Figure 1. Profile view of our Galaxy (ESA image).

Our Galaxy, like most spiral galaxies, is composed of a bulge (highest stellar density, at the central part), the spirals that are mostly included in the same plane with a higher stellar density when we approach this plane (thin and dense disc) and less dense thick disc surrounding it. Moreover, the whole is included in an even lower stellar density sphere or halo shown in Figure 1.

We can clearly see in Figure 1 the plane of symmetry of our Galaxy (z = 0).

If we were able to see our Galaxy from the outside, we could see the spiral shape of it on a face view, as shown in Figure 2.

Figure 2. Face view of our Galaxy (ESA image).

Our Galaxy is composed of several arms, linked to an Inner Ring, itself surrounding a bar-shaped Bulge at the center.

Our Sun is located at approx. 8.2 kpc from the center of the Galaxy [17] and can be considered as part of the symmetric plane of the Galaxy (it is lying 25 pc above the galactic plane [18]). Our Sun mass, M_☉, is estimated to 1.99 × 10³⁰ kg [19].

Also, our Galaxy turns clockwise (for Figure 2) in a very regular move: we estimate the galactic period of the Sun to be equal to approx. 240 million years [20], so our Galaxy has already completed dozens of turns since its creation.

Although there are vertical motion and an asymmetric drift slightly impacting the rotation curve [21], we will assume in this paper that the rotational motion is being fully planar (in the z = 0 plane of our Galaxy).

We will, as a main approx. consider that the Milky Way is a disc of 15 kpc radius and 2 kpc height and we will neglect the mass outside this disk for the study of the rotational motion of our Galaxy (the mass for 15 kpc < r < 26.5 kpc is estimated under 25 × 10⁹ M_☉, when the mass for r < 15 kpc is estimated to 95.9 × 10⁹ M_☉, see detailed explanations at the end of §2.4).

DM “disc” models are often using linear relation between mass and radius to determine the mass of our Galaxy represented as a disc of radius r. For example, this equation, used in [22] and [5], is directly coming from Newton’s law:

M(r) being the mass for the disc of radius r;

V(r) the circular speed at radius r;

G the gravitational constant.

This method has a lot of disadvantages. First, when the circular speed curve is flat (this is the case for Milky Way for 7 kpc < r < 15 kpc), the mass gradient d M d r is constant. It means that the mass grows linearly with the radius inside our Galaxy, up to immense radius values.

This unbalanced distribution is clearly not optimal and appears to be illogical after billions of years of circular motion of our Galaxy.

Also, it means, for the DM disc model, that the Mass gradient d M d r will drop abruptly for r > 20 kpc (when squared circular velocity drops far faster than r increases) and even become negative.

We expect a mass gradient to be decreasing slightly starting the moment when circular velocity is decreasing, and this is not the case with this model.

If you consider the mass distribution as only dependent on squared circular speed and radius (as shown Equation (1)), it means that when the circular speed drops, mass will also decrease.

For example, with this equation, if we consider that the circular speed is equal to 197.56 km·s⁻¹ at r = 21.5 kpc, then 175.68 km·s⁻¹ at r = 26.5 kpc (following [5] data), it means that:

Following this model and calculation method, the mass decreases when volume increases.

As explained at the end of §2.1, we will consider the rotational motion of our Galaxy as being fully planar.

It means that our model will be a 2d-model located inside the symmetry plane of our Galaxy, representing the volume masses of our Galaxy with bodies and allocated masses.

To do this, we needed the most accurate face view of our Galaxy, based on Gaia DR2. In 2019, for the first time, Khoperskov [23] provided evidence (see Figure 3) of the imprint left by spiral arms and resonances in the stellar densities not relying on a specific tracer, through enhancing the signatures left by these asymmetries, using coordinate space densities and radial velocity distributions.

Figure 3. Milky Way spiral arms and bar resonances revealed by Gaia DR2. Based spiral arms shown by symbols with error bars (Khoperskov, 2019).

So, for the spiral and Inner Ring area, we decided to locate all the bodies of our model on the local center of gravity lines, i.e., the arms axis (dotted lines below, Figure 4, MilkyWayGalaxyrenderingbyRobertHurtandNickRisinger [23]).

Figure 4. Location of local centers of gravity (Khoperskov, 2019).

Our model has 3324 bodies, distributed equidistantly, represented hereafter in Figure 5.

Figure 5. Inner Ring, Bulge and arms bodies (distances are in m).

We then need to allocate masses to each of those bodies.

In this article, we will not consider Dark Matter mass (or consider it as negligible).

We will compare our simulation data with a “DM + Baryonic” mass model (inspired by [5]) and a “Stellar + Gas” mass model (inspired by [24]).

For the “DM + Baryonic” mass model, as in [5], we choose a power-law with a power of α = −2.25:

where r is the distance to the galactic center and ρ 0 , the surface mass density at the galactic center (at r = 0), here equal to 41 × 10⁹ M_☉ kpc⁻² (following the value used by Miyamoto [25]).

For the “Stellar + Gas” mass model, we take as a reference the Surface density as a function of radius as measured by SEGUE G-dwarf samples and consider the gas mass, with an additional Surface Density (SD) of 13 M_☉ pc⁻² (0.013 × 10⁹ M_☉ kpc⁻²) for all r < 15 kpc, following [24].

After several iterations (method explained in §2.9), we arrived at a surface density profile (the horizontal lines in Figure 6) slightly different from the “Stellar + Gas” mass model (red left curve in Figure 6), with areas having a growth factor linked to the arm’s width (see explanations in Appendix1). The simulation surface density profile is yet always lower than the DM + Baryonic surface density profile (yellow right curve in Figure 6).

Figure 6. Surface Density (SD) simulation data Versus “Stellar + Gas” and “DM + Baryonic” models.

Hereafter in Table 1 is summarized the mass distribution in the Milky Way for r between 4.2 and 15 kpc (following surface density profile from Figure 6).

Table 1. Stellar & Gas mass distribution outside the Bulge & Inner Ring.

The total dynamical mass M_total of our model is then equal to:

M_Bulge is given in §2.5 while M_InnerRing is given in §2.6 and M_Spiral in §2.7.

The Milky Way mass for r > 15 kpc is not above 25 × 10⁹ M_☉, following our power-law (calculation given in Appendix3).

The Bulge has a “bar-shaped” aspect [26] that will increase gravitational forces along that bar’s axis (asymmetrical shape impact).

We assume that Bulge mass M_Bulge is equal to 19.6 billion M_☉, following Bulge E values used in [5].

From Portail [16], we can also observe a clear gap between surface density profile following the minor axis and the one following the major axis of the bulge.

Figure 7. Surface Density (SD) profile of the Bulge, following minor and major axes (Portail etal., 2017) [16].

We will consider the mass distribution of Figure 7 for the Bulge meshing cartography.

We will consider 4 different zones as described below in Figure 8.

Figure 8. Bulge bodies (distances are in m).

Zone 1, with a “major axis” diameter of 1.15 kpc and a “minor axis” diameter of 0.66 kpc, will be represented by one body, at the center of the Galaxy with a mass equal to 3.2 × 10⁹ M_☉ (SD = 5 × 10⁹ M_☉/kpc²).

Surrounding Zone 1 is Zone 2 with a “major axis” diameter of 3.63 kpc and a “minor axis” diameter of 1.87 kpc, will be represented by 4 bodies, each with a 1.25 × 10⁹ M_☉ mass (SD = 1 × 10⁹ M_☉/kpc²).

Surrounding Zone 2 is Zone 3 with a “major axis” diameter of 6.71 kpc and a “minor axis” diameter of 2.97 kpc, will be represented by 8 bodies, each with a 0.5375 × 10⁹ M_☉ mass (SD = 0.4 × 10⁹ M_☉/kpc²).

Surrounding Zone 3 is Zone 4 with a “major axis” diameter of 11.66 kpc and a “minor axis” diameter of 5.06 kpc, will be represented by 16 bodies, each with a 0.44375 × 10⁹ M_☉ mass (SD = 0.25 × 10⁹ M_☉/kpc²).

The total mass for the Bulge is then:

M B u lg e = M Z o n e 1 + M Z o n e 2 + M Z o n e 3 + M Z o n e 4 = 3.2 + 5 + 4.3 + 7.1 = 19.6 × 10 9   M ☉ , as expected.

We will consider a mass of the Inner Ring M_InnerRing equal to 21 × 10⁹ M_☉ (calculation explained in Appendix2).

We place the bodies equidistantly on the center of gravity line representing the Inner Ring (thanks to GMesh tool), as shown in Figure 9.

Figure 9. Inner ring and bulge bodies (distances are in m).

Inner Ring is represented by 167 bodies, with 0.22 kpc between the bodies.

For the arms, we will take a more refined value of 0.07 kpc for the inter-body distance. This is far above the 0.0013 kpc median distance between stars in our Galaxy and far under the 1 kpc minimal distance between arms.

The total mass for the arms M_Spiral will be 55.28 × 10⁹ M_☉ as explained in Table 1. The allocated mass per body follows strictly the distribution per arm and per area described in Table 1.

A single body represents roughly all the mass around including stars systems, rogue planets, dust, gas, … (in a box delimited by the half-distance with others nodes of the arm.)

Our main hypothesis is to consider that our Galaxy is not axisymmetric.

While studies already exist regarding the asymmetric distribution of gas in our Galaxy and its impact [27], this study extends this concept to all baryonic matter.

Even if the variations of stellar density are low (See Figure 10 from Khoperskov [23]), we can observe many non-axisymmetric areas in our Galaxy.

We can observe that there are many “junction areas” (between arms and between arms and the Inner Ring), and even the arms do not have all the same shape or length.

As the gravitational forces are mostly responsible for the global rotational movement of our Galaxy, we can imagine that in some areas, the circular movement will be almost uniform, while in other areas it will not.

For the “spiral area” (outside the Inner Ring), we will consider the rotational movement as being uniform (radial speed being negligible), but inside the “junction areas” we will consider that the radial speed is too high to only consider the tangential movement (see Figure 11, based on Milky Way Galaxy rendering by Robert Hurt and Nick Risinger [23]).

Figure 10. Stellar density in the Milky Way (Khoperskov, 2019).

Figure 11. Areas without uniform circular motion (Khoperskov, 2019).

We will exclude those areas in our model post-processing.

We will only consider the areas in the spiral zone, where the inner and outer arms are not too close (in order to avoid gravitational impact on radial velocity and rotational motion).

Also, we will exclude the nearby bodies (assuming the average distance between arms is equal to 2 kpc, we will exclude the contribution of all bodies closer than 1 kpc).

This is only to avoid having data “fog” and to better see the velocity trend on the curves, as shown in Figure 12.

Figure 12. Results with nearby bodies (left side) and without nearby bodies (right side).

For the uniform circular motion outside those areas, we will use the equation of uniform circular motion:

a_r being the radial acceleration (in m·s⁻²);

v_rot the rotational speed (in m·s⁻²);

r the radius (in m).

At each body of our model (with position I [x(i); y(i)] and mass m(i)), we will calculate the contribution to radial acceleration given by each other body (with position J [x(j); y(j)] and mass m(j)) by

G being the gravitational constant, equal to 6.7 × 10⁻¹¹ m³·kg⁻¹·s⁻².

Positions of I [x(i); y(i)] and J [x(j); y(j)] are given relatively to the origin point O [0; 0], in m.

m(j) the mass of the point J, in kg.

O I J ^ the angle between IO and IJ (allowing us to know the acceleration projected onto radial axis).

By summing the radial accelerations contributed by each distributed element, we can thus find total radial acceleration at point I (“N” being the number of bodies in the model):

Following Equation (3), the associated speed of rotation is:

Then,

The iterative calculations will allow us to reach the results obtained by Gaia data (rotation speeds of stars). Gaia data are given at z = 0 plane with median values of rotational speeds.

The simulation results will also be median values for an area equivalent to a square with the size of inter bodies distance (0.07 kpc).

As we can see for the Local Arm, the arm’s stellar density is maximum for a width that can be above the inter bodies distance, so we will have to apply a growth factor on arms with a spread stellar density (see Appendix1).

The iteration calculations principle is simple: we will target the desired results and try to reach them by iterating on our model (bodies locations or mass distribution).

Hereafter, in Table 2, are the rotational speed median values for 6 kpc < r < 15 kpc area coming from Gaia DR3 summarized in [5]:

Table 2. Milky Way Gaia DR3 measurements data.

Those data are then incorporated into our results graph, in order to see the gap between the obtained simulation results and the real data from Gaia DR3 (crossed lines in Figure 13).

Gaia Data

Systematic uncertainties from Gaia data DR3 rotational speeds are limited, especially for r < 15 kpc area, as last studies on Milky Way Rotational Curve show gaps of less than 5% between studies [5].

Mass distribution models

We have also decided to choose a model for Bulge mass distribution, given by [5], which could have lot of uncertainties for r < 4 kpc, but the shape impact of the Bulge will decrease with r > 4 kpc (as the impact of the Bulge on the results, see §4.2).

For the Disc (including Inner Ring) surface density profile, we have used, as a basis, a power-law that contains all the baryonic components (Stellar, Gas and Dust). We estimate global uncertainties of ~10% in the overall mass estimation for the disc.

Figure 13. Gaia Results incorporated in the simulation results for different iteration calculations.

Uniform circular motion

We consider the rotational motion as being sufficiently “slow” (one turn every 250 million years) to conduct a static study on rotational speed. This means we consider the object’s location as static even while studying the velocities of those objects.

As we consider a 0.4 kpc = 1.23 × 10¹⁶ m approximation for distances, dividing the distance by an average rotational speed of 200,000 m·s⁻¹ yields an average time of 6.15 × 10¹⁰ s, which is approx. 1950 years of motion without exceeding the distance uncertainties.

However, even with this good approach, the radial component of speed is never negligible (see §4.2) and could introduce an uncertainty in rotational speed comparable to the ratio between radial and rotational speeds (up to 20%).

2D model

We do not consider the vertical velocities (i.e., along the z axis), nor the vertical locations of the Milky Way objects.

We also do not consider the vibrations or modal motion that impact the Milky Way motion.

Furthermore, we exclude from our model the dwarf galaxies or other star clusters (and more broadly, everything that is outside |z| < 3 kpc and |r| < 15 kpc). We consider their masses as being negligible and having no impact on the |z| < 3 kpc and |r| < 15 kpc disc.

This adds another < 5% uncertainty to the results.

A solution against uncertainties: iterative calculations

All those uncertainties combined could lead to an impact of approx. 30% on the rotational speeds.

Nevertheless, the iterative method helps us to converge to the level of uncertainties due to Gaia Data, because our calculations and iterations will be driven by that data.

Then, we can state that we have the uncertainty level of [5] study, i.e., around 10%.

Scilab is a free and open-source platform that allows large-scale simulation.

This tool is a good solution for N-body simulation, as it allows coding without using declarative programming, and offers a large choice of post-processing graphics.

It is one of the most well-known alternatives to MATLAB tool.

We divided the code files by areas (5 files for the 5 arms of our Galaxy: Local, Perseus, Centaurus, Sagittarius and Cygnus).

For all the files, we will first establish the constants (gravitational constant, masses, coordinates of the bodies …), as displayed exemplarily in Figure 14:

Figure 14. Constants used in Scilab files.

Then we declare the radial acceleration, rotational speed, and distance matrix (initially set to 0). We then launch a double loop: for all the points located in the arm (for example, hereafter the Local Arm in Figure 15), we will calculate the contribution to radial acceleration given by each of the other bodies in the model:

We can also see in Figure 15 the calculation of the cosine that retrieves the radial acceleration for all bodies.

Figure 15. Calculations in Scilab files (Local Arm file).

As shown in Figure 16, for all the arms, the Gaia observed circular speeds are reached, even if the curve allure is not matching everywhere (non-axisymmetric behavior explained in §4.2):

Figure 16. Milky Way circular velocities from N-body simulation.

Rotational speed with Axisymmetric model

By using Table 1 and Table 2 data and by using Equation (1), we obtain the results given in Table 3:

Table 3. Axisymmetric circular velocity with simulation data.

From Figure 17, we clearly see that the results are far lower with an axisymmetric model:

Figure 17. Milky Way Circular velocities from axisymmetric model with N-body simulation data.

With an axisymmetric model, we lose on average 20% on the circular velocity and need an additional 36% on global mass (following Equation (1)).

In the “disc model” with Dark Matter, all the contributions on the radial acceleration are positive, i.e. all the parts of the Milky Way have a positive contribution to the rotational speed, and the total velocity is basically the sum of all the “partial velocities” (Sum of DM “Einasto” + Baryon or sum of Bulge + Disc + Dust + Gas = Total velocity, show in Figure 18 taken from [5]):

Figure 18. Milky Way circular velocities from Jiao etal. 2023 [5].

We can do exactly the same with our calculations, knowing that our bodies are numbered together per area. For example, in Figure 19 we show the bodies used for the Bulge (ID between 1 and 29).

Figure 19. Meshing file extract, Bulge bodies’ ID.

In order to calculate the rotational speed of a specific arm, we will first plot the contribution due to Bulge and Inner Ring (+ curve), then plot the contribution due to Bulge + IR + Perseus + Local (o curve), then the contribution due to Bulge + IR + Perseus + Local + Sagittarius (*curve) then the same with Bulge + IR + Perseus + Local + Sagittarius + Centaurus (x curve) and finally adding Cygnus contribution to retrieve the total rot speed.

For example, for the Local Arm, the results are shown in Figure 20 (arrows are added to clarify the contributions).

Figure 20. Local arm circular velocities.

We can see in Figure 20 that all the arms have a positive contribution to the Local Arm radial acceleration, except the Perseus Arm (the outer arm of the Local Arm), which globally decelerates the Local Arm.

In contrast, the Sagittarius Arm and the Centaurus Arm (which are the inner arms of the Local Arm) contribute significantly to radial acceleration and, consequently, circular velocity.

It is also interesting to note that in the case of the Local Arm, it is the Cygnus Arm that helps maintain the circular velocity for r > 9 kpc, because the “start” of the Cygnus Arm is located near this area (see Figure 21).

Figure 21. Cygnus and local arms location.

We also see a little “bump” in the Sagittarius contribution for r ≈ 11 kpc, which is due to a rapprochement between the two arms in this area.

For the Perseus Arm, all arms will have a positive contribution to rotational speed, as shown in Figure 22, except for the r < 9.5 kpc area, where the Cygnus Arm has a small negative impact:

Figure 22. Perseus arm circular velocities.

This is also consistent with our model, as we can see in Figure 23: the Perseus Arm is the outer arm of all the other arms, except for the Cygnus Arm and for r < 9.5 kpc.

Figure 23. Cygnus and perseus arms location.

For the Sagittarius Arm, we see in Figure 24 that Local and Perseus Arms have large negative impact (outer arm), whereas Centaurus gives an important contribution and Cygnus starts with a positive contribution for r > 8.5 kpc.

Figure 24. Sagittarius arm circular velocities.

Again, this positive impact of the Cygnus Arm is explained in Figure 25 by the location of the start of the arm.

Figure 25. Cygnus and Sagittarius arms location.

For the Centaurus Arm, the situation is less easy to analyze. The positive contribution of the Cygnus Arm (the inner arm) is clearly visible for r > 8.5 kpc. For 7 kpc < r < 8.5 kpc, the results are not considered, as we are in a “junction area” (junction between Cygnus and Centaurus Arms, indicated by the gray area in Figure 26). For the area r < 7 kpc, the main contributors to radial acceleration and circular velocity are the Bulge and Inner Ring, while the rest of the Galaxy tends to decelerate the Centaurus Arm:

Figure 26. Centaurus arm circular velocities.

We can also see that the Sagittarius contribution to radial acceleration starts to be positive for r > 10 kpc, which corresponds to the tail location of the Sagittarius Arm, as shown in Figure 27.

Figure 27. Centaurus and sagittarius arms location.

At last, for the Cygnus Arm, for r > 12 kpc we can see in Figure 28 that the Cygnus contribution to itself drops:

Figure 28. Cygnus Arm circular velocities.

In fact, this is where the “secondary Cygnus Arm” is ending as shown in Figure 29:

Figure 29. Cygnus arm location.

For the mass distribution in the spiral area, we have started using a power law with a power of α = −2.25 with this mass distribution, summarized in Table A1:

Table A1. Power law mass distribution in the spiral area.

After iterations, we arrive at this mass distribution, summarized in Table A2:

Table A2. Simulation mass distribution in the spiral area.

This important change, especially for 6.8 kpc < r < 11 kpc area, could be explained by a growth factor linked to the arm’s width (if the mass density is spread equally in the arm’s width, then the bodies’ location of inner arm is less beneficial to increase the rotation of outer arm).

If we consider all the arms with a width e and with also inter-arm distance equal to e, the best-case scenario would be to have half the mass of inner arm at e − e 4 and the other half at e + e 4 .

Then the new radial acceleration contribution from inner arm to outer arm will be:

It means the grow factor on spiral masses could be up to 1.21 ( ≈ 8 9 + 8 25 ).

Even with this growth factor, the mass distribution does not fit with our power law for 6.8 kpc < r < 11 kpc area, and does not allow us to fully use the Baryonic mass Surface density profile.

For the determination of Inner Ring mass, we have taken into account the spiral mass for r < 6.8 kpc (equal to 5.1 × 10⁹ M_☉, see Table 1) and deduced it from the total mass between 4.2 and 6.8 kpc (9.4 + 9.4 + 7.3 = 26.1, see Table A2), then we obtain an Inner Ring mass equal to 21 × 10⁹ M_☉.

For the determination of the mass of the Galaxy for r > 15 kpc, we keep the same power law (Equation (2)), integrated between r = 15 kpc and r = 26.5 kpc.

This mass m 15   to   26.5 is defined as (integration of Equation (2) between 15 and 26.5 kpc):