[Paper Review] Mass Distribution of Spiral Galaxies in a Thin Disk Model with Velocity Curve Extrapolation

Valentin Kostov|arXiv (Cornell University)|Apr 18, 2006

Scientific Research and Discoveries1 references3 citations

TL;DR

This paper proposes a thin disk model that directly calculates the surface mass density of spiral galaxies from their observed rotational velocity curves, using a Keplerian extrapolation beyond the measured radius. The method reconstructs mass distributions with high precision for galaxies like the Milky Way and NGC 3198, showing that mass estimates within 30 kpc are robust to uncertainties in the flat part's extent, while total masses scale with the unknown outer velocity curve extension, implying significant dark matter presence via high mass-to-light ratios.

ABSTRACT

We model a spiral galaxy by a thin axially symmetric disk that includes both visible and dark matter. The surface mass density of the disk is calculated directly from the rotational velocity curve without extra assumptions. We simplify the standard application of the model. Since most velocity curves are known out to some radius, r_{max}, we extrapolate them by attaching a Keplerian tail. The numerical procedure and the extrapolation are tested with a known toy mass density and shown to reconstruct it with a good precision if r_{max} includes a sufficient part of the velocity curve. Mass density curves are calculated for Milky Way and NGC 3198. We vary the extent of the flat part of the velocity curves from 30 kpc to 200 kpc and show that does not affect appreciably the calculated mass density inside r_{max}=30 kpc. The reconstructed masses for Milky Way are 15 x 10^10 solar masses inside the visible disk and 23 x 10^10 solar masses inside 30 kpc. For NGC 3198, the reconstructed mass inside the visible disk is 6.5 x 10^10 solar masses and 11 x 10^10 solar masses inside 30 kpc. The total galactic masses are roughly proportional to the extent of the flat part of the velocity curves which is currently unknown. The high light-to-mass ratios obtained for the visible disks of the galaxies - 11 solar units for Milky way and 9.3 for NGC 3198 - suggest presence of dark matter. The method is also applied to NGC 3031 - a spiral galaxy with a declining velocity curve in which case it is able to reconstruct both the mass density curve and the total mass (14 x 10^10 solar masses).

Motivation & Objective

• To develop a simplified, model-independent method for deriving surface mass density in spiral galaxies from observed rotational velocity curves.
• To test the accuracy of mass reconstruction using a known toy mass density and varying extrapolation radii.
• To estimate the mass distribution and total mass of real spiral galaxies (Milky Way, NGC 3198, NGC 3031) under different assumptions about the flat part of the rotation curve.
• To assess the sensitivity of inner mass estimates to the unknown extent of the flat rotation curve, particularly beyond the measured radius.
• To evaluate the implications of high mass-to-light ratios for the presence of dark matter in spiral galaxies.

Proposed method

• The surface mass density σ(r) is derived directly from the rotational velocity curve v(r) using Toomre's (1963) integral formulation involving Bessel functions and the inverse Hankel transform.
• The method assumes axial symmetry and uses Newtonian gravity, ignoring general relativity and non-axisymmetric features like spiral arms or bars.
• For radii beyond the measured r_max, the velocity curve is extrapolated using a Keplerian decline (v ∝ r^(-1/2)), consistent with expectations from isolated mass distributions.
• The Bessel spectrum of the potential is derived from v²(r), which is then inverted to obtain the surface mass density via the relation σ(k) = -kA(k)/(2πG).
• The accuracy of the reconstruction is tested numerically using a known toy mass density profile, varying r_max to assess sensitivity.
• The method is applied to real galaxies by assuming different extents for the flat part of the rotation curve (from 30 kpc to 200 kpc) and comparing resulting mass densities and total masses.

Experimental results

Research questions

• RQ1 Can the surface mass density of a spiral galaxy be reconstructed directly from its observed rotational velocity curve without assuming a specific dark matter halo profile?
• RQ2 How sensitive are the reconstructed mass densities inside r = 30 kpc to the assumed extent of the flat part of the rotation curve?
• RQ3 What are the implications of the derived mass-to-light ratios for the presence of dark matter in spiral galaxies?
• RQ4 How well does the model perform when applied to galaxies with declining rotation curves, such as NGC 3031?
• RQ5 To what extent do total mass estimates depend on the unknown outer behavior of the rotation curve?

Key findings

• For the Milky Way, the reconstructed mass inside the visible disk (r < 15 kpc) is 15 × 10¹⁰ M☉, and 23 × 10¹⁰ M☉ within 30 kpc, with a V-band mass-to-light ratio of 11 M☉/L☉ for the visible disk.
• For NGC 3198, the mass inside the visible disk (r < 14 kpc) is 6.5 × 10¹⁰ M☉, and 11 × 10¹⁰ M☉ within 30 kpc, with a V-band mass-to-light ratio of 9.3 M☉/L☉ for the visible disk.
• The mass density profiles inside r = 30 kpc remain largely unchanged when the flat part of the rotation curve is extended from 30 kpc to 200 kpc, indicating robustness of inner mass estimates.
• Total mass estimates increase significantly with the assumed extent of the flat rotation curve—e.g., from 15 × 10¹⁰ M☉ to 100 × 10¹⁰ M☉ for NGC 3198—confirming that total mass is highly uncertain due to incomplete outer data.
• For NGC 3031, which shows a declining velocity curve, the method successfully reconstructs both the mass density profile and a total mass estimate of 14 × 10¹⁰ M☉, with 79% of the mass inside the measured r_max = 21 kpc.
• The high mass-to-light ratios (11–22 M☉/L☉) for the visible disks strongly suggest the presence of dark matter, even when extrapolation assumes a Keplerian decline beyond r_max.