Orbits and Timed Rotations

Suppose we had a situation where the orbital angular velocity equals that of rotational spin in some two-body system such as you have in the earth-moon case. It would have the same meaning as “the same side of a moon always faces the planet”.

Suppose now that in such a situation as m with time is receding in a larger and larger orbit due to dissipation of mechanical energy and we find (prove) that irrespective of size of the orbit the same situation still holds with m always facing M throughout its orbital trajectory. It would be very convenient and for very good reasons. However it is not so.

A description of the model and my mathematical analysis is given in,

https://files26716869.files.wordpress.com/2020/07/synchronism.pdf

Physical evidence

For the Earth-Moon case there is additional significant data and history available for confirmation or negation of such theories.

Such a result would be desirable and could answer a number of questions. In the Earth-Moon case many material properties might be explained such as chemical composition and physical structure, detailed in both large and small scale, from the microscopic to macroscopic, all of which very much is known.

The asymmetry of the physical Moon on all scales together with intuition to me gives historically the simplest and most convincing arguments. Convenient speculations and explanations of observations are now many and there are many examples. In the end it all just appears a mystery and misleading just adding to confusion, but a discussion is not the purpose of this essay.

Observations on the problem

I had hoped to prove that in special cases the orbital period (month) and rotation (stellar day) stay mutually equal irrespective of the value r. This would then have hopefully cleared up some issues and answered a few questions. Originally I had hoped the moon's origin could be explained. Personally I am now convinced this hypothesis is not true and that the theory crashed right at the start already.

It has to be assumed the reader has basic knowledge and understanding of high school physics and elementary dynamics of classical mechanics. This here is good old solid mathematics.

The phenomenon is fascinating and we discover examples from observations in astronomy apparently from all over. Usually instances are just dismissed as a case of “gravitationally locked” I would really like to know what this is and is supposed to be, to me it sounds like only a convenient and plausible phrase. I do not have access to academic journals it could be that I'm wrong.

Sounds all like nonsense but what is it? What on Earth is it? It is much too common to be accidental there has to be much more to this it is not just coincidence.

The M-m System

In my highly simplified model it has to be absolutely clear what we actually want to achieve. The original (hoped for!) result turned out wrong. It is untrue. Attempting to be clear and together with the sketch I formulate the next question:

Given the model, find the values of the orbital radius r for which m's angular velocity omega_m, around its own axis is equal to the orbital velocity Omega_m around the system center of mass c.

These would also be the values of r for which the angular rotation and orbital velocity of m are synchronized. As common knowledge this is “The same side of m always faces M”. There are other known and similar situations physically observed while related examples are still being discovered.

Popular explanations

We want to study the behaviour of solutions. This orbit “gravitationally locked” story I don't believe, I think the term is thumb-suck and devoid of meaning. I would really like someone to factually explain and quantitatively prove the existence of such a thing. This too is wishful thinking at best.

For a start there is always dissipation of energy rapidly or slowly so that in the longer run in reality the rotation radius will have to keep on increasing gradually and cannot “lock”. You would need to have some fixed and static situation in an external kind of balance.

Any given constants m and M and I_m will determine the numbers r, they could routinely be calculated numerically given the constants, it is very basic stuff but ideally we want a formula. We want an explicit, closed form solution, ideally a mathematical formula in terms of the constants.

Fundamental equations and Calculations

The constants are the respective masses and moments of inertia. Our variables are the (rotational) angular velocities all of which are in the same plane. This simplifies matters considerably because you don't have vector equations everything is scalar. The orbital radius r is the independent variable.

From the geometry of the situation it can be assumed that orbital angular velocities are equal. Considerations of  centrifugal and gravitational force gives the orbital angular velocity as a function of the orbital radius.

Applied then basic mechanics formulas the system's total angular momentum is calculated the resulting value is constant because of the conservation law. With equations simplified resulting in an equation of motion for m as a function of r after a substitution the equation of motion is obtained in quartic (4th degree polynomial) form.

Subsequently simplifying, our quartic equation is obtained after which the roots will provide the appropriate radius values. To find the roots and then actual solutions finally this quartic equation must be solved. With the given constants the polynomial coefficients are known.

Note that in this model of an orbiting two body system there is no coupling between M's spin and orbital rotation speeds the rotations rates are independent. I have provided the actual mathematics the workings are detailed and I think are necessary to understand this work.

Simplifications

For the next step now the mathematics is basic algebra and of totally different nature, it concerns the quartic polynomial in x and finding the roots and subsequently the corresponding r values. The data we need in the end are just three constants, the masses M and m and m's moment of inertia I_m.

For our problem eventually we obtain a considerably simplified quartic equation p(x) = 0 where two of the five coefficients are zero, two coefficients are positive with the leading coefficient as 1 (one), and one is negative.

These kinds of observations should already simplify matters considerably and make it a lot more feasible to find properties of solutions for our quartic equation.

However we may easily solve for the real roots of the equation numerically to any desired accuracy and speed for instance Newton's method would work well with given actual numerical values for coefficients. This method is simple it converges very rapidly and is easy to implement. It would work especially well since polynomials are very smooth. But other than the specific numbers for a situation you will gain very little insight, what you actually want are general analytic closed form solutions for the roots.

Polynomials and solutions

The solutions of a quartic are more like algorithms pages full of it there do exist formulas similar the very well known one for quadratics (second degree polynomials), but the formulas for higher degree polynomials it turns out are not very helpful at all.

Calculating the quadratic roots is a simple matter with a simple formula but the general cubic (third degree polynomial) is already more than hard enough to solve. Similar to the very well known one for a parabola's x-intercepts, there exists for the cubic roots also formulas but are already very complicated and in themselves not of much practical use.

Whereas solving a quartic with the roots as known solutions is in principle futile and practically impossible, in contrast the 1st degree (linear) root of a straight line is trivial.

Ideally one would want an algebraic formula for the roots of a quartic in terms only of the given constants the coefficients, and such are known but algorithms are pages and pages prohibitively complicated and the equation is unassailable.

Valuable insights can however be deduced without actually solving. For example we know the roots can be given in algebraic form. For instance too there are a maximum of four real roots but for that matter no (zero) real roots at all, or repeated roots.

As a further example x and r can't be negative the roots have to be positive seen from the initial physical problem, and so negative and complex values must be eliminated since in which case such a number would be meaningless in the physical situation. There could also be all kinds of other properties to be known, for example since the fourth power is dominant eventually p(x) is unbounded positive as x (and r) increases.

Information involved such as continuity turning points, interceptions and intermediate value properties can be profitably investigated.

Mechanics and mathematics

Thus the algebra turns out to be a formidable undertaking but even so meaningful study of the equation can provide partial results and answers and give insight on properties and more qualitative information of the problem itself both of specific and general nature.

New ideas would arise as questions and explanations in future for instance assuming we have more relevant independent information from another source also as with our Moon. In the mathematical model as given here there are some restrictions, intrinsic simplifications and related physical properties to be made which can be very useful.

Findings would be of further interest and applied might explain some other and even unexpected things. This was thus only half the story the mechanical side, now the algebra has to be addressed. It is a considerable challenge and highly nontrivial in it's own right. Investigating this quartic's roots is a different type of mathematics problem altogether.

Motivations and continued investigation

Certainly there is possible broader usefulness and practical applications of this work and the quartic equation. It might bring new insight in celestial mechanics in general and knowledge for understanding problems for example in space travel and astronomy.

With given information one may unexpectedly find useful calculations and evaluations. For instance we might test and verify seemingly unrelated ideas or make predictions or confirm    information independently.

Situations as very universal potential candidates for application, specifically in our solar system are planet and moon, star and planet, also further possibilities such as binary stars. One could imagine other possibilities like rotating and colliding stars, black holes and galaxies.

Finally,

For the future I hope to analyse the polynomial equation and see what more we can learn. The current calculations are in the pdf but only the mathematics there isn't any discussion however this essay would probably not make a lot of sense without it.

I don't have anybody to proofread and verify my work and my calculations there are no collaborators please let me know if there are any serious problems.