Baryonic Mass Inventory for Galaxies and Rarefied Media from Theory and Observations of Rotation and Luminosity

Round 1

Reviewer 1 Report

The authors present a study of total baryonic mass in the Universe, focusing on gravitational interactions, updated measurements and implementing different approaches. Overall, the manuscript is well written and presents an interesting updated analysis. These novel results are important for understanding how our Universe and its structures evolve. I suggest only few minor modifications, before I recommend the paper for publication:

- line  111-112: please add a reference from the literature about this result, which will be useful for the reader.

-line 239-246: please quantify the referred uncertainties.

-left plot of figure 3: please add error bars, even just a representative point with the mean errors for radius and velocity. 

Author Response

Rev #1. Comments and Suggestions for Authors   

The authors present a study of total baryonic mass in the Universe, focusing on gravitational interactions, updated measurements and implementing different approaches. Overall, the manuscript is well written and presents an interesting updated analysis. These novel results are important for understanding how our Universe and its structures evolve. I suggest only few minor modifications, before I recommend the paper for publication:

Response: Thanks for your input!

- line  111-112: please add a reference from the literature about this result, which will be useful for the reader.

Agreed: We now mention the blackhole mass of 4 x 10⁶ suns provided by new ref [39]

10. D. Wang et al., Dissecting X-ray–Emitting Gas Around the Center of Our Galaxy.Science341,981-983(2013).DOI:10.1126/science.1240755

 

total authors:  Q. D. Wang , M. A. Nowak, S. B. Markoff, F. K. Baganoff, S. Nayakshin, F. Yuan, J. Cuadra, J. Davis, J. Dexter, A. C. Fabian, N. Grosso, D. Haggard, J. Houck, L. Ji, Z. Li, J. Neilsen, D. Porquet, F. Ripple, and R. V. Shcherbakov

 

We also now note that the total mass of Milky Way exceeds the 2 x 10¹¹ solar masses locked in its stars, because of non-luminous matter of varying types, and cite [https://www.nasa.gov/feature/goddard/2019/what-does-the-milky-way-weigh-hubble-and-gaia-investigate] as new ref [40]. 

 

These new citations required renumbering our subsequent references, with those necessary changes being highlighted in yellow.

 

-line 239-246: please quantify the referred uncertainties.

Done. We expanded the paragraph to discuss known uncertainties in RC of Andromeda and Triangulum. Fig. 4 shows these for the Milky Way

-left plot of figure 3: please add error bars, even just a representative point with the mean errors for radius and velocity.

Done, a point and some explanation were added. 

Most importantly, the reviewer's helpful comments motivated us to improve our discussions of uncertainties not only in the RC, but in the inverse model, and in the baryon density results, and to set a minimum for the density in the local Universe.

Reviewer 2 Report

The claimed aim of the study, to inspect the baryonic inventory of the (nearby) Universe, is not achieved due to shortage of approaches: The formula (5) gives the gravitating density within disk galaxies and not the density of the circumgalactic or intergalactic medium. It is the reason why the 'baryonic density' found by the authors (in fact, the gravitating, or dark matter density) is larger than the cosmological estimates. The cosmological estimates of the baryonic density are due, e.g., to the Big Bang nucleosynthesis analysis and not to gravitational effects. The most content of the paper, till 20th page, is trivial and well formulated in 60th-70th of the XX century by many authors (see Bosma 1981 for review e.g.). Moreover, even particular results such as volume density of galaxies in the nearby Universe (Figs11-14) are unreliable because are calculated from the very inhomogeneous extragalactic database NED. Such estimates must be usually made in the frames of dedicated surveys including much more galaxies than the consideration undertaken by the authors. I can recommend to inspect the papers Davis & Huchra ApJ 254, 437, 1982 (Harvard redshift survey); Yasuda et al. AJ 122, 1104, 2001 (SDSS); Karachentsev et al. AJ 127, 2031, 2004 (Local Volume) to see how it must be done. And once more about the main goal of the present study: the baryon inventory in the present Universe can be estimated only from their radiation (or absorptions of other sources).

Author Response

Rev #2 Comments and Suggestions for Authors

1) The claimed aim of the study, to inspect the baryonic inventory of the (nearby) Universe, is not achieved due to shortage of approaches: The formula (5) gives the gravitating density within disk galaxies and not the density of the circumgalactic or intergalactic medium.

Response: The reviewer’s underlined statement is refuted by 8 figures in the paper, all of which have an arrow pointing to the visible edge (defining the disk) and show that the RC data extend much further. Section 2.2.4 discusses CGM and IGM, based on these figures, another 8 galaxies with RC beyond the edge in [38], and summary figures 5 and 6.

In detail: 

i) RC data for Andromeda (Fig. 3) extend to 500 kpc which is 20 times the radius associated with the disk of stars. Obviously circumgalactic media are explored, which grade into intergalactic media. Equation 5 describes Andromeda out to 500 kpc.

ii) Fig. 3 with Fig. 4 shows that RC data extend center-to-center between the Milky Way and Andromeda, and so include intergalactic media. Moreover, Fang et al [75 original number] derived IGM density from the Sculptor wall of the Milky way. Equation 5 describes the Milky Way out to its intersection with Andromeda, and thus includes GCM and IGM.

iii) 6 more figures demonstrating the GCM was indeed sampled are in the appendix (Triangulum, WLM, NGC 6822, And VI, Large Magellanic Cloud, Ursa Minor, NGC 185.)  Given large RC radii for a few of these, as well as their isolation, IGM was sampled.

   Ref [38] analyzes Messier spirals numbers 51, 61, 81, 83, 98, 100, 101, and 106, which have RC data at radii at or beyond the visible edge, and thus sample CGM.

iv) Figure 5b shows that the maximum radius sampled is on average 1.5 times the visible edge so CGM is well sampled. This includes data from RC measurements of ~72 galaxies.

v) The text around Eq. 5 explains that this formula is consistent with RC of diverse galaxies.

Changes made: About 10 lines below Eq. 5, we added

“Equation (5) was obtained by applying Equation (4) to measured rotation curves. The basis of Equation (4) is gravitational stability and the Virial Theorem.”

and some additional discussion.

We clarified Section 2, where possible.

2) It is the reason why the 'baryonic density' found by the authors (in fact, the gravitating, or dark matter density) is larger than the cosmological estimates.

Response:  The Reviewer apparently based this remark on either a misconception and/or an incomplete reading of the ms, as follows:

Perhaps the reviewer missed that 4 independent observational measurements of baryons (Table 1) validates Equation (5) in our model, see Section 2.2.6 for discussion.  In summary, Eq. 5 describes the star-rich region at small distance, ISM at medium distance, GCM at large distance, and IGM at immense distance. Another explanation is that the reviewer believes dark matter exists around galaxies, despite the flaws in the multi-parameter, ambiguous fitting models that assume halos, and does not realize that our model is parameter-free and unambiguous.

Changes made: We inserted subsubsection titles in Section 1.1 to differentiate fitting models from inverse models.

We moved discussion of the central flaw in the halo model from below Eq 3 to Section 1.1.1, as a bullet point, which states:

• “Specifically, any RC can be fit in these forward models by considering that only an NBDM halo exists because a spherical distribution acts as a point mass per Newton [42].Although fitting approaches minimize the NBDM component [43=old 54], this does not quantify the baryonic component, since zero baryons is a valid solution. ”

References were renumbered to address reviewer #1. We shortened Section 2.2.5 for clarity and emphasis.

We streamlined discussion of validation near Table 1.

3) The cosmological estimates of the baryonic density are due, e.g., to the Big Bang nucleosynthesis analysis and not to gravitational effects.

Agreed: We made our descriptions of the cosmological estimates more precise, revising Table 4 and Fig. 15b, and adding 2 citations.

1. i) https://en.wikipedia.org/wiki/Friedmann_equations#Density_parameter
2. ii) “Big-Bang Nucleosynthesis and the Baryon Density of the Universe” by Craig J. Copi, David N. Schramm, and Michael S. Turner 1995 Science Vol 267, pp. 192-199

4) The most content of the paper, till 20th page, is trivial and well formulated in 60th-70th of the XX century by many authors (see Bosma 1981 for review e.g.).

Response:  The reviewer’s description is inconsistent with the contents of our manuscript, as well as with Bosma’s papers, and with the 1960-1970 halo models that he is referring to. 

i) Most of the first 20 pages concern post-2000 papers and new results. In particular, the inverse model that we use here was devised by us in 2015 to 2020. The 1960 to 1980 papers have many problems, as summarized in Section 1.1.1 (see the above bullet point).

ii) Bosma’s two papers from 1981, uploaded in supplementary information, discuss departures of spiral rotation from axial symmetry (warps). We are concerned with the multitudinous objects exhibiting axial rotation, as were most works from 1960-1980.

iii) Bosma states in 1981b that he could not analyze his data presented in 1981a. The reason is obvious: he collected data on objects that differ in all 3 Cartesian directions and so Newton’s spherical formula used in 1960-1980, is not relevant. 

iv) Contrary to the reviewer's assertions, searches using both google scholar and the nasa/harvard databases did not provide any review headed by A. Bosma. The only review that we know of which discussed all models analyzing RC as well as how RC are extracted from Doppler measurements is our 2020 paper [29]. All other reviews focus on their preferred approach.

5) Moreover, even particular results such as volume density of galaxies in the nearby Universe (Figs11-14) are unreliable because are calculated from the very inhomogeneous extragalactic database NED.

Response: Although NED has its limitations, the reviewer has misrepresented the contents of our manuscript.  Figures 11-13 concern luminosity whereas Figure 14 concerns count.  None of these figures are based on volume density (mass per length cubed). 

The reviewer probably means  the number of galaxies per volume, which is a different measure than used in these figures.  Furthermore, we address the uncertainties, focusing on how observational data only sample galaxies with luminosity exceeding some minimum value that increases distance. 

Changes made: To clarify, the first paragraph in Section 3 was rewritten.

6) Such estimates must be usually made in the frames of dedicated surveys including much more galaxies than the consideration undertaken by the authors. I can recommend to inspect the papers Davis & Huchra ApJ 254, 437, 1982 (Harvard redshift survey); Yasuda et al. AJ 122, 1104, 2001 (SDSS); Karachentsev et al. AJ 127, 2031, 2004 (Local Volume) to see how it must be done.

Response: None of his list discuss how luminosity and count change with distance, which was the focus of figs 11-14 in Section 3. His 3 cited papers are uploaded in the supplement.

The content of these papers is unlike the reviewer’s description:

Davis & Huchra 1982 is not a dedicated survey, but instead used the 1968 Zwicky catalog, which is far out of date. They culled this data set by 1/3, to 800 bright galaxies within 80 Mpc.  These data are in two preferred directions, which disagree.  In addition, it is skewed to the Virgo cluster which is extremely dense and does not represent the universe. Importantly, culling prohibits ascertaining the number of galaxies per volume.

In contrast, our ms addressed close-in galaxies (the local group and virgo cluster) in Figures 9 and 10, using modern measurements (McConnachie 2012, and multiple recent studies of the Virgo cluster). There is no reason to include the outdated study of Davis and Huchra.

Karachentsev et al.  2004 measure 451 galaxies within 10 Mpc.  They find a local baryon density lower than the cosmological estimate by a factor of 2.  So they did not solve the missing mass problem, but rather went in the opposite direction.  Why?

They assume the mass luminosity relationship of Roberts and Haynes 1994, which is based on the forward fitting models and thus assumes dark matter exists, while not account for gas in CGM.  The low baryon density of Karachentsev et al.  2004 is a consequence of using flawed RC fitting models from the 1960-1970s. Their study has the same flaws of the 1999 baryonic inventory, and measures few galaxies, and thus was not added to our citations.

 Yasuda et al. 2001 is now added, we thank the reviewer for mentioning this paper.  However, Yasuda et al. 2001 count only bright galaxies, and do not address the observational limitations on brightness vs distance. They cannot do that, as distance is not even considered in their analysis. They assume Schechter’s 1976 function (old ref [80]), that is again based on bright galaxies without accounting for observational limitations.  As stated on p. 1119, Yasuda et als’ survey concerns “100 to 200 Mpc” and the directions sampled disagree.  The number of galaxies the used, after culling, is only 2269 (ibid).  

In contrast, our figures explore the effect of distance, includes large objects, and we analyzed 13399 galaxies, without culling.., We also made a thorough analysis of the observational limitations.

Changes made:  A comparison is useful, so Yasuda et al. conclusion is discussed at the end of Section 5.2.1.  We stated:

“Other assessments exist, but used the pdf of [old 80] which does not account for the size of the smallest galaxy that can be observed depending on its distance squared. For example, Yasuda et al. [2004] deduce from observations extending to ~100-200 Mpc that luminosity is (2.4±0.6)´10⁸ L_sun per Mpc³ (which value includes 7% uncertainty in their fitting parameter h). From Table 3, our count and average L are ~20´10⁸ L_sun per Mpc³, using data out to 50 MPc, which are insignificantly affected by the minimum L required to detect a galaxy. Because the minimum L increases with distance squared (Fig. 13), consistent with the angle subtended (Section 3.4.1), Yasuda et al. and other previous work greatly underestimated L per volume.”                                                                                                      

7) And once more about the main goal of the present study: the baryon inventory in the present Universe can be estimated only from their radiation (or absorptions of other sources).

Response:  This remark contradicts much recent research, including 17 studies that are  noted detailed in the first paragraph of our introduction: 

This introductory paragraph concludes with ref [17] from 2008, who stated that the absorptions are not of WHIM, although detecting this material was the goal. So the absorption approach has failed. It is well known that emissions (radiation) of IGM are inadequate, as described in many of the 16 papers cited in this first paragraph.

Inadequacy of the emission and absorption measurements, as well as models of heating, to resolve the mass discrepancy motivated our direct study of gravitationally derived mass.

Forward fitting models done 50 years ago  cannot meet this need, because these cannot describe CGM or IGM. The assumed NBDM halo constituent overrides all possible baryonic contributions: see [43] (old [54]) and the above bullet point.

 

Reviewer 3 Report

See the attached file.

Comments for author File: Comments.pdf

Author Response

Rev #3 provided a pdf, here converted to ms word

Summary statement: The manuscript inventories the total baryonic mass in the Universe using updated measurements on the rotation curves and luminosity of galaxies. Using an analytic inverse method, the authors analyze rotation curves of 72 nearby galaxies with different sizes and morphologies and show that the mass density of these galaxies within and outside the visible edge follow a power law of 1/r2.

Based on this analysis, the authors compute the total mass contained in a galaxy and its surrounding CGM and IGM and argue that a large amount of baryonic mass is contained in the CGM and IGM gas, which is largely underestimated in previous studies. Therefore, there is no need to introduce non-baryonic dark matter to explain the flat rotation curve at large radii. Then the authors make use of the luminosity data of galaxies within hundreds of Mpc to estimate the representative luminosity and spacing between representative galaxies. Together with mass and density measured from rotation curve analysis, the mean baryon density is  estimated on different length scales. The authors claims that the mean baryon density they obtain lies within the uncertainty of the cosmology

critical density. The idea of this work is interesting. However, I have a few concerns that I would like the authors to address before considering it for further publication in Galaxies.

Response: thank you for the positive remarks. Below we address each comment.

Comment 1. Firstly, in the analysis of rotation curves, the authors assume that the galaxy spin in a way that it can be considered as nested coaxial homeoids. Each homeoid rotates with the same angular velocity. Then the virial theorem is applied to each of the homeoids, i.e. Eq. (3) in the manuscript. Similar approach has been used in several of the authors’ previous work, e.g. Refs. [38,46]. But I  am a bit confused with this treatment:  (1) To get Eq. (3), only the rotational kinetic energy is included. But to maintain the oblate shape, the fluid element must also be subject to a “pressure” term originated from the random motion to balance the gravitational force along z direction (the rotational axis). Otherwise the fluid element in the ellipsoid will oscillate in the z direction. Taking into account the kinetic energy from the random motion, how will Eq. (3), thus the enclosed mass obtained in the manuscript, be affected?

Response:  As summarized in Section 2, Newton and MacLaurin addressed the problem of the shape of spinning oblates - it is solely a gravitational effect, and was derived for fluids, although it holds for deformable solids, like the Earth.  In summary, the rotational motion effectively provides a radial force outwards, whereas gravity goes in all directions inwards. So spin stretches the sphere into an oblate shape whose surface is equipotential.  Pressure at any interior point is irrelevant because fluid is necessarily hydrostatic; otherwise, hydrodynamics requires that the fluid would flow and this would necessarily change object shape.

How much the object is flattened depends on how fast it is spinning. A quantitative relationship of angular speed and flattening was derived by Todhunter, using MacLaurin’s geometrical diagrams. Shape was recently addressed by balancing energy (old number [49]), who showed the historic relations holds for layers and gradational density, and confirmed this against seismic data on the Earth’s layers.

There is no need to invoke pressure or random motions. Pressure is isotropic.  To respond anisotropically to forces, the body must have strength (rigidity), i.e, must be solid.   

Regarding oscillations perpendicular to the axis, nothing like this has been observed for galaxies, and it is not expected.

Changes made:  Many interested in the inventory may be not familiar with literature on galactic rotational motions.  So at the beginning of Section 1.1.1, we added:

 “These Doppler shift “snapshots” document organized, slow rotational motions about the special axis of these essentially uniaxial bodies. Gravitational stability is implicated.”

The shape of spinning bodies is well-known.  We think the text (Section 2.1, e.g. the first pp) is adequate, but we added a sentence to the abstract: “The inverse model is based on the observed axial symmetry of the body and its dynamical consequences.”  

We also read over Sections 1 and 2 and deleted non-essential material, to help with focus.

(2) The Virial theorem is assumed to be satisfied by each homeoid. How accurate is this assumption?

Response:  This is not an assumption. A homeoid is a thin layer bounded by two equipotential surfaces, or else matter would slide around the homeoid until a dynamically stable shape is created. This was understood by Newton, see Binney and Tremain.  Since the homeoid is equipotential, the virial theorem holds for these shells.

(2a) For example, consider a isolated virialized spherical halo, |2K/W| = 1 is usually only satisfied for the whole halo instead of each spherical shell.

Response:  If the halo is spherical, then Newton’s law for spherical symmetry holds.  From Newton, and as described in elementary physics textbooks, the halo can be treated as if it were point mass, and likewise for each and every spherical shell.

That a spherical halo acts equivalently to a point mass is the basis of the dark matter halo proposal.

Perhaps the reviewer was confused by these 4-9 parameter NBDM models, which are much more complicated than is justifiable.  Confusion may also result because external potentials differ from internal, yet external potentials were inappropriately utilized in NBDM models. Stars are part of the galaxy (a continuous distribution) whereas planets are discretely distributed and move independent of each other, except for small perturbations (e.g. Hill circa 1900).

Changes made: We added a subsubheading “1.1.1 Analysis of RC through multiparameter fitting models” to help guide the reader.  We moved discussion of the central flaw in the halo model from below Eq 3 to Section 1.1.1, as a bullet point, which states:

• “Any RC can be fit in these forward models by considering that only an NBDM halo exists because a spherical distribution acts as a point mass per Newton [Halliday and Resnick]. Although fitting approaches minimize the NBDM component [54], this does not quantify the baryonic component, since zero baryons is a valid solution. ”

Below Eq. 4 we reiterate this point, as a prelude to discussing uncertainties. References were renumbered to address reviewer #1 call for MW references.

(2b) Secondly, when estimating the mean baryon density in Section 4, the authors assume that the power-law density profile, ∼ 1/r², can be extrapolated to radii much larger than the visible edge of a galaxy.

Response:  This behavior is not assumed. For Andromeda, data go out to 20 times the visible radius of 20 to 25 Mpc, and likewise for the Milky Way, which together compose the best RC data available.

Six more figures on RC, Figures 5 and 6 summarizing the data, and the independent observational results in Table 1 demonstrate that Eq. (5), which indicates that density goes as 1/r², holds beyond the visible radius.  Our earlier paper has 8 more demonstrations (Messier spirals). 

(2c) For example, for the Local Group, the density profile of Andromeda (Milky Way) obtained in Section 2, Eq. (5), needed to be extrapolated to the representative radius ∼ 1.2Mpc. More discussions are needed to justify this assumption. Otherwise, a caveat should be added.

Response:  Going from 0.5 Mpc to 1.2 Mpc is a small change, given astronomical scales.  The trends in Figures 3 and 4 support our 2x extrapolation. Figures in the appendix show that all spiral galaxies behave similarly, which provides support.

More importantly, Fang et als [76] study of the Sculptor wall shows that Eq. 5 gives too low a density (Table 1).  We improved discussion of  IGM by adding the distance of 127 Mpc to the Sculptor Wall.

(2d) Although the authors have shown in Section 2 that the density profile of Andromeda is well described by the power-law relation up to ∼ 500kpc, without a theoretic explanation or confirmation from simulations, it is difficult to know how well this extrapolation will work at ∼ 1.2Mpc.

Response: We agree and added a theoretical basis below Eq 5, with a subheading so it stands out:

“2.2.4 Physics underlying our r(r) formula

Equation (5) was obtained by applying Equation (4) to RC. The basis of Equation (4) is gravitational stability and the Virial Theorem.

Distant from the galactic center, tangential velocities become constant. The dv/dr term in Equation (4) is thus zero, and the v²/r² term dominates. Because v is constant, r in Equation (4) goes as 1/r², yielding Equation (5). This integer power is associated with each homeoidal shell at large distance having the same mass. Equation (5) describes a gradual decline in density of a stable galaxy out to infinity, or until the dilute media surrounding the next galaxy is encountered.”

 The above provides verification. To make our steps towards validation (against data) clearer, Section 2.2.5 now lumps our calculation of ISM at the visible edge with CGM and IGM densities  to precedes validation in 2.2.6.  The comparison (Table 1) shows that the mass from Eq. 5 is baryonic.  Section 2.2.6 was clarified and slightly reorganized to cover validation from galactic centers outward.

(2e) For example, at large radii, there might be mass infalling due to gravity or mass outflows due to baryonic feedback, which may make the density profile very different from the inner part.

Response: The reviewer’s postulates contradict the accepted interpretation of RC, which are considered to represent gravitational stability of these large and ancient objects. 

Changes made:  At the beginning of Section 1.1.1, referring to Doppler shifts, we added:

 “These Doppler “snapshots” document organized, slow rotational motions about the special axis of these essentially uniaxial bodies. Gravitational stability is implicated.”

(2f) Furthermore, the measurement of Andromeda’s rotation curve have large error bars beyond its visible edge. If possible, an error analysis on the density profile  derived from the rotation curve will be helpful.

Response: We agree that further discussion of uncertainties is needed.

Changes made:  Error bars were added to Fig. 3, which represents about 8 studies. We expanded the discussion on RC uncertainty.  Roughly, 10% uncertainty describes the RC literature (e.g. Triangulum, in the appendix).  Density would also have a similar error.  The power law should not have a larger error, considering the best datasets behave in this manner, and that most datasets have similar powers (Fig. 6).

Our clarified discussion of Eq. 5 is covered above. To Table 1, adding the distance to the IGM probed by Fang et al. In Section 2.2.6 on validation, the discussion is now limited to data and estimates of critical densities are not discussed.

(2g) Thirdly, in Section 4, to estimate the mean baryon density, the authors assumed that the representative galaxies are close-packing. In reality, galaxies form at the intersection of cosmic filaments. Between the filament structure, there are vast spaces (cosmic void) with low densities. In the central region of the void, the density can be even be underdense. Assuming the representative galaxies fill in the space uniformly and the IGM density between galaxies described by the 1/r² profile is likely to overestimate the mean baryon density.

Response: We think the reviewer misunderstood that our close packing includes CGM which grades into IGM of the galaxies. Close packing was invoked because this explains why dwarfs and very small galaxies are unimportant to the inventory.

Regarding filaments, we appreciate the Reviewer alerting us to this material. Here is information on wikipedia:

“In cosmology, galaxy filaments are the largest known structures in the universe, consisting of walls of gravitationally bound galactic superclusters.” Filament width is described as being ~100 Mpc, which is 100 times the spacing of representative galaxies.  With this scale being imaged, individual galaxies cannot be resolved.

Filaments around galaxies also exist [ https://en.wikipedia.org/wiki/Galaxy_filament  ]. Widths are similar.

Filaments are detected by Lyman-alpha line, which is in the vacuum ultraviolet region. Filaments are described as immense clouds of the H gas.  UV line emissions are a response to stimulation from some source, such as stars or galaxies.

Filaments are not part of the 1999 galactic inventory [5] or ours. Both focus on the local universe, due to RC measurements being limited to shorter distances.

The existence of filaments points to additional gas density between galaxies and between superclusters (Section 4.1.4), and emphasizes that our calculation is a minimum.

Changes made: We clarified the caption of Figure 15a, noting that the CGM is an integral part of the galaxies, and that this grades into IGM. Filaments are now mentioned in Section 4.1.4.

 (3) Lastly, in several places, the units of the cosmological critical density quoted by the authors might be incorrect. For example, in Table 4, the cosmological critical density is quoted as 10 × 10−25kg/m3, which is 2 order of magnitude larger than the correct value, i.e. ρc = 3H20 /(8πG) ∼ 10−26kg/m3. Similarly, the density from Cosmology models given in Table 4 seems also 2 order of magnitude too large. The authors should check through their manuscript to make sure the density units are used consistently.

Changes made: Thanks for catching this mistake. We corrected this value throughout the paper, and added primary sources. Table 4 and Figure 15b were accordingly revised.

(3a) With the above correction, the mean baryon density obtained by the authors, 6.2 × 10−25kg/m3, is more than 60 times of ρc. This seems to be unrealistically high and also far exceed the value predicted by the primordial nucleosynthesis theory. Either the units of mean baryon density given in the manuscript are also incorrect, or the issues mentioned in previous points leads to an overestimation of the mean baryon density.

Response: The transcription error (previous point) stems from incorrectly copying a critical density from the literature, which propagated into the (3 to 4%) value for the Big Bang Nucleosynthesis.  So our own values are correct and are indeed much larger than these two estimates from modelling.  Before submitting the paper, we redid the calculations of paper [38] and added more galaxies (the Appendix).

We do not delve into cosmologic or big bang models because the subject of our paper is inventorying the baryons from direct observations and from their gravitational interactions, as expressed in the rotations of galaxies.

Changes made: To address the much larger density of our inventory, some additions were made:

First, we added (to Table 4) a column for a local universe that has no IGM, and discussed this and filaments:

       “4.1.4 Baryon Inventory with and without IGM

Although our calculations set a minimum on baryon density (Sections 2.2.6 and 4.1.2), our values are much larger than previous estimates. Because our calculation of the mass of galaxies and their atmospheres (GCM) from their organized rotational motions (Section 2) is the most robust part of our inventory, we now isolate this contribution, as described in the footnote to Table 4, and list the baryonic density inferred from the stable rotations of galaxies in the rightmost column in Table 4.

Regarding the Local Group, Andromeda and the Milky Way carry virtually all the galactic mass, but this contribution is small compared to the IGM. Larger regions behave likewise. Although IGM is highly rarified (Equation (5); Table 1), these regions are vast and so store much of the local universe’s baryons.

In addition, filaments surrounding galaxy clusters and superclusters have unconsolidated gas [new reference]. Their contribution is not included in the middle column of Table 4 because our study, as previously [5], has focused on matter detected through gravitational interactions.“

Second, we added a new Section 4.5 on the assumptions underlying critical density of 10^-26 kg/m³, and clarified in Sections 5.1.3 and 5.2.2.  Uncertainties in our estimates (called for above) pertain, so this discussion (Section 5.1 and earlier) was expanded a bit. Likewise discussion of uncertainties in the previous inventories (5.2.1) was expanded.

(4) In conclusion, I think the results in this manuscript are interesting. But the author should check through their manuscript for possible typos and clarify the above concerns.

Summary Response:  We fixed the unfortunate error regarding the critical density and cosmological estimates, and believe that we have clarified all parts of the ms that concerned the reviewer. 

The bottom line is that accounting for non-spherical shapes of spirals leads to a higher density than found in the previous inventories, which assume the existence of halos constituted of hypothetical matter of a kind that has yet to be detected in space or in laboratory experiments.  Models assuming halos replace baryon mass with NBDM. The density we find, even if uncertain, far exceeds the critical density of the expanding universe. Given the importance of our finding, our discussions of uncertainties were improved and we also provide a minimum density, by pretending the IGM contribution is negligible.  Even with ignoring the substantial gas in IGM  our densities are close to the critical values.

Thanks for your helpful remarks.  We believe that the ms was substantially improved, and hope you agree.

Round 2

Reviewer 2 Report

The reply of the authors demonstrates that they don't want to recognize the main caveat of the referee concerning their paper submitted to Galaxies. Their approach implies the calculation of the dynamical (gravitating) mass of the galaxies, whether the radial range of the rotation curves covers 20 or 200 kpc. They believe that the source of gravitation is limited only by baryons (stars+ISM+CGM...) but they cannot prove this hypothesis; and the belief is quite insufficient to get a scientific result. The pretension claimed by the title of the paper - "Baryonic Mass Inventory for Galaxies..." - is not supported by the approaches and calculations within the paper. So the aim formulated by the authors is not achieved - and cannot be achieved in the frame of their approach.

English is rather good.

Author Response

Reviewer #2 provided 6 sentences, listed in bold type below.

 Below, each of the reviewer’s sentences is addressed and changes to the ms are described. Changes to the ms were highlighted in green.

 

Reviewer point 1. The reply of the authors demonstrates that they don't want to recognize the main caveat of the referee concerning their paper submitted to Galaxies.

 Response: We provided a detailed, specific response to every comment made by this reviewer in the previous review round, so his claim that we do not “recognize” his criticism is untrue.  Instead, we showed that his criticisms are misguided, on the basis of 1) his misunderstandings of Newton’s findings;  and 2) his belief that NBDM must exist, even though it has never been observed either in space or in the laboratory, despite numerous expensive attempts.

 Changes: we slightly expanded the bullet point in Section 1.1.1 to emphasize short comings of the halo models.

 

Reviewer point 2. Their approach implies the calculation of the dynamical (gravitating) mass of the galaxies, whether the radial range of the rotation curves covers 20 or 200 kpc.

Response: First, some rotation curves extend far beyond 200 kpc. Second, his sentence states that we assume that the dynamic rotational motions of galaxies originate in gravitational forces. All models of RC assume this.

So, this point of the reviewer does not describe any shortcoming in our ms. 

 

Reviewer point 3. They believe that the source of gravitation is limited only by baryons (stars+ISM+CGM...) but they cannot prove this hypothesis; and the belief is quite insufficient to get a scientific result.

Response:   Contrary to the reviewer’s claim, we did not assume (believe) that gravitation is limited to baryons.  Rather, we applied Newton’s laws to the measured RC data, accounting for the oblate shape which first was mathematically described by Gauss.  Table 1 compares our results for density vs distance with density values that were independently and direct ascertained from 8 kpc to 127 Mpc.  The comparison in Table 1 shows that baryons are sufficient to produce the observed RC. We never claimed that NBDM does not exist, but we clearly demonstrated that it is not needed to explain galactic dynamics.

Changes: To make validation of our model even more clear, we added “independent” and “baryon” to the Subsection title where Table 1 resides. This now reads:
“2.2.6 Validation via Comparison with Direct, Independent Measurements of Baryon Densities”

We also make Table 1 more comprehensive by adding another paper (Nicastro et al., 2018) which is widely cited for IGM density.

 

Reviewer point 4. The pretension claimed by the title of the paper - "Baryonic Mass Inventory for Galaxies..." - is not supported by the approaches and calculations within the paper.

Response: Our paper uses the scientific method, and available observations, to explain dynamic behavior, and to count what can be seen.  This time-honored methodology is clearly and correctly reflected in the title “Baryonic Mass Inventory for Galaxies and Rarefied Media from Theory and Observations of Rotation and Luminosity”

 

Reviewer point 5. So the aim formulated by the authors is not achieved - and cannot be achieved in the frame of their approach.

 Response:   There is nothing to address in this comment, as it rests on 1) an incomplete understanding Newton’s work (described above); 2) not realizing that Table 1 validates our extracted density from 8 kpc to 127 Mpc; 3) Unflinching beliefs that NBDM, must exist or the fitting models must be true, without being open to other approaches or ideas.

 

 

Reviewer 3 Report

The manuscript is much improved compared to the previous version. But I am still not fully convinced by the authors' explanations for some of my previous comments.

 

1.    (Response 2b)

As mentioned by the authors, the rotation curves suggest that the mass distribution decreases approximately as 1/r^2 beyond the visible edge of galaxies, but my concern is that this conclusion is based only on gravitational effects. In other words, it does not prove that the large amount of mass measured beyond the visible edge is baryonic matter, e.g. it is still possible that it is composed of dark matter, or the rotation curves can be explained by modified gravity theories. Other observations are needed to determine whether the baryon density still behaves as 1/r^2, rather than has a steeper cutoff, beyond the visible edge of galaxies.

 

2.    (Response 2c)

Assuming the density decreases as 1/r^2, the total mass increases by a factor of 2 when doing 2x extrapolation. So the validation of the extrapolation is not obvious.

 

3.    (Response 2e)

My previous comment may not be very accurate. Let me rephrase it in another way. For the Milky Way and Andromeda, the rotation curves approach to a constant of ~200km/s. If one assumes the rotation velocity keeps constant up to 1.2 Mpc, then a test particle rotating together with the galaxy will take ~6Gyr to complete just one orbit. Considering the age of the Universe, the baryon matter at 1.2 Mpc may not have enough time to reach equilibrium with the inner part of the galaxy, which is a necessary assumption for the calculations done in the manuscript.

 

4.  (Response 2g)

My main point here is that galaxies are distributed along filament-like structure (overdense regions). Therefore, the baryon density measured from rotation curves may not be representative for the mean baryon density on large scales. As mentioned in my previous comment, the matter density is much lower in cosmic voids. A more accurate model should take the realistic spatial distribution of galaxies into account, which may result in a lower mean baryon density that better match the cosmological expectations.

Author Response

Reviewer #3 Comments

The manuscript is much improved compared to the previous version. But I am still not fully convinced by the authors' explanations for some of my previous comments.

Response: Thanks your efforts and prompt response. Below we discuss each comment (shown in bold) and changes made to the ms (blue highlights)

 

1. (Response 2b)

As mentioned by the authors, the rotation curves suggest that the mass distribution decreases approximately as 1/r^2 beyond the visible edge of galaxies, but my concern is that this conclusion is based only on gravitational effects. In other words, it does not prove that the large amount of mass measured beyond the visible edge is baryonic matter, e.g. it is still possible that it is composed of dark matter, or the rotation curves can be explained by modified gravity theories. Other observations are needed to determine whether the baryon density still behaves as 1/r^2, rather than has a steeper cutoff, beyond the visible edge of galaxies.

Response:  Our previous work shows that the 1/r^2 dependence extends well beyond the "visible edge", which is located at a much smaller radius than many Doppler measurements, or of the observed radio edge of H atoms (Figure 1). Our dependence of density on distance is indeed based on gravitation.

Most importantly, this result is validated in Table 1 far beyond the visible radius by comparing our values to independent observations of density (absorption and/or star count).   The 1^st measure is slightly inside the visible edge (the solar neighborhood); the 2^nd is at the visible edge of the Milky; the 3^rd is circumgalactic media; and the 4^th is intergalactic media beyond 100 Mpc. 

“Other observations” describe (1) the middle columns of Table 1 on independent densities, and  (2) Section 3 on luminosity. This section covers behavior of galactic luminosity with distance, which is  independent of our gravitational assessment of RC.

Since IGM seems to be the concern, here is a consensus number: (1.7 to 17) x 10^-27 kg/m^3 from https://www.cfa.harvard.edu/research/topic/intergalactic-medium.  This value comes from the added reference: Nicastro et al. 2018 [77].  Their density for IGM is 10 to 100 times our value at 127 Mpc, so it suggests that we have underestimated the baryon density, as discussed. The postulated sharp drop off in density is not supported by independent measurements of IGM of [76,77].

Along this vein, from the kinetic theory of gas, a sharp drop of is unexpected.  Gas atoms have finite temperature and an associated velocity.  So the H is space will bounce about, reversing direction during a collision, which smooths any imposed discontinuity.

Changes: Table 1 now includes the most recent density determination of IGM in [77]. The subsection 2.2.6 title was modified to stress that our model is verified against independent measures of density. Renumbered references are highlighted grey.

To Table 4, we added a newer (2021) reference re average baryon density for clarity and emphasis. This is from [10], cited in the introduction.

Additional Response:    MOND was addressed just above Section 1.1.1.  This model arbitrarily alters Newton’s gravitational constant, while still assuming spherical symmetry describes a flat spiral.  As in the halo models, unconstrained parameters permit fits to RC.  MOND is not relativistic and there is no evidence that G is altered at slow speeds. 

Changes:  We now mention MOND in the conclusions.

 

2. (Response 2c)

Assuming the density decreases as 1/r^2, the total mass increases by a factor of 2 when doing 2x extrapolation. So the validation of the extrapolation is not obvious.

 Response: The reviewer’s point #3 rephrases this point, so we address this below.

 

3. (Response 2e)

My previous comment may not be very accurate. Let me rephrase it in another way. For the Milky Way and Andromeda, the rotation curves approach to a constant of ~200km/s. If one assumes the rotation velocity keeps constant up to 1.2 Mpc, then a test particle rotating together with the galaxy will take ~6Gyr to complete just one orbit. Considering the age of the Universe, the baryon matter at 1.2 Mpc may not have enough time to reach equilibrium with the inner part of the galaxy, which is a necessary assumption for the calculations done in the manuscript.

 Response: Equilibrium is not part of our model or any model of RC. Our paper does not mention this word. The assumption of all RC analyses is that galaxies rotate stably, since rotation curves depict circular motions, not contraction or expansion.

            Regarding time, gravitation is denoted “action at a distance” as this force is effectively instantaneous.  This behavior has puzzled many great minds.  Our paper cannot address this historic issue, but we note that the time taken for the galaxies to revolve is immaterial to RC analyses.

Changes:  We now note that RC models do not assume equilibrium, 5 lines below Section 1.1 heading. The word "stable" is directly linked to RC in several places in our previous version: we added one more. Dynamics was mentioned in several places.

 

4. (Response 2g)

My main point here is that galaxies are distributed along filament-like structure (overdense regions). Therefore, the baryon density measured from rotation curves may not be representative for the mean baryon density on large scales. As mentioned in my previous comment, the matter density is much lower in cosmic voids. A more accurate model should take the realistic spatial distribution of galaxies into account, which may result in a lower mean baryon density that better match the cosmological expectations.

 Response: This comment motivated us to take a closer look at filament literature.

Filament-like structures are model results, as stated by this reviewer in the first round. Density measurements are spots on what is considered to be walls around galaxies at ~120 Mpc. Beyond ~1000 Mpc or z~0.4, even redshifts are equivocal (e.g., Nicastro et al [1]).  So evidence for galaxy distribution at huge distance is equivocal due to this ambiguity and minimum observed luminosity (Figure 13). Thus, as in previous inventories, we probe the local universe.

Regarding “voids,” observations in the large NED database indicate a spacing of ~1 Mpc between Andromeda size galaxies.  Additionally, Section 4.3 discusses the work of Conselice et al. 2016 whose count suggests a spacing of 0.2 Mpc out to z~8 (~8000 Mpc), but this is for smaller galaxies and is based on Schechter’s mass function. They also evaluated Hubble’s ultra deep field image in their appendix which provides a similar spacing, within uncertainties. Both spacings are incompatible with vast “voids” in the (local) universe that we explore here, as did previous inventories.  Lastly, no region can be devoid of matter as this permits attaining 0 K, in violation of the 3^rd law of thermodynamics.

Thus, the spacing we use is realistic as it is based on observations of the distribution of the galaxies in the local universe certainly out to ~1000 Mpc and possibly out to ~8000 Mpc.

Lastly, Table 4 shows that ignoring IGM (i.e., just inventorying galaxies and CGM) provides more mass than the big bang estimates. The problem with earlier inventories was replacing mass of baryons in and around galaxies with NBDM.  The result disagreed with Big Bang models, motivating much work, including our inventory.

If there is extra mass between galaxies, our inventory would be even larger.

Regarding galaxy interactions, the Milky Way and Andromeda are only 0.765 Mpc center to center, yet their RC are similar to many spirals that are isolated [39] and appendix figures.  Given this behavior and the (3-dimensional)  ~1 Mpc spacing of galaxies our approach reasonably represents baryon density associated galaxies in the local, observable universe.

Changes: We made sure Sections 4 to 6, which discuss the inventory and uncertainties, are clear. We now mention Hubble data in Sections 4.3 and 5.1.2.  Above the list preceding Table 3, to address interactions, we added “Considering isolated galaxies is reasonable because Andromeda and the MW are closer together than r_{representative} yet have RC similar to isolated galaxies probed in [38].”

 

We believe that the ms was improved and hope you agree.