But it actually turns out to be flat…ish

We are bounded in a nutshell of Infinite space: Blog Post #7, Worksheet # 3.1, Problem #4: But it actually turns out to be flat…ish

4. We actually observe a flat rotation curve in our own Milky Way. (You will show this with a radio telescope in your second lab!) This means \(v(r)\) is nearly constant for a large range of distances.

(a) Lets call this constant rotational velocity \(V_c\) . If the mass distribution of the Milky Way is spherically symmetric, what must be the \(M( <  r)\)  as a function of \(r\) in this case, in terms of \(V_c\), \(r\), and \(G\) ?

(b) How does this compare with the picture of the galaxy you drew last week with most of the mass appearing to be in bulge?

(c) If the Milky Way rotation curve is observed to be flat \((V_c \approx 240 km/s ) \) out to 100 kpc, what is the total mass enclosed within 100 kpc? How does this compare with the mass in stars? Recall the total mass of stars in the Milky Way, a number you have been given in your first assignment and should commit to memory.

(a) Now that we redefined \(v(r)\) as \(V_c\), and \(M_{enc}\) will now be \(M(  <  r)\) , these new description will appear in the formulas used previously. Following this, we now start to understand the problem, we simply need to re-express the values set forth by the previous problem with new definitions and understand the interpretations. These equations now leave us with: \[V_c  = \left(\frac{G M( <  r)}{r}\right)^\frac{1}{2} ,\] \[\frac{V_c^2 r}{G} = M( < r)\] the form of the equation in its new iteration with the new definitions for the variables and outputs.

(b) However, as we start comparing the view of the Milky Way, as we saw in previous blog posts and readily available as part of your night sky if you have low light pollution and are in the southern hemisphere, and the mathematical models we have been using the last couple of problems, there seems to be a problem. We have been assuming the Milky Way can be considered a spherical galaxy when it clearly is not, rather it is a Spiral Barred galaxy that has become flat because of billions of years of rotations and conservation of angular momentum. This also conflicts with the fact we have been assuming symmetrical mass distribution when we are also aware of the fact of varying densities of the galaxies depending on the distance from the center.

(c) Now, taking into account the distance of 100 kpc where most of the mass of the galaxy is found, and the apparent speed of the curvature of the galaxy \((V_c \approx 240 km/s ) \), these values can simply be inserted into the equation to find the mass of the galaxy. First, a few constants: \[G_{(gravitational constant)}=6.674\times10^{-8}\frac{cm^3}{g s^2}\] \[2 \times 10^{23} \frac{g}{M_\odot }\] \[ 1 pc = 3.1 \times 10^{18} \frac{cm}{pc}. \]

Using these values, we may start to plug into the general equation, adding in some dimensional analysis conversion factors:

\[\frac{V_c^2 r}{G} = M( < r),\]

\[M ( < r)=\frac{(2.4 \times 10^2 \frac{km}{s} \cdot 10^5 \frac{cm}{km})^2 \cdot 10^2 kpc \cdot 3.1 \times 10^{18} \frac{cm}{pc} }   {6.674\times10^{-8}\frac{cm^3}{g s^2}} , \]

\[M( < r)= 2.7 \times 10^{45} g , \]

\[M( <  r)= \frac   {   2.7 \times 10^{45} g  } {2 \times 10^{23} \frac{g}{M_\odot} } , \]

\[M( <  r)= 1.4 \times 10^{12} M_\odot .\]

Thus we have the calculation for the mass of the galaxy where its majority accumulates, however, it is a known quantity that the amount of stars in the galaxy equates to a mass of \( 10^{10} M_\odot \), so there is a difference of 2 orders of magnitude in this scale. Therefore, there is an enormous amount of matter in the galaxy that is not stars, it could be dust, planets, gases and many other forms of matter, but even high estimates of all the visible matter in the universe does not account for the total mass of the galaxy. Therefore, there is mass in the galaxy that, so far, cannot be observed, but its effects are evident and necessary to explain the speed the galaxy spins at the distances it does. This is dark matter, some unknown substance* in the universe that acts upon it without our being able to ascertain what it is, it could be dark, un-ignited stars, it could be huge amounts of rocks and dust that are in some way invisible, or it could be accumulations and existence of massive quantic particles that seemingly never interact with what we perceive as normal matter. And I thought we were just talking about flat galaxies instead of spherical ones, yet the spherical now starts to make sense when you think where all the extra matter might be.