Difference between revisions of "Forest for the trees"

Let $n$ be the mean density of stars in the Universe and $L$ be the mean luminosity of stars. The observed luminosity (energy flow's surface density) on the Earth from a star with luminosity $L$ at distance $r$ is \[F(r) = {L \over {4\pi r^2 }}.\] Consider a spherical shell of radius $r$ and thickness $dr$ centered at Earth. The radiation intensity of stars inside this shell (the power which reaches a unit surface from one steradian) is $$ dP(r) = F(r) \cdot n \cdot r^2 dr = {{nL} \over {4\pi }}dr. $$ It is important to note that total intensity of radiation from the shell is independent on the distance to it. Thus, the total intensity of radiation from all stars in the Universe \[P = \int_0^\infty {dP = {{nL} \over {4\pi }}\int_0^\infty {dr} }\] diverges in the case of infinite stationary Universe. Olbers' paradox is an example of the so-called "law of incorrect naming", which states that no law is called after a person who in fact discovered it. This paradox had been known 150 years before Olbers' formulated it (Diggers (1576)).

Let's draw a circle of radius $x$ around an observer. Part of an observers' viewfield $s(x),\;0 \le s(x) \le 1$ is hidden by trees. Next, expand the observation region by $dx$, passing to a circle of radius $x+dx$. There are $n\times 2\pi xdx$ trees in a ring between the circles. They overlap the portion $nDdx$ of observed circle. Since portion $s(x)$ was already overlaped, the additional overlap is $\left[ {1 - s(x)} \right]nDdx$. Hence, $$ s(x + dx) - s(x) = \left[ {1 - s(x)} \right]nDdx $$ or $$ {{ds} \over {dx}} = \left[ {1 - s(x)} \right]nD. $$ This equation could be easily integrated as $$ \int_0^s {{ds} \over {1 - s}} = \int_0^x {nDdx}, $$ thus obtaining a solution $$ s(x) = 1 - e^{ - nDx}. $$ This value is a probability that looking at random direction, sight could reach no more than $x$, i.e. it reach a tree at distance $x$. Hence, the mean length of "free sight" is $$ \lambda = \int_0^\infty x {{ds(x)} \over {dx}}dx = {1 \over {nD}}. $$ If there are $1000$ trees on one hectare and diameter of trunk is $20 \mbox{cm}$, the mean length of sight is $50 \mbox{m}$. This result could be generalized to 3-dimensional case. There are $n$ stars in unit volume and each star has diameter $D$. If the surface, perpedicular to the line of sight is $A=\pi D^2/4$, then $$ s(x) = 1 - e^{ - nAx} $$ and $$ \lambda = {1 \over {nA}}. $$ If the Universe was infinitely old and infinitely extended, then line of sight would reach a surface of a star in finite time (and at finite distance). This is the meaning of Olbers' paradox.

$\displaystyle \lambda = {1 \over {nD}} \approx 333\mbox{m}. $

$\displaystyle \lambda = {1 \over {n_{st} \pi R_ \odot ^2 }} \approx 6 \cdot 10^{15} Mpc. $

Let's make some simplifying assumptions. First, assume that total energy of a pair of stars conserves when they converge due to gravitational interaction. Second, the masses of stars and their velocities are equal at the start of convergence (formally, at infinitely large distance). These assumptions are sufficient for estimate. Energy conservation in the center of momentum frame reads $$ mV^2 - G{m^2\over r}=mV_0^2, $$ where $V$ is current velocity of stars (which velocities are equal due to symmetry of the system), $r$ is the distance between them and $V_0$ is velocity at infinite distance. Considering collision as a moment, when kinetic energy of stars doubled, we obtain $$ G{m\over r}=V_0^2, $$ and, hence, collision cross--section is $$ \sigma = {\pi G^2m^2\over 4V_0^2}. $$ Characteristic time before first \underline{(sic!)} collision is $$ t_{st} = (nV_0\sigma)^{-1}. $$ For the Milky Way we have $n \simeq 3\cdot 10^{-56}\mbox{cm}^{-3}$, $V_0\simeq 20\mbox{km/s}$, so that $\tau_{st} \simeq 10^{21}\mbox{s}$. This time is three orders larger than the age of the universe. This estimate allows to consider stars as collisionless medium.