# Miscellaneous Science

## Size of the Universe

The size of the universe is a quantity which is estimated based on a large number of other quantities which in turn depend on other estimates and so on. It is often called the "Cosmological Distance Ladder". Each distance depends on the smaller, closer ones. First we had to measure the size of the Earth, then we could use that to determine the distance of the Moon and Venus, which allowed us to compute the size of Earth's orbit and the rest of the Solar System; then we could measure the distance to nearby stars, and so on. Here is an approximate representation of what depends on what, listed in order of discovery. Each item is followed (in parentheses) by a description of how it was determined. If it's computed from earlier items on the list, the item numbers are given; otherwise a description of the experiment is given:

dec1 Relative angular altitude of Sun at midday (easily measured since antiquity)

dist1 Distance between two places on the Earth's surface (easily measured since antiquity)

dist2 Diameter of the Earth (dec1,dist1)

ang1 Angular size of Moon's orbit as seen from Sun (Aristarchus)

ang2 Angular size of Sun

for1 Gravitational force between the Earth and an object of known mass (easily measured since antiquity)

tim1 Length of year (easily measured since antiquity)

ang3 Parallax of moon over 8 hours versus 24 hours

dist3 Distance of moon (dist2,ang3)

rat1 Ratio between orbital speed of Earth and speed of light (Measured accurately since 1725)

tim2 Transit times of Venus observed from different latitudes on Earth (first measured accurately in 1761)

ang4 Solar parallax (angular size of Earth as seen from Sun) (dist2,ang2,tim2)

dist4 Distance between Earth and Sun (dist2,ang4)

tim3 Phase shift of observed eclipses of Jupiter's moons (known by 1676, measured since 1729)

tim4 Time it takes light to cross Earth's orbit (tim3)

vel1 Speed of light (tim4,dist4)

for2 Gravitational force between two objects of known mass (measured accurately since 1798)

k1 Gravitational constant (for2)

mass1 Mass of the Earth (dist2,for1,k1)

den1 Density of Earth (mass1,dist2)

acc1 Centripetal acceleration of the Earth in solar orbit (vel1,tim1)

mass2 Mass of the Sun (k1,vel1,acc1)

dist5 Distance of other planets relative to distance of Earth from Sun (computed since by triangulation)

dist6 Distances of other planets (vel1,dist5)

ang5 Parallax of nearby stars (measured accurately since 1838)

dist7 Distance of nearby stars (vel1,ang5)

lux1 Apparent brightness of nearby stars (measured accurately since )

lux2 Absolute brightness of nearby stars (dist7,lux1)

func1 Mass as a function of brightness for "main sequence" stars (supported by theory)

func2 Mass distribution of stars in stellar neighborhood (lux2)

lux3 Distance and brightness of Delta Cephei, the nearest Cepheid variable (dist7,lux2)

data1 Observations of many Cepheids in the Small Magellanic Cloud (first performed in 1912)

func3 Calibration of Cepheid period-luminosity correspondence (lux3,data1)

data2 Apparent brightness of Cepheid stars around the galaxy (measured accurately since )

data3 Distances of Cepheid stars around the Milky Way (func3,data2)

data4 Size and shape of Milky Way (data3)

data5 Relative radial velocities of stars around center of Milky Way (measured accurately since by redshift method)

data6 Orbital velocity of stars at various radii from center of Milky Way (data5)

data7 Mass distribution of Milky Way (data6)

k2 Percentage of dark matter within Milky Way (data7)

data8 Observations of Cepheid variables in nearby galaxies (known since 1923)

dist8 Distance of Andromeda Galaxy and other nearby galaxies (data8)

data9 Differential redshift of 21-cm hydrogen line between opposite edges of spiral galaxies

data10 Rotation velocities (and rotation curves, giving relative mass distribution) of spiral galaxies (data9)

data11 Calibration of Tulley-Fisher relation

data12 Sizes and distances of spiral galaxies (data11,data10)

Aristarchus of Samos, c.310-230 BC

Aristarchus of Samos estimated the Solar Parallax as being 3 degrees, by trying to guess when the Moon was exactly at first or third quarter and measuring the angle between the Moon and the Sun at that moment. (1 AU = 20 lunar units)

Eratosthenes of Cyrene, c. 200 BC

Eratosthenes of Cyrene calculated the circumference of the Earth,
using the known distance
between two cities in Egypt that are situated on the same longitude
line (Syene and Alexandria) combined with the knowledge that the sun's
angle in the sky is 82^{o}48' in Alexandria and directly overhead
(90^{o}) in Syene on the summer solstice. The distance of 5000 stadia
and angular difference of 7.2 degrees gives 5000 × 360 / 7.2, or
250,000 stadia.

I have also read that he estimated the distance of the Moon as 780,000 stadia and the distance of the Sun as 804,000,000 stadia: "Plutarch reports that Eratosthenes gave the range of the Sun as 804 million stadia and the Moon 780 thousand.", but cannot find more details or corraboration.

Nicolaus Copernicus proposed the theory that the planets go around the Sun, in circular orbits. This explains much more simply why the planets weave back and forth as they move through the sky. It was a good first approximation, which was much refined by Kepler years later.

Once Copernican heliocentrism was accepted, it was possible to measure the relative sizes of the orbits of the planets by their parallax: as the Earth moves through its orbit, the planets appear to be in different parts of the sky as compared to stars (which are assumed to be nonmoving and much further away). Of course the planets are moving too so you have to make lots of measurements and do some trigonometry. Also, for the inferior planets (Mercury and Venus) the measurement can be made a lot easier because all you really have to do is note the maximum elongation (maximum angular distance between the planet and the Sun).

It was soon found that the planets Mercury and Mars had notable discrepancies, as if their orbits were off-center. If you make more accurate mesaurements you can see that they're not quite circles either, more like ellipses.

Johannes Kepler saw a relationship between the planets' orbital radii and their periods of revolution:

R^{3} * K = P^{2}

and also guessed that the planets were in elliptical orbits with the Sun at one focus. The observed data fit these theories quite well. It wasn't until Newton that anyone came with a more fundamental explanation of why the planets followed these paths.

Sir Isaac Newton developed calculus and hypothesized the force of gravity. Using calculus he found that the formula

F = G M_{1} M_{2} / D^{2}

precisely explained the laws of planetary motion set out by Kepler.

G is the Gravitational Constant, and Newton did not know its value.
The value of G could be computed by measuring F, M_{1}, M_{2}
and D for a given pair of bodies attracting each other. Until
Cavendish the best anyone could do was to estimate the
mass of the Earth and arrive at a value of G based on the weight of
objects at sea level and the Earth's radius. The accepted estimate,
based on the assumption that the Earth is solid rock, was too high.

Repeated Aristarchus' method and arrived at a value of 15 arc-minutes, (1 AU = 229 lunar units).

Calculated the angle subtended by Venus as viewed from the Sun (and got 28 arc-seconds), by measuring the width of Venus as it transited the Sun. He then assumed that the Earth subtended the same angle, which gives 1 AU = about 59 million miles.

Pointed out that the phase shift in the eclipses of Io could be explained by a finite speed of light.

Wrote to future generations of astronomers about the importance of observing the transits of Venus in 1761 and 1769. He accurately described the method which was to be used.

Measured the relative angles between stars 90 degrees apart in the sky (one in the direction of Earth's motion, the other perpendicular to it). The angle varies by as much as 41 arc seconds over the course of 6 months, due to the fact that the Earth is moving through space at 0.02% of the speed of light. To actually repeat the experiment you have to use a baseline somewhat shorter than 6 months in order to have both stars visible at the same time at night at two different times of the year. In modern times the easiest way to reproduce the experiment is with the help of a prism or half-silvered mirror permanently fixed to the telescope so that both stars can be seen at once. This effect is called stellar aberration. Stellar parallax is significantly smaller; the closest stars have a parallax of less than one arcsecond. Since stellar aberration has an equal effect on all stars in a given area of the sky, it doesn't interfere with tasks like measuring stellar parallax or proper motion.

Confirmed Roemer's work.

An intense worldwide effort following the method outlined by Halley in 1677 to measure the solar parallax. These astronomers observed transits of Venus across the Sun from many different locations on the Earth (at different latitudes) and accurately measured the time it took for the transit to occur. From this it is possible to calculate the diameter of the Sun in terms of the distance between the different observing points. They got values between 8.55 arc-seconds and 8.88 arc-seconds for the Solar parallax. (The modern value is 8.794148)

Once this crucial value was known it was possible to immediately compute the sizes of all of the planetary orbits, the radii of the planets and the speed of light, all of which had hiterto been known only as ratios to the size of the Earth's orbit.

Cavendish Torsion Balance experiment

Cavendish first measured the gravitational attraction between two known masses and thus derived the value of the gravitational constant (6), from which the mass of the Earth (7) can be calculated. The value he found was twice as great as the previously held estimate, which was based on the assumption that the Earth is solid rock (it actually has a metal core).

Two equal weights are attached to either end of a horizontal rod. The rod is suspended by a very thin fiber. When another mass is brought close to one of the weights, the rod twists very slightly (usually a microscope must be used to see how far it has twisted).

An accurate value of the gravitational constant immediately led to accurate estimates of the masses of the Sun and the known planets with satellites (Mars, Jupiter, Saturn and Uranus).

In the 200+ years since Cavendish no significantly different method has
been found for estimating the value of the gravitational constant. The
current best estimate is 6.6726 ^{.} 10^{-11} m^{3} / kg s^{2}, with
an uncertainty of 0.01%.

Measured the brightness curves and periods of many Cepheid variable stars in the Small Magellanic Cloud. Assuming the stars were all at the same distance (which is still considered a reasonable assumption) she found a strong correlation between the maximum brightness and the period of oscillation. This, in combination with the absolute brightness of Delta Cephei (whose distance was already known by parallax) allowed her to establish the correlation between period and absolute brightness, which in turn allows the determination of the distance of other Cepheid variables of known period (but hitnerto unknown distance).

This made it possible to accurately estimate the distances of the globular clusters and the Small Magellanic Cloud itself, and in following years the distances of many nearby galaxies.

Estimated the distances of the globular clusters by measuring the brightness of their RR Lyrae variables, and assuming they were short period Cepheids. He found that they were distributed in a more or less spherically symmetric way, centered at a point about 40000 light years from us, which he argued to be the galactic center. The estimate was inaccurate for two reasons. The first was the "interstellar absorption" also called "interstellar extinction", the darkening of starlight by the gas and dust that is found everywhere there are stars, which was discovered in the 1930's by Robert Trumpler. The second reason is that there are two classes of Cepheids, the original Cepheids and the RR Lyrae variables, with a different formula relating period to actual brightness. This discovery was made in the 1950s and caused all of our estimates of galaxy distances to double.

Up to this point the furthest known stars we had been able to see were about 5000 light years away (in all directions in the plane of the Milky Way) and it had been assumed by most that this was the extent of the galaxy. (Dark gas and dust obscures the rest of the galaxy from us).

Discovered a Cepheid variable in the Andromeda galaxy, enabling the first accurate determination of its distance.

Using the Hubble Space Telescope, discovered a Cepheid variable in the galaxy M100. It is the furthest known Cepheid variable to date, at 56 million light years.

This page was written in the "embarrassingly readable" markup language RHTF, and was last updated on 2018 Aug 27. s.11