Free sections II.II.IV and II.II.VI — Book: “Philosophy Unscrambles Dark Matter”

In response to my recent blog post that briefly explains how scientists applied irrelevant Newton’s Theorem XXXI to the galactic rotation problem and missed to apply correct Theorem XXXIII, a PhD physics person pointed out that even if Theorem XXXIII is applied, the discrepancy remains. To this, my reply was that blog post was brief. The remaining discrepancy has been acknowledged by the book. Two things directly come from Theorem XXXIII i.e. (i) Inverse square distance law is replaced by inverse (linear) distance law within the disc of galaxy and; (ii) For any depth ‘r’ within the disc of galaxy, the outer layers have no gravitational effect. Now only these two things really do not cover the full discrepancy. Third thing, which has been explained in sections II.II.IV and II.II.VI of the book “Philosophy Unscrambles Dark Matter” is that luminosity is not actually reduced towards the edges. Yes along the line of diameter, or radially, the luminosity is decreased … but total luminosity of the complete circumference of outer layers is not decreased. These sections did satisfy him and he only raised one further objection to which I satisfactorily replied (will share it in next blog post).

II.II.IV. Flat Rotation Curves of Galaxies – Proper Interpretation

After having seen that galactic rotations should have been described in the light of Theorem XXXIII of Newton’s Principia Mathematica, let us therefore try to do it now.

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Expected (A) and observed (B) star velocities as a function of distance from the galactic center. (Credit: Wikimedia: Commons)

The prerequisite here is that we must completely forget the so-called ‘Expected’ line (A) because within the right context of Theorem XXXIII, we simply do not expect line (A). Line (B) is the actually observed line and the same is anticipated by applying Theorem XXXIII. Regardless of what official theory tells us about the existence of super massive black hole at the center of galaxy, this graph is actually telling that closer to the center, orbital speed is lowest. Within a disk of uniform density of mass, we should expect zero orbital velocity at the center of the disk. The lowest orbital velocity at point close to the center is consistent with this theory which means that law of inside of sphere (or disk) is being demonstrated. Non-zero but lowest orbital velocity near the center of disk may or may not indicate the presence of super massive black hole at the center. Afterwards, over a very short distance, there is substantial increase in the orbital velocity as the velocity curve moves up quite sharply. Our interpretation is that this area is the central bulge of the galactic disk and over this short distance, actual mass is substantially increasing layer upon layer such that density of each layer almost remains the same. Following actual graph confirms the idea that area of sudden increase of orbital velocity approximately relates to central bulge of the galaxy M33.

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M-33 Galaxy Rotation Curve, Credit: Wikimedia Common – Source link[i]

M33 is not very large galaxy as the diameter of galactic disk is only about 60000 light years. We see (or assume) in this picture that radius of the central bulge of the disk spans about 5000 light years and within this distance of 5000 light years, there is sharp jump in the velocity curve. This actual graph is showing gradual upward movement of velocity curve even beyond this point but for the sake of simplicity, we shall assume that after this point, velocity curve becomes flat.

Basically there are two distinct portions of the Rotation Curve of Galaxy. Up to the distance from center towards the edge of the central bulge, there is sharp increase in orbital velocity of stars within disk. The lowest orbital speed is found in the area closer to the center of the disk. It means that area close to the center is subject to lowest gravity and this thing is in harmony with the Shell Theorem as applicable within the sphere (or disk). In the example of galaxy M33, we see that radius of central bulge is almost 5000 light years. For the sake of our analysis regarding why orbital velocity is increasing very sharply over this distance, we suppose that there are 5 layers within the radius of central bulge and the width of each layer is 1000 light years. Our interpretation will not depend on the existence or absence of super massive black hole at the center of galactic disk. So the interpretation goes that for the five layers of central bulge, a huge quantity of mass, let’s say 1 billion solar masses, is concentrated in the innermost layer that may or may not include super massive black hole. The second layer is orbiting around inner most layer with the lowest velocity. The second layer has same width of 1000 light years but due to being outer layer of the circle, the area is far greater than the innermost layer. The second layer has almost equal density of mass which means that total mass of the second layer may be around, let’s say, 8 billion solar masses i.e. just approximate number only to explain the point.

Now the third layer is orbiting a total mass of 9 billion solar masses. Therefore, within the third layer, orbital velocity has increased quite sharply. Width of third layer is also same 1000 light years but area is still far larger than that of second layer. And again, the density of mass remains the same and thus total mass of this layer may be let’s say 16 billion solar masses.

Now this setup repeats up to the fifth layer which is subject to the highest orbital velocity of stars within the disk so far and also marks the boundary of the central bulge of the galactic disk. The central bulge area is therefore the first portion of the Rotation Curve of Galaxies. The important thing of the first portion is that mass is considerably increasing layer upon layer and reaches to, let’s say, 32 billion solar masses for the fifth and outermost layer of the first portion.

The central bulge area was characterized by layer upon layer successive and substantial increase in mass such that overall density of the bulge remained uniform. The outermost layer of the central bulge contains greatest quantity of mass so far which is 32 billion solar masses (i.e. approximate number just to explain the point). Next to the central bulge area, the second ‘flat’ portion of the Rotation Curve of Galaxies begins.

If the radius of M33 galaxy is 30000 light years wide then this second portion starts from 5000 light years from center of the disk and ends at 30000 light years from the center of the disk. For the sake of simplicity, here again, we divide this second portion into 25 layers each having width of 1000 light years.

We know that outer layer of central bulge had mass of 32 billion solar masses. Now we interpret the start of flat curve portion by saying that inner layer of this portion contains almost same mass i.e. 32 billion solar masses. In this way, the innermost layer of the second portion is having same mass as the outer layer of the central bulge had. However due to larger area, the density and luminosity (per unit area) of this layer is lower than that of central bulge. Due to the fact that previous layer i.e. the outer layer of the central bulge had the greatest mass, our present layer i.e. the inner layer of outer area has the greatest orbital velocity and the rotation curve moves still higher. Therefore, flat portion of curve has not actually started yet.

Now comes the second layer of the outer portion of galactic disk. Again mass will remain the same i.e. 32 billion solar masses and due to larger area, there will be slight reduction in the density and luminosity (per unit area) across this layer. Because previous layer had augmented a constant mass, therefore, keeping in view the applicable inverse distance law of gravity, orbital velocity curve will remain horizontally flat across the current layer.

If this pattern repeats up till 25th outermost layer, each successive layer will get equal quantity of mass however slightly lesser and lesser density and (per unit area) luminosity will be added and the overall galactic rotation, keeping in view the simplified assumptions, should show up as a flat curve on graph. It is possible that same pattern of successive layers, up to few more, may continue even after 25th layer but that outer portion of galaxy may remain invisible or normally undetectable due to low density and (per unit area) luminosity over there.

An important thing to be noticed is that let’s say when an object moves from 10th layer to 11th one, the object will be subject to gravity of the mass available in all the inner layers including central bulge and up to 10th layer (or even 11th layer). Objects placed in 11th layer will not be subject to gravitational effects of still outer layers i.e. 12th and rest of the outer layers because according to Theorem XXXIII, an object placed at certain depth within sphere (or disk) will not be affected by the gravity of outer surface area. With this setup, availability of constant mass in each successive outer layer will give the result of flat rotation curve because law of inverse square distance is also replaced with the law of linear inverse distance within the sphere (or disk).

The following is the graph of mass available in successive layers and it is similar to the rotation curve graph of galaxies.

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The above scheme of the things is actually based on oversimplified assumption of two dimensional setup of mass. In reality, galactic disk has thickness that is usually more or less or almost 1000 light years. Thus within central bulge, in reality, there should be far greater increment of available mass than by the factor of just 8 which is being presented in this scheme. Moreover, onward from central bulge, the quantity of mass may get slightly increased layer upon layer i.e. only as much that density of the layer should remain lower than that of previous layer and the net effect may be slightly upward velocity curve which is the case we have seen in the diagram of M33.   

However, for the purpose of our analysis, we carry on with the simplistic two dimensional assumption and constant increase of mass for area onward from the central bulge. Following schematic diagram with inner five layers of central bulge with uniform density and outer (only) eleven layers each having mass equal to the outermost layer of central bulge shows that such a structure not only explains the observed flat rotation curves of galaxies, it also develops the spiral structure of galaxies.

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Blocks placed in successive layers

The above diagram is made up of equal size squares or blocks. The central yellow mark is the innermost layer of the central bulge and the other prominent yellow square is the outermost layer of the central bulge such that this layer consists of 32 small blocks which means that outer layer of the central bulge is 32 times massive than the innermost layer. Following is close up view of above diagram up to only the fifth layer and covers the complete central bulge area.

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Blocks placed in successive layers – Bulge area close up. Outermost layer is 32 times massive than central layer.

In this schematic diagram, each small square represents equal quantity of mass let’s say 1 billion solar masses. If there is mass of 1 billion solar masses in the innermost layer, then second layer contains 8 billion solar masses and overall density remains the same. The fifth layer is the outermost layer of the central bulge.

Following close up shows what would eventually look like spiral structure from a far-view:

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Blocks placed in successive layers – Outside of Bulge area close up

Here we see that outer layer of central bulge had mass of 32 billion solar masses whereas the total mass of the central bulge was (1+8+16+24+32) = 81 billion solar masses.

Next to the yellow layer starts the second portion of galaxy whose just eleven layers are shown in the image that starts looking like a spiral galaxy. In this portion, each layer contains 32 billion solar masses. While density remained uniform throughout the central bulge but beyond the central bulge, now mass is constant per layer and density per layer is getting reduced layer upon layer. A random placement of 32 blocks in each successive layer would give the overall shape of a spiral structure.

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Blocks placed in successive layers – total 11 layers after Central Bulge

Note that this schematic diagram is based on square blocks and yet the basic shape of spiral has been achieved. Here, equal number of blocks have been randomly placed in each successive layer of the second portion of galaxy which is outside of the central bulge and the result is a crude or basic shape of galaxy. In a real galaxy, matter is not randomly arranged as the actual shape is determined by the overall scheme of the larger structure as well as quantity and placement of nearby mass or the availability of local structures. After eleventh layer, if we add next layers up to 25th layer by placing the blocks in accordance with the already emerging shape, the following final shape is achieved.

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Blocks placed in successive layers – total 25 layers after Central Bulge

The real galaxies are often arranged in spiral shapes such as following.

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Pinwheel Galaxy. Image Credit: NASA/ESA

In real galaxy, there is no empty space between spiral arms. But it does not mean that Spirals are merely illusions. In the schematic diagram, one billion solar masses was represented by just one square box. But in a real galaxy, mass of one billion solar masses is spread out in the form of fog of stars. Secondly, one box actually represents the compacted mass of central bulge area. For the outward area, mass should remain the same layer upon layer but one billion solar masses, being non-compact area, actually takes space of more than one box and this would be the reason why in-between spiral arms areas are not empty for the real galaxies. The in-between spaces of spiral arms are not empty or devoid of matter but however spiral arms are the places where greater mass is concentrated and thus spiral arms are real (i.e. not illusion) and assume their shape due to slightly greater mass but overall reduced density of the successive outer layers of galaxy. Within an actual galaxy, each successive layer may get more than slightly greater mass which seems to be the case with M33 galaxy where flat rotation curve is actually a slightly upward curve. It is also possible that in any galaxy, each successive layer may get slightly reduced mass than the previous layer and dark matter regime ‘scientists’ may identify such a galaxy as ‘dark matter free’ galaxy. Scientists do have identified two such galaxies so far but firstly they have not measured the rotation speed of stars within galaxies rather they have taken the velocity dispersions of globular clusters around them therefore inside of sphere or disk rule does not apply. Secondly, they also assert that these are not the confirmed cases of dark matter free galaxies as with ‘latest’ observations, they have considerably reduced the distance of those galaxies[ii] and have started saying that these are not dark matter free galaxies. Therefore it seems appropriate to not discuss this issue here at length.

As for as mainstream Astrophysics goes, standard interpretation accepts that there seems to be increase of available mass as one moves from inner parts of galaxy towards the outer ones. But within the standard interpretation, the total mass of galaxy is theorized to be concentrated at the center and test particles (stars) are orbiting around the center. Test particles are facing full gravity subject to inverse square distance law while the source of gravity is the central point of galaxy and there is no distinction between inner or outer layers and also it is not deliberated that mass belonging to outer layers has no actual gravitational bearing on this setup and thus, due to non-consideration of important factors, Keplerian drop-off is expected for this system. But since actually observed rotation curve is flat therefore they theorize (or hypothesize) that extra mass, over and above the total mass of galaxy is increasing with increase of distance from the center and to this supposed extra mass they assign the name ‘Dark Matter’.


Following is section II.II.VI  …. The relevant part where I openly disagree that luminosity decreases towards edges of disk is written in bold font. My point is that TOTAL LUMINOSITY of every complete outer layer or BAND remains the same … then flat curves are obtained. And also … then Spiral Structure is obtained.


II.II.VI. Is Dark Matter the failure of Theory?

We conclude that Newton’s Theory, subject to correct application, would have rightly described the rotation pattern of galaxies. Accurate theory already existed but problem of rotation curves of galaxies was never interpreted in the light of relevant part of the available theory. By 1920, when on the basis of famous 1919 solar eclipse experiment, Arthur Eddington and co-authors wrote in their paper that Einstein’s General Theory of Relativity was found superior theory of gravity to Newton’s theory, at that point in time, Relativity Theory did not even have Shell Theorem. Relativistic Shell Theorem was presented in year 1923 or as early as 1921[i]. Yes – it should mean that relativistic shell theorem was available at the time when scientists were dealing with the problem of dark matter. But it seems like the Birkhoff’s Theorem i.e. the Relativistic Shell Theorem does not consider the specific case of gravity field experienced by a test particle which is placed inside a sphere having uniform density which mean that till date relativistic counterpart of Newton’s Theorem XXXIII does not exist. But overall implication of this Birkhoff’s Theorem is that general relativity reduces to Newtonian gravitation in the Newtonian limit[ii].

The problem of rotation curves was within the Newtonian limit and the theory to be applied was Newton’s Theory thus we can accept that, in principle, theory was complete; rotation patterns could have been rightfully interpreted without invoking the need of dark matter. But – it did not happen; rotation curves were not rightfully interpreted. Theoretical Physicists did apply Newton’s theory but missed an important aspect i.e. Theorem XXXIII of the theory. Instead, they applied irrelevant Theorem XXXI. The wrong application of theory was dubbed as incredible discovery of ‘dark matter’ which was basically a ghost object; an unprovable hypothesis that was also found out to be seemingly supportive of few other unprovable conjectures relating to the Big Bang Cosmology and credit of those farfetched findings was assigned to the ‘more accurate’ theory of General Relativity. In this way, Theoretical Physicists extended the wrong application of (Newton’s) simple theory to their so-called ‘precise’ theory (GR) without realizing that they merely interpolated the results of incorrect application of simple theory to their ‘precise’ theory and this thing casts serious doubt on their claim that they do understand their counterintuitive theories.

Dark Matter was thus not the failure of theory. Precisely, it was the failure of correct application of the theory whereas the theory itself was capable for the task. What happened was like that while first time noting the rotational pattern of galaxies, scientists were naturally anticipating Keplerian drop-off in the rotation curves because by that time, it was the only observed pattern. But deviation of actual finding from the expectations did not spark the willingness to review the dynamical considerations even though Babcock (1939) had pointed out the need for the same. Scientists focused their attention towards getting better accuracy of observed data regarding rotation of galaxies but no one questioned in official papers concerning why Keplarian drop-off should be expected at all when galaxy is a whole different structure than solar system. Experimental Scientists were doing their job well as their task was really to gather correct observational data. But Theoretical Physicists were not using their commonsense because commonsense is a despised thing which they officially do not use. At least they should have seriously reviewed the relevancy of Keplerian drop-off for the dynamics of the galaxy.

Experimental scientists were doing their job well and they were presenting their findings along with judgments regarding what they had observed. In 1939, Horace Babcock reported in his PhD thesis that measurements of the rotation curve for Andromeda suggested that the mass-to-luminosity ratio increased radially[iii]. Yes – it was accurate judgment because at least gravitational mass does increase radially in terms of Theorem XXXIII. Babcock was accurate also because he pointed out that new dynamical considerations were required; a right proposal that was not taken seriously. Off course, whole new theory was not required; only requisite thing was to get rid of the Keplerial drop-off anticipations and to reach to the relevant Theorem XXXIII of the already available theory. Likewise, following quote out of Wikipedia article titled “Galaxy Rotation Curve”[iv] also informs that MS Vera Rubin (1970) not only reported her observations but also came up with accurate judgment that observations had the implication that galaxy masses grow approximately linearly with radius well beyond the location of most of the stars.

In the late 1960s and early 1970s, Vera Rubin, an astronomer at the Department of Terrestrial Magnetism at the Carnegie Institution of Washington, worked with a new sensitive spectrograph that could measure the velocity curve of edge-on spiral galaxies to a greater degree of accuracy than had ever before been achieved.[13] Together with fellow staff-member Kent Ford, Rubin announced at a 1975 meeting of the American Astronomical Society the discovery that most stars in spiral galaxies orbit at roughly the same speed,[14]and that this implied that galaxy masses grow approximately linearly with radius well beyond the location of most of the stars (the galactic bulge). Rubin presented her results in an influential paper in 1980.[15] These results suggested that either Newtonian gravity does not apply universally or that, conservatively, upwards of 50% of the mass of galaxies was contained in the relatively dark galactic halo. Although initially met with skepticism, Rubin’s results have been confirmed over the subsequent decades.[16]

Here we note that MS Vera Rubin said in year 1970 that galaxy masses grow approximately linearly with radius well beyond the location of most of the stars.

We know that according to Theorem XXXIII, a test particle placed at a particular depth within a sphere of uniform density will not be gravitationally affected by the outer layers of the sphere (or disk). It means that ‘gravitational mass’ of outer layers can be regarded as non-existent. Now suppose that test particle was placed at the edge of the galactic bulge and then starts moving towards outer area of the disk. This movement towards outer surface will cause regular ‘growth’ in the gravitational mass which according to MS Rubin, will be approximately linear with increase in radius. And yes, MS Rubin was talking about regular linear growth in mass over and above the total luminous mass that, for the purpose of determining the influence of gravity, was already theorized to be located at center. Theorem XXXIII, on the other hand, have the implication of regular linear growth in gravitational mass such that at every depth, the available (gravitational) mass is exerting full gravity from the center. This gravitational mass is not over and above the luminous (observable) mass. One thing Experimental Scientists missed was that they only radially determined the luminosity of disk. Yes radially the luminosity decreases over large distances but great distance with low (per unit distance) luminosity when projected in complete circumference of the outer belt, band or layer then ‘total’ luminosity also should remain the same layer upon layer just like total mass also remains the same layer upon layer. For example, Roberts and Whitehurst (1975)[v] also concluded the same that mass increases linearly towards the outer edge of the M31 galaxy. They had studied southern end of M31 and observed rotation and luminosity, off course, relating to only that southern end and observed, for that part of the galaxy, that luminosity decreases with no decrease of rotational velocity. The Astrophysical Journal (Aug:2011) has published a paper titled “The Luminosity Profile and Structural Parameters of The Andromeda Galaxy”[vi]This paper presents bell shaped graphs of luminosity of Andromeda as recorded along major and minor axis of the disk. Thus luminosity is decreasing only along the line of diameter and so far there is no realization that total luminosity of the outer bands or layers should be almost equal to total luminosity of inner bands or layers. Therefore, there may actually be no increase of mass to luminosity ratio taking place for the outer parts of galactic disks.  

Now we can recall our schematic diagram where mass increased linearly with radius well beyond the central bulge.

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Outer edge of central bulge have 32 square boxes (representing mass). Each succeeding outer layer also has 32 boxes which means that mass is increasing linearly with radius i.e. exact wording of MS Vera Rubin.

This schematic diagram is based on idea that in accordance with Theorem XXXIII, after central bulge, mass should linearly increase so as to give flat rotation curve like graph. The outer layer of the central bulge consists of 32 equal size boxes. Now onward mass should increase linearly therefore each succeeding layer also consists of exact 32 boxes. By random placement of boxes in succeeding outer layers up to 11th layer (after bulge), the basic shape of spiral started to emerge. Rest of the layers, up to 25th, were arranged by placing the boxes in accordance with already emerging shape of spiral.

Here basic spiral shape was achieved but actual spirals of real galaxies are denser and in-between spiral areas are also not empty. Therefore, in real galaxies, mass increases more than linearly and ‘flat rotation curves’ may actually be slightly upward curves throughout most of the disk as we see in the case of M33 which seems to be usual case and these curves are accomplished due to offsetting caused by the inverse distance (from center) law of gravity as applicable within the sphere (or disk). The galaxy rotation is actually an excellent confirmation of the astonishing accuracy of Newton’s Theory. Here we are dealing with the inside of sphere or disk scenario and if we wrongfully consider inverse square distance law, we shall get Keplerian drop-off even though gravitational mass grows linearly. The flat or slightly higher rotation curves and usual spiral structures of galaxies are in great harmony with Theorem XXXIII of Newton’s Principia.

‘Dark Matter’ is thus not the failure of the Theory but can be regarded as failure of counterintuitive regime. It is failure of overrated understanding level of the theory and it is the failure of the idea that counterintuitive ideas are correct and are actually understood when they, intrinsically being ‘counterintuitive’ were not actually comprehensible. Failure was in the unscientific method that assigns reality status to ghost objects. For example, following paragraph from Wikipedia article titled ‘Dark Matter’ shows that they do not treat this ghost object just as a placeholder only to denote a shortage of proper explanation but they take it for a real object that cannot be traced in the real world.

Dark matter is a form of matter thought to account for approximately 85% of the matter in the universe and about a quarter of its total energy density. The majority of dark matter is thought to be non-baryonic in nature, possibly being composed of some as-yet undiscovered subatomic particles.[a] Its presence is implied in a variety of astrophysical observations, including gravitational effects which cannot be explained by accepted theories of gravity unless more matter is present than can be seen. For this reason, most experts think dark matter to be abundant in the universe and to have had a strong influence on its structure and evolution. Dark matter is called dark because it does not appear to interact with observable electromagnetic radiation, such as light, and is thus invisible to the entire electromagnetic spectrum, making it undetectable using existing astronomical instruments.[1]   


[i] https://en.wikipedia.org/wiki/Birkhoff%27s_theorem_(relativity) – This Wikipedia article about Relativistic Shell Theorem says following:

“The theorem was proven in 1923 by G. D. Birkhoff (author of another famous Birkhoff theorem, the pointwise ergodic theorem which lies at the foundation of ergodic theory). However, Stanley Deser recently pointed out that it was published two years earlier by a little-known Norwegian physicist, Jørg Tofte Jebsen.”

[ii] https://en.wikipedia.org/wiki/Birkhoff%27s_theorem_(relativity)

“Thus, this part of the theorem is just what we would expect from the fact that general relativity reduces to Newtonian gravitation in the Newtonian limit.”

[iii] Babcock, H. W. (1939). “The rotation of the Andromeda Nebula”.

[iv] https://en.wikipedia.org/wiki/Galaxy_rotation_curve

[v] http://adsabs.harvard.edu/full/1975ApJ…201..327R

[vi] https://iopscience.iop.org/article/10.1088/0004-637X/739/1/20/meta


[i] https://en.wikipedia.org/wiki/Galaxy_rotation_curve#/media/File:Rotation_curve_of_spiral_galaxy_Messier_33_(Triangulum).png

[ii] https://en.wikipedia.org/wiki/NGC_1052-DF2 — “A more recent study on NGC 1052-DF2 suggests the previously reported distance of the galaxy was greatly exaggerated. Consequently, the galaxy now looks “normal” in every way. Using five independent methods to estimate distances of heavenly bodies, a team of researchers from the Instituto de Astrofísica de Canarias (IAC) found the correct distance of NGC 1052-DF2 to be 42 million light years (13 MPc), not some 64 million light years (19 MPc) from the Earth.[10] The total mass of the galaxy is around one-half of the mass estimated previously, but the mass of its stars is only about one-quarter of the previously estimated mass. This implies a significant part of NGC 1052-DF2 could be made up of dark matter, like any other galaxies.”
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