Kepler’s 3rd law was not relevant to Galactic Rotation:
Idea of Dark Matter is like an offshoot of wrongfully expecting the relevancy of Kepler’s 3rd law for rotation of Galaxies.
In our solar system, Mercury is closest to Sun and has greatest orbital velocity. (Let’s say) Pluto is farthest from Sun and has lowest orbital velocity.
In simple terms, Kepler’s 3rd law is that a planet farther from (central) Sun revolves slowly in orbit than the one closer in. In other words, a planet closer to the center has greater orbital velocity and there is ‘drop-off’ in the orbital velocity with increase of distance from the center.
Off course, this behavior of planets was due to underlying general principles that finally Newton extracted but the Kepler’s 3rd law itself was a specific law that was applicable to systems where greater mass is concentrated in the center whereas test masses orbit around that central mass.
Galaxy is a different system where greater mass is not concentrated at the central point. In principle, there was no genuine reason for scientists to expect Keplerian-drop-off for the galactic rotations.
Why actually did scientists expect Keplerial drop-off for galactic rotation had been a mystery for me because galaxy was a different system and Keplerian drop-off was specific to only Solar Systems or Plant-Moon Systems etc. But somehow scientists wrongfully linked Keplerian drop-off with rotation of galaxies.
For example please consider from Dark Matter article of Wikipedia:
Galaxy rotation curves
The arms of spiral galaxies rotate around the galactic center. The luminous mass density of a spiral galaxy decreases as one goes from the center to the outskirts. If luminous mass were all the matter, then we can model the galaxy as a point mass in the centre and test masses orbiting around it, similar to the Solar System.[d]
At the end of this quote, there is reference to footnote [d] which reads as follows:
This is a consequence of the shell theorem and the observation that spiral galaxies are spherically symmetric to a large extent (in 2D).
Galaxy was modeled in the light of irrelevant part of Shell Theorem:
Now it is clear as to what scientists actually did with the galactic rotation problem and how did they reconcile galactic system with 3rd law of Kepler. They modeled galaxy as a point mass located at center of galaxy where stars were orbiting around that central mass and they did it on the basis of ground or justification that this is a consequence of shell theorem.
In this way, for the purpose of studying the gravitational effects, they modeled galaxy exactly like solar system and justified the expectation of Keplerian drop-off.
But in reality, galaxy is a different system and could not be equalized with Solar System. Yes – Newton’s Shell Theorem was applicable to the disc of galaxy but there are various independent components of the so-called Shell Theorem. Basically various Theorems of Newton’s Principia relating to gravitational effects of spherical bodies are collectively known as ‘Shell Theorem’ such that name ‘Shell’ is not assigned to them in the Principia.
The investigation into the matter revealed that (which I have explained in details in book “Philosophy Unscrambles Dark Matter”) scientists, by modeling galaxy as point mass located at center and test masses orbiting around the central mass, actually applied Newton’s Theorem XXXI that was irrelevant part of the Shell Theorem as it was applicable to Solar System or any Planet-Moon System and NOT to Galaxy.
The official Gravity Model for Galaxy was wrong. Galactic rotation means rotation of stars within the disc of galaxy. There was another relevant part of Shell Theorem i.e. Theorem XXXIII that deals with the cases where test masses are located within sphere (or disc) of uniform density.
What scientists actually did was that they applied irrelevant Theorem XXXI in following way:
Scientists modeled gravity of Galaxy within the above depicted meaning and scope of Theorem XXXI. With this setup, they were expecting that Keplerian drop-off will be observed but that was not observed. Thus they were trapped into a fictitious problem to which they eventually settled with ghost solution of ‘dark matter’. Following is the snapshot of Newton’s Theorem XXXI that was wrongfully applied by the scientists:
Real scientists can make mistake of applying irrelevant theorems. But real scientists should never settle with ghost solutions.
Scientists Terribly Missed to apply relevant Theorem XXXIII:
Following diagram shows the scenario of Theorem XXXIII which was rightfully applicable to the galactic rotation but that was terribly missed by the scientists.
Following is the snapshot of Newton’s Theorem XXXIII that was applicable to galactic rotations but was terribly missed by scientists:
Theorem XXXIII is not a usual case of gravity. There are two important points:
- The test mass is NOT subject to inverse square distance law. Now the test mass is subject to inverse (linear) distance law and;
- If test particle is located at a particular depth within sphere (or disk), the outer area of sphere (or disk) will have NO GRAVITATIONAL EFFECT on test mass. Its meaning is that if test particle moves from center to outer edge, the (gravitational) mass will seem to increase.
Given the above two points along with few others which I have explained in the book (Free PDF link), Theorem XXXIII naturally gives flat rotation curves for galaxies; Kepler’s 3rd law becomes totally irrelevant and Dark Matter problem is solved from root.