An alternative explanation for the shape of spiral arm galaxies (part 1)

We are familiar with spiral arm galaxies. The 'problem' is that to maintain their shape, the rotational speed of the outer stars must be the same as inner stars.

That is in stark contrast to smaller systems like the solar system where the innermost planet Mercury goes round the Sun every 88 days, the rotation period gets progressively longer the further a planet is from the Sun, so the outermost planet Neptune (sorry, Pluto!) goes round every 165 years. This follows the inverse square law - see 2. below.

The still-fashionable explanation - Dark Matter - has been debunked endless times, most recently in the last few days.

Density wave theory is a contender, but the real front runner must be Modified Newtonian Dynamics (MOND), which just says that people get Newtonian Gravity slightly wrong.

In such cases, I find it helpful to write down everything you know and then draw the obvious conclusions.

1. The surprising similarities between behaviour of light and gravity

Yes of course, gravity doesn't really exist as an independent force, but for simplicity we might as well assume it does. The behaviour of light is well studied and understood, so let's use it as an analogy:

* The speed of light = the speed of gravity waves (I remember vividly reading about some fairly conclusive experiment/measurement in 2002 or so and thinking "Well, yes, obviously...")

* Photons = gravitons

* Light waves = gravity waves

* Ripples in electromagnetic field = ripples in space time

2. The inverse square law

The brightness of light is inversely proportional to the square of the distance. This stands to reason. The source is emitting the same number of photons every second and they travel in straight lines, so the surface ares of a hypothetical sphere with radius one light second (with its centre at the source) contains as many photons as the surface area of a sphere with radius two light seconds.

But the surface area of the two-light-second-radius sphere is four times as large (surface area of a sphere = 4 Pi r^2) as the one-light-second-radius sphere, so the light (number of photons) is only one-quarter as bright. Real life example: because of perspective, a light a certain distance away also only looks one-quarter as big as one half as far away, so if you look at a row of street lights stretching into the distance, they all appear to have similar brightness.

The same thing happens with gravity - the force of gravity you feel is also inversely proportional to the square of the distance. To continue the analogy, there are a quarter as many gravitons per unit area of the second sphere.

3. Gravity bends light waves, and...

That light waves appear to bend when they pass near massive objects is also undisputed (I hope); gravity bends light waves. Although photons have no mass so shouldn't respond to gravity. General relativity explains this.

The massive object bends light and so it changes the shape of the hypothetical sphere considered in 2. If the observer is at A, the massive object at B and the light source (star) at C, the light sphere emitted by C is stretched a bit. It is more like a boiled egg shape, with the star at the centre of the yolk and the observer at the pointy end. The observer sees brighter light than they 'should'; the star appears larger/closer than it really is; the observer receives more photons than they 'should' etc. The massive object acts like a lens.

Galaxies bend light on a much larger scale than a single massive object (h/t Dyson and Eddington in 1919), hence the term galactic lens. If the light-gravity analogy is to hold, then gravity must also bend gravity waves. I'd guessed this all along, but Googled it this morning to check and yes, they do. For sure that's 'only' a blogpost, but she's a proper qualified scientist and her post is full of links to official stuff.

UPDATE, a few days later a physicist explained in Forbes:

This tells us, unambiguously, that gravitational waves, as they travel through the Universe, are affected by the warping, curvature, and stretching of space.

There’s another piece of evidence, too. The kilonova event of 2017, where we observed the merging of two neutron stars in both gravitational waves and in electromagnetic light, had these two signals arrive nearly simultaneously: with less than a 2.0 second difference between them. 

Traveling from a distance of over 100 million light years (and given that there are over 30 million seconds in a year), we can state that the speed of light and the speed of gravity are equal to within better than 1 part in a quadrillion (1015).

This tells us another important piece of the puzzle: whatever time delays take place for photons as they travel through the Universe owing to the curvature of space also occur for gravitational waves. Whenever you enter or leave an area where gravitation is strong, you have to follow the path set forth by the curvature of space. Around a massive galaxy, for example, like the one we observed the kilonova in, space is curved, and all massless particles have to climb out of that potential well.

The fact that photons and gravitational waves arrived simultaneously tell us that they had to experience the same effects as one another from the curved space they passed through.

Here's the kicker:

Every star's gravity field bends, and is bent by, the other stars' gravity fields. Start with two stars (sources of light AND sources of gravity). Each light/gravity sphere is bent egg-shaped, so we end up with a two overlapping light/gravity spheres shaped like a Rugby ball. Which is like an American football but a bit less pointy.

Add more stars and the gravity fields merge and flatten out into something the shape of an Olympic discus; add a whole galaxy and the galaxy's gravity field gets flatter and flatter.

4. Modified Newtonian Dynamics

Enter stage left, towering giant of non-bullshit astro-physics, Mordi Milgrom. What his MOND (link at start of this post) says is that up to certain radius (about 5,000 light years, 5 kly for short), the force of a galaxy's gravity on stars follow the normal Newtonian inverse square law; beyond that certain radius, the force of gravity diminishes inversely proportional to distance i.e. pull of gravity on outer stars is stronger than expected, meaning they spin round faster than expected, so have the same rotational speed as inner stars, maintaining the spiral arm shape, the problem we are trying to explain.

The problem I have when I read up on his MOND (whether he deliberately chose an acronym that spells the German word for 'moon' is unknown) is that nobody explains why there is jump from normal Newtonian gravity nearer the centre of a galaxy to MOND gravity beyond a certain radius, it all seems a bit arbitrary.

Observations fit his equations because he tweaked his equations to fit observations etc. Which is why I have had to work out (reverse engineer?) the actual explanation myself.

5. A worked example

Let's start with a star 5 kly out from the centre where Newton's rules stop applying. It is pulled towards the centre with a gravitational force of X (whatever unit that is).

* Under normal Newtonian inverse square root rules, a star 15 kly out - at the other end of a spiral arm - only feels a pull of X/9 of that (15/5 = 3, 3^2 = 9)

* MOND says a star 15 kly out feels a pull of X/3 (15/5 = 3) not X/9, which seems like a huge discrepancy.

It's not such a big discrepancy really. The star 15 kly out just 'thinks' it's only 9 kly out (as it feels the same pull as if it were only 9 kly out).

Maths: The pull of gravity towards the centre of gravity on the surface of a hypothetical perfectly shaped sphere with radius 9 kly centred on the centre of gravity is (approx) 1/3 of that on the surface of a sphere with 5 kly radius. 9/5^2 = 3.24, = 1/3.24 = close enough to 1/3 for our purposes. (The correct number is 8.56 kly but let's go with 9 kly).

Possible explanation: the 15 kly star is actually sitting on the surface of a 9 kly-radius sphere... which has been stretched out in every horizontal direction, so it has height +/- 18 kly and width of 30 kly (see 3). The star is 15 kly from the centre, but in gravity terms, it is only 9 kly from the centre.

Bonus: Newton wasn't wrong, it's just that you can't expect his inverse-square-law spheres to be perfectly spherical in all conditions. They are at the small scale of a solar system where the central Sun is somewhere between 99.8% and 99.9% of the total mass of the solar system anyway; not in a large galaxy where mass is more evenly distributed and the cumulative effects are much greater.

6. If anybody has access to the right telescope...

... and somebody else knows how to do the calculations, I'm happy to split the Nobel Prize money three ways. Get to it!
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Part 2 to follow, including diagrams and ways that we can test this theory i.e. what sort of results it predicts and how to observe and measure them.