[first posted: 1.24.17/most recent edits & additions: 3.14.17]

Preamble: this conjecture relates foremost to the cell cycle and to the possible cause for the shared circadian rhythm found in all plants, insects and animals. But it also relates to the origins of why cell mechanisms have a biological preference for a specific chiral order in molecules. The discussion is extended in Topics VI: Molecular Precession, Gravitation, and Circadian Rhythms and XI: Jet-Lag, Artificial Accelerations, & Phase Shifts in Cell Rhythm? Experiments designed to test this conjecture will be posted here in the spring.

Hypothesis: A body standing on the surface of the earth is subjected to a host of accelerations that include, not only the centrifugal and gravitational accelerations produced by the earth's mass and rotation, but also those produced by the earth's orbital motion around the sun. When the kinetic energy of an earthbound body is defined relative to the reference frame of the sun, the magnitude is a function of the combined orbital and rotational velocity of the earth. The kinetic energy is, therefore, a function of the position around the earth relative to the position of the sun, and therefore a function of the time of day. What is intriguing is that - in the sun's frame - the kinetic energy of the body at midnight (when the rotational velocity is parallel to the orbital velocity) is significantly higher that the kinetic energy of the body at noon (when the rotational velocity is anti-parallel to the orbital velocity). Accordingly, the body must undergo an acceleration between noon and midnight and undergo a deceleration from the midnight to noon. Analogy may be found with the gravitational slingshot effect which NASA and other space agencies use to boost the velocity and kinetic energy of a satellite relative to the sun. The magnitude of this acceleration for a body on earth is not trivial, and it results in a torque induced into and through the body with circadian rhythm. Conjecture is offered to not only link this torque with the circadian rhythm - known since 1998 to be expressed at the cellular level - but also with mechanisms that drive cell activity. Lastly, it can be shown that the induced torque is chiral in nature (defined by the earth's orbital geometry) and that a biological energy incentive can be found when directing motion in patterns that reflect this chirality.

This majestic shot of the Earth was taken by the crew of Apollo 8 just as they emerged from the darkside of the moon (1968); it serves as our starting point for a discussion about how large-scale gravitational effects influence our biology. However, in order to enjoy the scientific side of this argument, we will need to expand our view a little further and place these orbital bodies into a reference frame that includes the sun. This will allow us to apply a lesson learned from relativity theory over a century ago and one which NASA applies each time it utilizes the gravity assist (slingshot) method to send a probe to a distance planet.

(Note: liberty was taken in the positioning of the sun here. In NASA's photo, we can tell from the shadow on the earth that the sun was positioned 'above' the earth and moon; but for graphic convenience the sun has been placed to the right.)

With reference to the above drawing, when we position ourselves in the reference frame of the sun and study the motion of a body that is standing on the surface of the rotaing earth, we observe that the velocity of the body varies depending on the time of day. This is due to the velocity of the body being a function of both the earth's rotation about its axis, u, and the earth's orbital velocity around the sun, v. Thus, for example, the relative speed of the body at midnight is (v+u) while at noon the velocity is (v-u). As shown above, in the frame of the sun, this gives the body a kinetic energy of .5m(v+u)^2 at midnight, and .5m(v-u)^2 at noon. This is not surprising, and neither is the computation of the energy difference between those intervals, (delta E)=2muv, as shown in the diagram.

What is surprising is what we can deduce when we ask: where does the kinetic energy get transferred to when the body decelerates in speed from its midnight to noon position, and where does the kinetic energy come from to accelerate the body as it moves from its noon to midnight position? That is not a semantic exercise - neither value for the kinetic energy is the proper energy the body should have to maintain the orbital position it possesses at those noon and midnight time intervals. Yet, the body does remain in orbit, because it finds itself gravitationally 'bonded' to the earth. This leads to the surprising deduction that the gravitational bond between body and earth must be the source for the transfer of orbital kinetic energy in the sun's system. And further, because the force that stabilizes the earth-body configuration while the body is standing, sitting or lying down can only be transferred through the points in contact (feet, tush, back, stomach, etc...) - that a torque must constantly be acting between the earth and the body as they each seek to balance the forces associated with the perpetually changing orbital positions! Such torque can be transferred up through the body by inducing a strain angle between contiguous vertical elements and enacted by a slight anharmonicity in the 10 billion collisions that occur each second between molecules.

If that does not surprise you, then read it again.

(Or, read it again and show us why this conclusion is wrong.)

To keep the change in energy balanced within the sun-earth system, kinetic energy must be shuffled in and out of the body with each day/night cycle. Since this argument can be extended to any two vertically separated points in the body, we reach the conclusion that this can only be achieved by the action of a torque that is driven downwards in scale into each cell of the body. That is both surprising and important, for this becomes an intriguing driving force for cell activity and circadian rhythms. Now, to be sure, the magnitude of the correcting torque needed to shuffle this energy back and forth and to maintain balance is small, but it is not trivial. How such torque is transferred and the patterns we expect to see relative to our day/night cycle becomes the focus of the discussion below.

[Note 1: Rotational velocity u is, of course, a function of latitude; for these discussions we will treat the body as being located at the equator which gives the body a velocity of roughly 460 meters/second. Orbital velocity v also varies depending on the position of the earth around its orbit; we will use the earth's mean orbital velocity of ~ 29,000 meters/second. We will also ignore the tilt of the earth's axis relative to the orbital plane of the earth. This tilt produces other important effects that will be explored later - for now it unnecessarily complicates the problem.]

[Note 2: The drawing above and the one below were part of the original posting on 1.26.17, and therefore retained for continuity. A more formal presentation is provided below.]

[Note: the argument below may be seen as a bit pedantic to anyone trained in the field. This is because the authors are not experts in the physics of orbital mechanics, and by showing each step that leads to the above conclusion, mistakes are more likely to be revealed. Further, although vector equations and tensor notation are more conventionally used for such derivations, we believe the argument is more tangible as presented.]

Let us begin with a few general relationships pertaining to (Newtonian) orbital mechanics. The speed of a body in a circular* orbit is derived by balancing the centrifugal force that tends to throw the body outwards with the gravitational force that pulls it inwards. Thus, letting (m=mass of the body; M=mass of the gravitating mass (the sun in this case); R=orbital radius; G=gravitational constant), then:

which gives the body an orbital velocity of:

* (In elliptical orbits, velocity changes with position - but the argument here holds for either circular or elliptical orbits.)

The velocity is independent of the mass of the orbiting body, and, thus, for a point mass orbiting the sun with a radius given by the mean radius of the earth's orbit we obtain a velocity of...

If, instead of using a single point mass located at the position R defined by the orbital center of the earth (CM), we consider two bodies separated in their orbital position by the diameter of the earth (twice the radius, r), then the proper velocity each body should possess for a stable orbit is given by:

..........

Using the value r=6.4x10^6 m for the radius of the earth, the proper orbital velocities are:

..........

The difference in velocities of 1.3 meters-per-second is not so trivial. If, for example, two such bodies were travelling straight for 12 hours, they would be over 56,000 meters apart (~35 miles). In orbital mechanics, even small differences become important over long intervals. However, this small difference in velocity is only part of the problem. Of much greater importance is that, due to the rotation of the earth, the actual difference in the velocity of a body on the 'near' and 'far' side of their orbits are significantly larger than this.

Since the far-side position corresponds to the position a body has at mid-night, and the near-side corresponds to the position of the body at noon, let us refer to them as such. Accordingly, the necessary velocities that at body located in a mid-night orbital position and one at an noon orbital position to sustain their orbits are:

...........

But these are not the velocities that our bodies possess at mid-night and noon. Rather, due to a rotational velocity u of roughly 463 meters-per-second, which - in the reference frame of the sun - can be seen to increase our speed at midnight and decrease our speed at noon, the actual velocities body are:

..........

This difference in velocity between daytime and nightime equates to a difference of

The momentum associated with this energy has to be shuffled into and out of each kilogram of mass in our body during the day/night cycle to maintain its helical path around the sun. The force needed to produce this momentum change must come from an external torque acting on the points of contact that the body makes with the the earth. The torque depends the orientation of the body, and thus is different for a standing body than one lying down. Such a torque clearly has a day/night circadian rhythm. Further, due to the handedness of the trajectory of our orbiting rotating body there is a chiral sense to the torque. Accordingly, it is reasonable to extend this conjecture into modelling how this torque might be used to drive the cell cycle and why a specific chiral handedness may be preferential when manipulating proteins.

The observations/conjecture is the previous paragraph certainly need greater support than that which has been provided above. To do so in any detail way, we will need to switch our description about path trajectory into the more sophisticated format used in Hamiltonian mechanics. [We welcome any assistance by the reader to place this conjecture into the more formal mathematical structure; otherwise, please return in a month or so, and we will try to derive and post the Hamiltonian equations.]

Before we switch to a more formal presentation which will define the torque experienced by a body as a function of position (time) relative to the sun, let us repeat much of what was presented above, but with slightly different images:

First, in Fig. 1.A, let us repeat the initial geometrical arrangement, which we view from the reference frame of the sun. The earth orbits the sun with radius R (we will approximate the orbit as a circle). The earth, with radius r and angular velocity w, causes bodies at the equator to rotate with velocity u=w.r (we ignore the 23.5 degree tilt in the rotational axis of the earth relative to the plane of the orbit and consider only on bodies at the equator). The rotational velocity of a body positioned in the 'Noon' orientation will be in the direction opposite to the earth's orbital velocity v(e); their velocity in the sun's frame will be (v-u). The rotational velocity of a body positioned in the 'Midnight' orientation will be in the direction of the earth's orbital velocity; the midnight body will be traveling with the velocity (v+u) in the sun's frame. Balancing the gravitational force of the sun acting on the earth with the centrifugal force of the orbit gives the velocity for the center of the eart; shown below. (For reference, the earth's orbital velocity v is approximately 29,000 m/s and the rotational velocity u at the equator is approximately 463 m/s).

In Fig. 1.B(below) the configuration is the same as in Fig. 1A, but we have masked out the earth. This allows us to focus on the noon and midnight bodies. The velocity that a body has at noon and midnight is highlighted in yellow. We can compare these velocities with the proper velocities such body would have if they were simply orbiting the sun without the earth. Those derivations are shown in blue, and the differences shown near the bottom of the figure. What is clear is that the velocity a body possesses at noon and at midnight - or, at most other times during the earth's ~ 24 hour day/night rotation - is not the proper velocity that a body must have to exist in a stable orbit around the sun.

When we calculate the kinetic energy possessed by a body in the noontime position and compare it with the kinetic energy possessed by that same body at midnight, shown Fig. 1.C (below), we find they are not the same. We recognize that this is not surprising, for these orbital effects were calculated long ago. But what is intriguing from a biological perspective, is to inquire about how that kinetic energy is transferred in and out of the body. In the sun's reference frame, kinetic energy must be shifted back and forth between the body and the earth. That can only be accomplished by the application of a torque acting between the body and the earth. This torque has a circadian period to it, and a chirality defined by the orbital configuration of the earth-sun system.

 

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(more about the biological aspects of this topic later this week)

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A few other comments that link this conjecture to discussions found on the home page:

When a force is applied to an achiral molecule through its center parallel to its axis of symmetry, no rotation is expected. However, when that same force acts on an chiral moleucule, it induces a rotation. That is explored in Topic VII; Isotopes, Nuclear Spin and Chirality?

When discussing acceleration cycles within contained fluids, there are some unexpected patterns that arise due to differences in density, viscosity and sedimentation coefficients. Those discussions are the subject matter in Topic IV: Asymmetry in the Cell Nucleus as a Means for Driving Torque? and Topic V: Protein Shape and Terminal Velocity in the Cytosol?

This discussion will be expanded in the coming months.
Please return in late spring 2017.

[sketch and continued argument to follow]

See Topic XII: ATP and Gravitational Potential Energy?

[v=at...gives an a/g diagram...indicating a torque induced strain angle of roughly 0.005 radians...which may conceptually be interpreted as: lean 'forward' to accelerate, lean 'back' to decelerate...]