The arctan functions are dimensionless in that each of them can be thought of as a distance divided by a distance. If this equation is in any way valid, I would guess that the Ln of the mass is an approximation representing some sort of integral of a mass density in space, so when the real integral is done, it will be the integral of a density function, so this Ln of density represents that approximately I’m assuming. When I say “density of mass” here, I am talking about adding up all the force differentials for that integral and sort of integrating it as a volume.
So if the results of the trig functions are dimensionless, and the lnM represents mass, then the G constant would be in units of acceleration, m/s^2
The argument of the arctan is not dimensionless, it’s in units of length. You can’t have this as the result is an angle, which is dimensionless. lnM is not the units of mass, but units of the natural log of M - it doesn’t matter if it’s an approximation. Approximations of a value has the same units as said value. You have to fix the dimensions first before this equation makes sense.
@PdotdQ I can see what you mean about ln of the mass. I changed that to something else that works, but with units such that the constant g can cancel easily. I am going to disagree with you about taking arctan of lengths:
The entire essence of my idea is that gravity is dependent upon the various angles that points in space take with the two masses. The fact that that portion of the formula comes out being expressed in radians makes sense in this context – the entire function delivers a radian-based dimensionless multiplier for the mass to tell you what the force is.
It also makes perfect sense because determining this multiplier is entirely dependent upon the ratio R/L – so again, taking arctan of this makes perfect sense.
Anyways, here is the adjusted equation for the centripetal acceleration due to gravity:
You cannot do this without adding a new free parameter - a new lengthscale = 1 meter denoting the unit being used to measure the distance. This equation will then change behavior depending on the “units you used to measure distance”. You also cannot take this out and absorb it in the G - this is because arctans and divisions do not commute. Just try and plot it with the distance measure being 1m, 10m, etc if you don’t believe me.
This is basic dimensional analysis; you can look online for resources to read up on it, but it should be in any first year physics textbooks.
I understand about dimensional analysis. I am just not following your reasoning as to why it is relevant here. If, instead of writing (L-R) , I wrote it instead (L-R)/1 … why does this not solve the issue?
I would only agree with what you’re saying if I would now get different answers by measuring with different length units ----- but I get answers that are consistent no matter what length units I use.
Because of this, I am doubting your statement that I need to add another parameter. The equation works now, as is, no matter what length units are used.
@PdotdQ How about if we forget what I said before? How about if we forget that I said to “relate it to the unit of measure.”?
You said you were ok with the second arctan. That is a distance in meters divided by a distance in meters. Just look at the first arctan in the same way. It is a distance in meters divided by 1 meter, which is also a distance in meters.
@PdotdQ oops, ok it hit me. I get it now. I was thinking of 1 as if it were L/L, and that means that the meters would cancel out. SO I get what you’re saying now. I will keep thinking about it.
I did not come to this conclusion based on what you said.
It doesn’t matter. The second arctan is dependent on the length scale L. The first one is also dependent on the length scale L’=1 meter. These two arctans each depend on a length scale. So there are now two length scales in your equation, L and L’. L is a physically meaningful length scale in your theory. L’ just come from the units you are using, and is not physically meaningful. The only way to remedy this is to now have a theory with two physical length scales, upon which it becomes odd that the second one essentially is promoted from a non-physical unit-measurement.
@Joe_Bakhos, this is getting unproductive. I am not here to teach you basic stuff like this, so I will just stop here. It was nice to talk to you, and I wish you the best.
Further, the silence from other members of this forum made me think that perhaps no one cares about this topic in this forum, and maybe this forum isn’t really the right place to talk about it in the first place anyway… After all, this is a forum on religion and science; what you want is to talk at a physics message board - you probably will find more people willing to help you there .
I would like to second that you and also @dga471 have been patient as well as knowledgeable. You obviously enjoy questions about cosmology and most importantly you care that people make good use of mathematics. I was drawn in to reading all of this. I only followed some of the math - I am just a programmer at work (and an amateur astronomer). There wasn’t anything I was qualified to comment on. But how you both were so patiently helpful was impressive.
So just before this topic dies naturally, or is closed, I am going to include here some of my remarks about why it is not surprising that in light of the astrophysics evidence we have, scientists prefer to posit the existence of dark matter rather than modify gravitational laws like @Joe_Bakhos or even MOND does. This is adapted from what I wrote to Joe in a PM in response to his more specific remarks - I thought it would be beneficial to have it accessible to everyone else.
This shouldn’t be taken as an authoritative opinion universal to all physicists, but merely one physicist’s take on why the idea doesn’t seem far-fetched. In the course of writing this post I came across this excellent recent review article (Bertone & Hooper, History of Dark Matter) which has a more complete picture of how this paradigm came to be developed. I learned several interesting facts from this paper. If Joe or anyone else interested in learning in greater depth how dark matter became the dominant paradigm, please read this paper instead.
Thank you for this writeup, @dga471! I think that this perhaps deserves its own thread. I am sure that there are people here who are interested about dark matter, but won’t go through this long thread to find your writeup.
This absolutely should be kept somewhere prominent as a reference and not buried at the end of a long thread. It was a pleasure to read. It ties everything together. It is a great “big picture” look at things.
With a great deal of help, I have substantially re-worked my entire hypothesis. It also has bearing on the recent Oumuamua mystery. For any that might be interested, here is an abstract along with a link to the full article:
ABSTRACT
Galactic rotation rates, the distribution of matter in the early universe shown by the scale of anisotropies in the CMB, and cosmological expansion present problems that current theory attempts to resolve by positing dark matter and dark energy. This paper posits that gravitational force is a dampened wave function dependent upon mass and distance. Therefore gravity reverses at regular dampened intervals. This reversal would also be in effect at smaller scales such as our own solar system, implying that current theory may have overlooked evidence of this in the data from various probes that have been launched.
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