<p>I assume what you are talking about is the &quot;rotation curve&quot; of a galaxy, which measures the tangential speed of stars around the centre as a function of galactocentric radius.</p>
<p>The answer to your question comes from simple Newtonian mechanics.
If we assume a spherically symmetric distribution of mass (this would not be a bad assumption within the central few thousand light years of our own Galaxy and others, but the assumption does not affect the principal point of the answer), then the centripetal acceleration is related only to the mass <em>interior</em> to a stellar orbit. If we further assume a circular orbit, with a tangential velocity <span class="math-container">$v$</span> (the thing that is plotted in a rotation curve), then
<span class="math-container">$$ \frac{v^2}{r} = \frac{GM(r)}{r^2} = \int 4\pi G \rho(r)\ dr\ ,$$</span>
where the left hand side is the centripetal acceleration and is equated to the gravitational acceleration due to the mass within <span class="math-container">$r$</span> and this is calculated by integrating the density <span class="math-container">$\rho$</span> over the volume contained within <span class="math-container">$r$</span>.</p>
<p>If the density were constant, you can immediately see that <strong><span class="math-container">${\bf v}$</span> would be proportional to <span class="math-container">${\bf r}$</span>.</strong>
<span class="math-container">$$ v = \sqrt{4\pi G\rho}\ r\ .$$</span>
In other words, the tangential velocity near the centre would be small and would then grow with radius as more and more mass is contained within that orbit and accelerates the star.</p>
<p>The exact dependence of <span class="math-container">$v$</span> on <span class="math-container">$r$</span> will depend on exactly how <span class="math-container">$\rho$</span> behaves with <span class="math-container">$r$</span>, since the assumption that <span class="math-container">$\rho$</span> is a constant, while roughly ok in the central parts of a galaxy, will certainly not work as we move further out into a galaxy.
For example if we imagine a density distribution that dropped to zero suddenly at some radius - then beyond this radius, the mass contained on the right hand side of the equation above would be a constant and <span class="math-container">$v$</span> would be proportional to <span class="math-container">$1/\sqrt{r}$</span>.</p>
<p>What is seen in most spiral galaxies however is that the <span class="math-container">$v \propto r$</span> behaviour rolls over to a flat rotation curve (<span class="math-container">$v$</span> is a constant). From the equation above, this suggests that the mass contained within <span class="math-container">$r$</span> increases linearly with <span class="math-container">$r$</span> and thus that the density decreases roughly as <span class="math-container">$1/r^2$</span>  beyond the central parts of a galaxy, even beyond where the visible matter appears to run out.</p>

CC BY-SA 4.0
2021-12-24T11:35:15.147Z
2021-12-26T15:41:19.673Z
2021-12-26T15:41:19.673Z
ProfRob
<p>Why are Galaxy Doppler Light Curves rising from a low point near the center of the Galaxy? Stars near the center of a Galaxy are moving faster than those near the edge. I understand that we probably can't see (measure the light from) the stars near the center of a galaxy that is exactly &quot;edge on&quot; to our view. But we should be able to measure Doppler effect of light from stars near the center of Galaxies that are tilted a bit to our view.
All the light curves I see in articles show the Doppler light curves indicating stars' velocities to rise from the Galaxy center. I'm not talking about the anomalous flat part of the curve nearer Galaxy edges. Thanks.</p>

CC BY-SA 4.0
2021-12-23T23:02:20.220Z
That’s one of the clues to the existence of so-called “dark matter.” The amounts of normal (baryonic) matter that are observed can’t explain the light curves of galaxies. Hence, it has been postulated that there is some form of matter that is invisible (“dark”) to us. Astrophysicists and particle physicists have been working hard to determine the composition of this dark matter.
Pierre Paquette
CC BY-SA 4.0
2021-12-23T23:25:14.220Z
Your premise is doubtful that the strongest red/blue-shift is and should be found at the core of galaxies. https://en.wikipedia.org/wiki/Milky_Way#Galactic_rotation (see the observed and the expected doppler curves) - there is little central mass compared to the overall galaxy - unlike in stellar systems where virtually all mass is in the star
planetmaker
CC BY-SA 4.0
2021-12-24T07:15:14.433Z
What is a "Doppler light curve"? And where have you seen that term used?
ProfRob
CC BY-SA 4.0
2021-12-24T08:47:16.330Z
@PierrePaquette She said she wasn’t talking about “the anomalous flat part of the curve near Galaxy edges”, so dark matter isn’t relevant. The rising part of the rotation curve in the inner parts of galaxies is mostly or entirely due to the local distribution of baryonic matter.
Peter Erwin
<galaxy><galactic-dynamics>
Why are Galaxy doppler Light Curves rising from the center of the Galaxy? Stars near the center of a Galaxy are moving faster than those near edge
Elsie
# Why are Galaxy doppler Light Curves rising from the center of the Galaxy? Stars near the center of a Galaxy are moving faster than those near edge

(Asked by: username_0 on 2021-12-23)

Why are Galaxy Doppler Light Curves rising from a low point near the center of the Galaxy? Stars near the center of a Galaxy are moving faster than those near the edge. I understand that we probably can't see (measure the light from) the stars near the center of a galaxy that is exactly "edge on" to our view. But we should be able to measure Doppler effect of light from stars near the center of Galaxies that are tilted a bit to our view.
All the light curves I see in articles show the Doppler light curves indicating stars' velocities to rise from the Galaxy center. I'm not talking about the anomalous flat part of the curve nearer Galaxy edges. Thanks.

**Tags:** galaxy, galactic-dynamics
> **Comments:**
> **username_1**: That’s one of the clues to the existence of so-called “dark matter.” The amounts of normal (baryonic) matter that are observed can’t explain the light curves of galaxies. Hence, it has been postulated that there is some form of matter that is invisible (“dark”) to us. Astrophysicists and particle physicists have been working hard to determine the composition of this dark matter.  
> **username_2**: Your premise is doubtful that the strongest red/blue-shift is and should be found at the core of galaxies. https://en.wikipedia.org/wiki/Milky_Way#Galactic_rotation (see the observed and the expected doppler curves) - there is little central mass compared to the overall galaxy - unlike in stellar systems where virtually all mass is in the star  
> **username_3**: What is a "Doppler light curve"? And where have you seen that term used?  
> **username_4**: @PierrePaquette She said she wasn’t talking about “the anomalous flat part of the curve near Galaxy edges”, so dark matter isn’t relevant. The rising part of the rotation curve in the inner parts of galaxies is mostly or entirely due to the local distribution of baryonic matter.  

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## 1 answers

### Answer by username_3 on 2021-12-24. Score: 2

I assume what you are talking about is the "rotation curve" of a galaxy, which measures the tangential speed of stars around the centre as a function of galactocentric radius.

The answer to your question comes from simple Newtonian mechanics.
If we assume a spherically symmetric distribution of mass (this would not be a bad assumption within the central few thousand light years of our own Galaxy and others, but the assumption does not affect the principal point of the answer), then the centripetal acceleration is related only to the mass *interior* to a stellar orbit. If we further assume a circular orbit, with a tangential velocity $v$ (the thing that is plotted in a rotation curve), then
$$ \frac{v^2}{r} = \frac{GM(r)}{r^2} = \int 4\pi G \rho(r)\ dr\ ,$$
where the left hand side is the centripetal acceleration and is equated to the gravitational acceleration due to the mass within $r$ and this is calculated by integrating the density $\rho$ over the volume contained within $r$.

If the density were constant, you can immediately see that **${\bf v}$ would be proportional to ${\bf r}$.**
$$ v = \sqrt{4\pi G\rho}\ r\ .$$
In other words, the tangential velocity near the centre would be small and would then grow with radius as more and more mass is contained within that orbit and accelerates the star.

The exact dependence of $v$ on $r$ will depend on exactly how $\rho$ behaves with $r$, since the assumption that $\rho$ is a constant, while roughly ok in the central parts of a galaxy, will certainly not work as we move further out into a galaxy.
For example if we imagine a density distribution that dropped to zero suddenly at some radius - then beyond this radius, the mass contained on the right hand side of the equation above would be a constant and $v$ would be proportional to $1/\sqrt{r}$.

What is seen in most spiral galaxies however is that the $v \propto r$ behaviour rolls over to a flat rotation curve ($v$ is a constant). From the equation above, this suggests that the mass contained within $r$ increases linearly with $r$ and thus that the density decreases roughly as $1/r^2$ beyond the central parts of a galaxy, even beyond where the visible matter appears to run out.


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