Observational Evidence of the Dark matter Existence - Revision history

Cosmo All: /* Problem 5 */

2012-10-16T09:34:18Z

‎Problem 5

Cosmo All at 11:45, 2 October 2012

2012-10-02T11:45:53Z

Cosmo All: /* Problem 1 */

2012-10-02T11:44:33Z

‎Problem 1

Cosmo All at 11:43, 2 October 2012

2012-10-02T11:43:44Z

Cosmo All at 11:38, 2 October 2012

2012-10-02T11:38:45Z

Cosmo All: /* Problem 1 */

2012-10-02T11:37:35Z

‎Problem 1

Cosmo All at 11:36, 2 October 2012

2012-10-02T11:36:59Z

Cosmo All: Created page with "1"

2012-10-02T10:47:04Z

Created page with "1"

New page

[[Category:Dark Matter|1]]

@@ Line 93: / Line 93: @@
 === Problem 5 ===
 Show that the age of the matter--dominated Universe contradicts observations.
-<div class="NavFrame collapsed">
+<!-- <div class="NavFrame collapsed">
    <div class="NavHead">solution</div>
    <div style="width:100%;" class="NavContent">
      <p style="text-align: left;"></p>
    </div>
-</div></div>
+</div> --></div>
@@ Line 105: / Line 105: @@
 <div id=""></div>
 <div style="border: 1px solid #AAA; padding:5px;">
 === Problem 6 ===
 A galaxy has the visible mass of $10^{11}M_\odot$ and a horizontal rotation curve up to distance of $30kpc$ at velocity $250km/s$. What is the ration What is the dark to visible mass ratio in the galaxy?

@@ Line 33: / Line 33: @@
 <div id=""></div>
 <div style="border: 1px solid #AAA; padding:5px;">
-=== Problem 1 ===
+=== Problem 2 ===
 What radial dependence of the density of a spherically symmetric galactic halo corresponds to the constant velocity of satellite galaxies?
 <div class="NavFrame collapsed">
@@ Line 49: / Line 49: @@
 <div id=""></div>
 <div style="border: 1px solid #AAA; padding:5px;">
-=== Problem 1 ===
+=== Problem 3 ===
 We have noted above that outside the optically observed range of a galactic disk the rotation curves of the dwarf satellite galaxies do not depend on the galacto-centric distance (the flatness of the rotational curves). External regions of the galaxies are mostly filled by cold neutral hydrogen. Evidently the rotation velocity of the gas gives important information about the mass distribution in the galaxy. How can one measure that velocity?
 <div class="NavFrame collapsed">
@@ Line 63: / Line 63: @@
 <div id=""></div>
 <div style="border: 1px solid #AAA; padding:5px;">
-=== Problem 1 ===
+=== Problem 4 ===
 Using the virial theorem, express the mass of a galaxy cluster through the observed quantities---the average velocity of galaxies in the cluster and its size. Estimate the mass of the Coma cluster (Berenice's hair) for $R\approx10^{23}m$, $\langle v^2\rangle^{1/2}\approx2\cdot10^6m/s$.
 <div class="NavFrame collapsed">
@@ Line 91: / Line 91: @@
 <div id=""></div>
 <div style="border: 1px solid #AAA; padding:5px;">
-=== Problem 1 ===
+=== Problem 5 ===
 Show that the age of the matter--dominated Universe contradicts observations.
 <div class="NavFrame collapsed">
@@ Line 105: / Line 105: @@
 <div id=""></div>
 <div style="border: 1px solid #AAA; padding:5px;">
-=== Problem 1 ===
+=== Problem 6 ===
 A galaxy has the visible mass of $10^{11}M_\odot$ and a horizontal rotation curve up to distance of $30kpc$ at velocity $250km/s$. What is the ration What is the dark to visible mass ratio in the galaxy?
 <div class="NavFrame collapsed">

@@ Line 115: / Line 115: @@
 $$
 where $M_{_{DM}}$ is the dark matter mass (the hidden mass),
-$R = 30 \mbox{\it kpc},$ $M_b = 10^{11} M_ \odot.$ Then one obtains
+$R = 30 \mbox{ kpc},$ $M_b = 10^{11} M_ \odot.$ Then one obtains
 $$
 \frac{M_{_{DM}}}{M_b} = \frac{Rv^2}{GM_b} -1\approx 3,4.

@@ Line 25: / Line 25: @@
 File:Rotationcurv.JPG|
 </gallery></p>
    </div>
 </div></div>

@@ Line 2: / Line 2: @@
-_NOTOC_
+__NOTOC__
@@ Line 22: / Line 22: @@
 \[v(r)\sim 1/\sqrt{r} \]
 The rotation velocity $v(r)$ can be in particular determined by the measured Doppler shift in the emission spectrum of the corresponding satellite galaxies. The behavior of the observed rotation curves measured for some thousands galaxies does not show the expected decrease $v(r)$ with radius growing. A pronounced approximate constance of $v(r)$ is observed instead for large radii (See Figure). This fact implies that the mass $M(r)$ must increase with the radius growing: $M(r)\sim r$. Then one can conclude that mass of the central is far from being exhausted by the one contained in the visible volume: it is distributed instead in much larger region. Unlike the stars, the latter additional mass does not emit light (it is invisible on pictures), but creates considerable gravity force, additional to that of the visible stars. That is how the concept of the hidden mass originates.
-<gallery widths=600px heights=500px>
+<gallery widths=600px heights=400px>
 File:Rotationcurv.JPG|
 </gallery></p>
    </div>
 </div></div>

← Older revision		Revision as of 11:36, 2 October 2012
Line 1:		Line 1:
	[[Category:Dark Matter\|1]]		[[Category:Dark Matter\|1]]
		+
		+
		+	_NOTOC_
		+
		+
		+
		+	<div id="DM01"></div>
		+	<div style="border: 1px solid #AAA; padding:5px;">
		+	=== Problem 1 ===
		+	Analyze the problem of a small satellite galaxy moving in the gravitational field of a large galaxy (the problem of non-decreasing behavior of the rotational curves.)
		+	<div class="NavFrame collapsed">
		+	<div class="NavHead">solution</div>
		+	<div style="width:100%;" class="NavContent">
		+	<p style="text-align: left;">Many galaxies have several small satellite galaxies. For the latter the second Newton law reads:
		+	\[G\frac{mM(r)}{r^{2} } =\frac{mv^2}{r} \]
		+	For the rotation velocity one then obtains
		+	\[v(r)=\sqrt{\frac{GM(r)}{r} } \]
		+	where $M(r)$ is the total mass of the matter inside the sphere of radius $r$. In the spherically symmetric case the effect of the outside mass vanishes, and it follows that
		+	\[M(r)=\bar{\rho }\frac{4}{3} \pi r^{3} \Rightarrow v(r)=\sqrt{\frac{4}{3} \pi G\bar{\rho }r^{2} } \sim r,\]
		+	where $\bar{\rho }$ is the distribution of the mass $M.$ Thus linear growth of the rotation velocity with the galactocentric radius is expected in the inner part of the galaxy. In the outer region of the galaxy the dependence of the rotation velocity on the radius corresponds to the case of point mass in the center of the galaxy:
		+	\[v(r)\sim 1/\sqrt{r} \]
		+	The rotation velocity $v(r)$ can be in particular determined by the measured Doppler shift in the emission spectrum of the corresponding satellite galaxies. The behavior of the observed rotation curves measured for some thousands galaxies does not show the expected decrease $v(r)$ with radius growing. A pronounced approximate constance of $v(r)$ is observed instead for large radii (See Figure). This fact implies that the mass $M(r)$ must increase with the radius growing: $M(r)\sim r$. Then one can conclude that mass of the central is far from being exhausted by the one contained in the visible volume: it is distributed instead in much larger region. Unlike the stars, the latter additional mass does not emit light (it is invisible on pictures), but creates considerable gravity force, additional to that of the visible stars. That is how the concept of the hidden mass originates.
		+	<gallery widths=600px heights=500px>
		+	File:Rotationcurv.jpg\|
		+	</gallery></p>
		+	</div>
		+	</div></div>