Rydberg”, and the “largest” energy separation between the energy levels found in the previous calculation.

We will take the values of the following fundamental constants:

    \[ \hbar=1\mbox{.}06\cdot 10^{-34}Js\]

is the reduced Planck constant.

    \[ m_p=1\mbox{.}67\cdot 10^{-27}kg\]

is the proton mass.

    \[ m_e=9\mbox{.}11\cdot 10^{-31}kg\]

is the electron mass.

    \[ G_N=6\mbox{.}67\cdot 10^{-11}Nm^2/kg^2\]

is the gravitational Newton constant.

Let R be the radius of any electron orbit. The gravitational force between the electron and the proton is equal to:

(1)   \begin{equation*} F_g=G_N\dfrac{m_pm_e}{R^2}\end{equation*}

The centripetal force is necessary to keep the electron in any circular orbit. According to the gravatom hypothesis, it yields the value of the gravitational force (the electric force is neglected):

(2)   \begin{equation*} F_c=\dfrac{mv^2}{R}\end{equation*}

(3)   \begin{equation*} F_c=F_g\leftrightarrow \boxed{\dfrac{mv^2}{R}=G_N\dfrac{m_pm_e}{R^2}}\end{equation*}

Using the hypothesis of the Bohr atomic model in this point, i.e., that “the allowed orbits are those for which the electron’s orbital angular momentum about the nucleus is an integral multiple of \hbar, we get

(4)   \begin{equation*} L=m_evR=n\hbar$ $ \forall n=1,2,\ldots,\infty\end{equation*}


(5)   \begin{equation*} v=\dfrac{n\hbar}{m_eR}\end{equation*}


(6)   \begin{equation*} v^2=\dfrac{n^2\hbar^2}{m_e^2R^2}\end{equation*}

From previous equations, we obtain

(7)   \begin{equation*} \boxed{v^2=G_N\dfrac{m_p}{R}}\end{equation*}

Comparing (5) with (6), we deduce that

(8)   \begin{equation*} G_N\dfrac{m_p}{R}=\dfrac{n^2\hbar^2}{m_e^2R^2}\end{equation*}

and thus

(9)   \begin{equation*} \boxed{R_n=R(n)=n^2\dfrac{\hbar^2}{G_Nm_pm_e^2}}\end{equation*}

This is the gravatom equivalent of Bohr radius in the common Bohr model for the hydrogen atom. To get the spectrum, we recall that total energy is the sum of kinetic and potential energy:

(10)   \begin{equation*} E=T+U=\dfrac{1}{2}m_ev^2-G_N\dfrac{m_pm_e}{R}\end{equation*}

Using the value we obtained in (5), by direct substitution, we have

(11)   \begin{equation*} E=\dfrac{1}{2}m_ev^2-G_N\dfrac{m_pm_e}{R}=-G_N\dfrac{m_pm_e}{2R}\end{equation*}

and then

(12)   \begin{equation*} E=-\dfrac{G_Nm_em_p}{2}\dfrac{G_Nm_pm_e^2}{n^2\hbar^2}\end{equation*}

and so the spectrum of this gravatom is given by

(13)   \begin{equation*} \boxed{E_n=E(n)=-G_N^2\dfrac{m_p^2m_e^3}{2n^2\hbar^2}}\end{equation*}

For n=1 (the ground state), we have the analogue of the Bohr radius in the gravatom to be:

(14)   \begin{equation*} R_1=\dfrac{\hbar^2}{G_Nm_pm_e^2}=1\mbox{.}20\cdot 10^{29}m\end{equation*}

For comparison, the radius of the known Universe is about R_U=4\mbox{.}4\cdot 10^{26}m. Therefore, R(gravatom)>R_U!!!!!! R_1 is very huge because gravitational forces are much much weaker than electrostatic forces! Moreover, the energy in the ground state n=1 for this gravatom is:

(15)   \begin{equation*} E_1=-G_N^2\dfrac{m_p^2m_e^3}{2\hbar^2}=-4\mbox{.}23\cdot 10^{-97}J\end{equation*}

The energy separation between this and the next gravitational level would be about 1-1/4=3/4 this quantity in absolute value, i.e.,

(16)   \begin{equation*} \Delta E=\vert E_2-E_1\vert =3\mbox{.}18\cdot 10^{-97}J=1\mbox{.}99\cdot 10^{-78}eV\end{equation*}

This really tiny energy separation is beyond any current possible measurement. Therefore, we can not measure energy splittings in “gravatoms” with known techniques. Of course, gravatoms are a “toy-model” or hypothetical systems (bubble Universes?).

Remark (I): The quantization of angular momentum provided the above gravatom spectrum. It is likely that a full Quantum Gravity theory provides additional corrections to the quantum potential, just in the same way that QED introduces logarithmic (vacuum polarization) corrections and others (due to relativity or additional quantum effects).

Remark (II): Variations in the above quantization rules can modify the spectrum.

Remark (III): In theories with extra dimensions, G_N is changed by a higher value G_N^{eff} as a function of the compactification radius. So, the effect of large enough extra dimensions could be noticed as “dark matter” if it is “big enough”. Can you estimate how large could the compactification radius be in such a way that the separation between n=1 and n=2 for the gravatom could be measured with current technology? Hint: you need to know what is the tiniest energy separation we can measure with current experimental devices.

Remark (IV): In  Verlinde’s entropic approach to gravity, extra corrections arise due to the change of the functional entropy we choose. It can be  due to extra dimensions and the (stringy) Generalized Uncertainty Principle as well.

Gravatoms and Dark Matter: a missing link

I will end this thread of 3 posts devoted to Bohr’s centenary model to recall a connection between atomic physics and the famous Dark Matter problem! The calculations I performed above (and which anyone with a solid, yet elementary, ground knowledge in physics can do) reveals a surprising link between microscopic gravity and the dark matter problem. I mean, the problem of gravatoms can be matched to the problem of dark matter if we substitute the proton mass by the mass of a galaxy! It is not an unlikely option that the whole Dark Matter problem shows to be related to a right infrared/long scale modified gravitational theory induced by quantum gravity. Of course, this claim is quite an statement! I work on this path since months ago…Even when MOND (MOdified Newtonian Dynamics) or MOG (MOdified Gravity) have been seen as controversial since Milgrom’s and Moffat’s pioneer works, I believe it is yet to come its “to be or not to be” biggest test. Yes, even when some measurements like the Bullet Cluster observations and current simulations of galaxy formation requires a component of dark matter, I firmly believe (similarly, I think, to V. Rubin’s opinion) that if the current and the next generation of experiments trying to discover the “dark matter particle/family of particles” fails, we should take this option more seriously than some people are able to accept at current time.

May the Bohr model and gravatoms be with you!

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