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Chapter 34: Q30PE (page 1238)

To get an idea of how empty deep space is on the average, perform the following calculations: (a) Find the volume our Sun would occupy if it had an average density equal to the critical density of\({\rm{1}}{{\rm{0}}^{{\rm{ - 26}}}}{\rm{kg/}}{{\rm{m}}^{\rm{3}}}\)thought necessary to halt the expansion of the universe. (b) Find the radius of a sphere of this volume in light years. (c) What would this radius be if the density were that of luminous matter, which is approximately\({\rm{5 \% }}\)that of the critical density? (d) Compare the radius found in part (c) with the\({\rm{4 - ly}}\)average separation of stars in the arms of the Milky Way.

Short Answer

Expert verified
  1. The volume our sun would occupy is obtained is \({\rm{2 \times 1}}{{\rm{0}}^{{\rm{56}}}}{{\rm{m}}^{\rm{3}}}\).
  2. The radius of the sphere is obtained is \({\rm{382}}{\rm{.66 ly}}\).
  3. The radius will be \({\rm{1040 ly}}\) times bigger.
  4. The radius of the sphere of luminous matter is approximately \(260\) times that of the average separation between stars in the Milky way galaxy.

Step by step solution

01

Expression for volume, mass and density

The expression for the volume of the sphere is give by,

\(V = \frac{4}{3}\pi {r^3}\)

Here\(r\)is the radius of the sphere.

The expression for the density is given by,

\(\rho = \frac{M}{V}\)

Here\(M\)is the mass of the sun,\(\rho \)is the density.

Then, the expression for the volume is,

\(V = \frac{M}{\rho }\)

02

Evaluating the volume

(a)

If the Sun had a density of:

\(\rho = {\rm{1}}{{\rm{0}}^{{\rm{ - 26}}}}{\rm{ kg/}}{{\rm{m}}^3}\).

The volume will be then obtained as,

\(\begin{align}V &= \frac{m}{p}\\ &= \frac{{{\rm{2 \times 1}}{{\rm{0}}^{{\rm{30}}}}{\rm{kg}}}}{{{\rm{1}}{{\rm{0}}^{{\rm{ - 26}}}}{\rm{ }}\frac{{{\rm{kg}}}}{{{{\rm{m}}^{\rm{3}}}}}}}\\ &= {\rm{2 \times 1}}{{\rm{0}}^{{\rm{56}}}}\,{{\rm{m}}^{\rm{3}}}\end{align}\)

Therefore, the volume is \({\rm{2 \times 1}}{{\rm{0}}^{{\rm{56}}}}\,{{\rm{m}}^{\rm{3}}}\).

03

Evaluating the radius

(b)

The volume of sphere is:

\({\rm{V = }}\frac{{\rm{4}}}{{\rm{3}}}{{\rm{r}}^{\rm{3}}}{\rm{\pi }}\)

Then, the radius of such sphere will be:

\(\begin{align}r{\rm{ }} &= {\left( {\frac{{{\rm{3V}}}}{{{\rm{4\pi }}}}} \right)^{\frac{1}{3}}}\\ &= {\left( {\frac{{{\rm{3 \times 2 \times 1}}{{\rm{0}}^{{\rm{56}}}}{\rm{\;}}{{\rm{m}}^{\rm{3}}}}}{{{\rm{4 \times 3}}{\rm{.14}}}}} \right)^{\frac{1}{3}}}\\ &= {\rm{3}}{\rm{.63 \times 1}}{{\rm{0}}^{{\rm{18}}}}{\rm{\;m}}\\ &= {\rm{3}}{\rm{.63 \times 1}}{{\rm{0}}^{{\rm{18}}}}{\rm{\;m}}\frac{{{\rm{1y}}}}{{{\rm{9}}{\rm{.5 \times 1}}{{\rm{0}}^{{\rm{15}}}}{\rm{\;m}}}}\\ &= {\rm{382}}{\rm{.66}}\,{\rm{ly}}\end{align}\)

Therefore, the radius is:\({\rm{380 ly}}\).

04

Explanation for part c

(c)

If a density was\({\rm{5 \% }}\)of density in the part a,

We have then obtained the volume twenty times larger than in the part a.

It means that the radius is \(\sqrt({\rm{3}}){{{\rm{20}}}}{\rm{ r}}\) times bigger

\(\begin{align}r' &= {\rm{ }}\sqrt({\rm{3}}){{{\rm{20}}}}{\rm{ r}}\\ &= {\rm{ 1040 ly}}\end{align}\)

Therefore, the radius is \({\rm{1040 ly}}\) times bigger.

05

Explanation for part d

(d)

The radius ratio,

\(\begin{align}Ratio &= \frac{r'}{r}\\ &= \frac{{1040\,{\rm{ly}}}}{4}\\ &= 260\end{align}\)

As the radius of the sphere of luminous matter is approximately \(260\) times that of the average separation between stars in the Milky way galaxy.

This shows that most of the region in deep space is almost empty.

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Most popular questions from this chapter

If some unknown cause of red shift鈥攕uch as light becoming 鈥渢ired鈥� from traveling long distances through empty space鈥攊s discovered, what effect would there be on cosmology?

Assuming a circular orbit for the Sun about the center of the Milky Way galaxy, calculate its orbital speed using the following information: The mass of the galaxy is equivalent to a single mass\({\rm{1}}{\rm{.5 \times 1}}{{\rm{0}}^{{\rm{11}}}}\)times that of the Sun (or\({\rm{3 \times 1}}{{\rm{0}}^{{\rm{41}}}}{\rm{ kg}}\)), located\({\rm{30,000 ly}}\)away.

The decay of one type of \({\rm{K}}\)-meson is cited as evidence that nature favours matter over antimatter. Since mesons are composed of a quark and an antiquark, is it surprising that they would preferentially decay to one type over another? Is this an asymmetry in nature? Is the predominance of matter over antimatter an asymmetry?

Discuss the possibility that star velocities at the edges of galaxies being greater than expected is due to unknown properties of gravity rather than to the existence of dark matter. Would this mean, for example, that gravity is greater or smaller than expected at large distances? Are there other tests that could be made of gravity at large distances, such as observing the motions of neighbouring galaxies?

Another known cause of red shift in light is the source being in a high gravitational field. Discuss how this can be eliminated as the source of galactic red shifts, given that the shifts are proportional to distance and not to the size of the galaxy.
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