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This chapter of the video

outlines how to measure the mass of a galaxy from its brightness (the Brightness Method).

As might be expected, the method that physicists actually use to calculate a galaxy's mass from its brightness is considerably more complicated than the approach presented in the video. However, the core principle underlying the method used is that galaxies with greater mass tend to contain more stars and thus tend to be brighter.

A more detailed breakdown of the steps involved in the Brightness Method is as follows:

1. 

Seven Billion Suns
The Brightness Method result of 7 billion Suns is the combined mass of all of the following within a radius of
4.0 x 10²⁰ m: (i) all of the stars, (ii) the hydrogen gas, and (iii) the helium gas. Helium is the second most abundant element in the universe after hydrogen and there is a significant amount of it within Triangulum.

How Accurate is the Brightness Method?
Like many calculations in astronomy, the Brightness Method contains an appreciable amount of uncertainty. However, this fact is not overly significant because the discrepancy between the Brightness and Orbital Methods is so large. Even if the actual mass of the stars and gas within a radius of 4.0 x 10²⁰ m was double the value of 7 billion Suns (a 50% error), there would still be a discrepancy of 32 billion Suns.

In addition to the numerical discrepancy between the Orbital and Brightness methods, the overall pattern of the orbital speeds of the stars within galaxies (constant orbital speed with increasing distance) is fundamentally different from the expected pattern (orbital speed declining with increasing distance). Thus, even if the actual masses of stars in distant galaxies were higher than current estimates, this would only have the effect of moving the plot for expected orbital speed in Figure 6 (chapter 1) upwards. It would not alter the plot's overall pattern so that it matched the observed plot. Thus, the stars and gas alone cannot explain the observed speeds of stars, no matter how large their combined mass is.

Mass-Luminosity Relationship
The graph shown in Figure 13 plots brightness (i.e., luminosity) against mass for individual stars. It shows the well-known mass-luminosity relationship. Although the relationship shown is linear, we have used logarithmic scales on both axes for the purposes of simplification. The actual relationship is

where M_S and L_S are, respectively, the Sun's mass and luminosity. M and L are the mass and luminosity of the star in question. (Note that the exponent 4 is only approximate and sometimes a different one is used, e.g., 3.5 or 3.9.)

Density of Dark Matter
Measurements of orbital speed can be made at distances much farther out than the outermost stars by looking at faint concentrations of hydrogen gas. Physicists have found that the speeds measured remain constant with distance and are much higher than expected far beyond where the stars end.

From the shape of the resulting graph of orbital speed against orbital radius, physicists have determined that the total mass of dark matter, M_dark, within an orbital radius of r increases linearly with r

As dark matter gravitationally attracts other dark matter, it tends to be found clumped together. As a result, in the image of dark matter (Figure 14) that appears near the end of this chapter of the video, the density of dark matter is greatest at the centre and gradually decreases as we move farther out.

Is Dark Matter the Same as Dark Energy?
Dark matter is distinct from dark energy, a recently discovered unseen energy that many physicists also think makes up a large fraction of the universe. Dark energy is anti-gravitational and is thought to be making the universe expand at an ever-increasing rate.

Dark Matter within Triangulum
As stated in the video, there are 7 billion Suns of luminous mass in Triangulum within a radius of 4.0 x 10²⁰ m and
39 billion Suns of dark matter within the same radius. There are very few stars beyond this point, although small quantities of hydrogen can be found farther out. Dark matter extends far beyond 4.0 x 10²⁰ m and so, overall, there is much more than 39 billion Suns of dark matter within Triangulum.