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Measurements of the Universe

04/22/2007 6:49 PM

I thought I would share another document with my friends. It was revised in 2002, so it's somewhat older. My best guess is that it came from the book The Universe by Isaac Asimov. Any discussion is welcome.

Measurements of the Universe

Diameter of Earth

The Greek philosopher Eratosthenes of Cyrene (276-196 B.C.) calculated the circumference of the Earth. He knew that on the summer solstice, June 21, when the sun was at the zenith in the city of Syene, Egypt (now Aswan), that a vertical stick would cast no shadow. At the same time, a vertical stick in Alexandria, 500 miles north, would cast a shadow a little over 7 degrees north of the zenith. He proved his calculations by geometry, but we can use a ratio: 360/7.2 = X/500. X = 25,000 miles. The diameter = circumference / pi. The latest figures are 24,902.4 miles at the equator, and diameter of 7,917.78. The surface area is 196,950,000 square miles.

Earth to the moon

The Greek astronomer Aristarchus of Samos (320-250 B.C.) was the first to make a serious attempt. He made observations during a lunar eclipse. The earth's curvature was compared to the curvature of the moon. Another Greek astronomer, Hipparchus of Nicea (190-120 B.C.) improved the method. He concluded that the distance from earth to moon was 30 times the diameter (about 240,000 miles). More recently, parallax was used. The angle from the edge of the moon to a particular star was measured by one observatory when the moon was directly overhead, and from another at the same time, when the moon was on the horizon. Trigonometry was used, with the base of the triangle being the radius of the earth. The difference in angles was 57.04 minutes of arc (0.95 degrees). Today the distance can be measured by reflecting a laser beam off of a reflector on the moon, and measuring the time it takes. The speed of light is well known. The distance = the speed of light times the time of travel of the light one way. The modern figure is 238,854.7 miles average, 221,463 perigee, and 252,710 apogee. Knowing the distance, one can calculate the diameter of the moon from it's apparent size (2160 miles).

Earth to the sun

Aristarchus realized that when the moon was exactly at the 1st or last quarter, the sun, the moon, and the earth were at the vertices of a right triangle. By measuring the angle separating the moon and the sun, as seen from the earth, one could use trigonometry to get the ratio of distances of the moon and sun. He came up with a ratio of 20, giving a distance of about 5 million miles (way off). The modern figure is 92,950,000 miles average, 91,400,000 perihelion, and 94,600,000 aphelion. (see next section)

The sun to the planets

No progress in astronomical measurements was made for 1800 years. Then the German astronomer Johannes Kepler (1571-1630) discovered three laws:

1. Planets and comets travel around the sun in ellipses, with the sun at one focus of the ellipse.

2. The straight line connecting the sun to a planet sweeps out equal areas in equal amounts of time.

3. The squares of the orbital periods of the planets are proportional to the cubes of their average distances from the sun.

The periods of the orbits were known from observation, so the relative distances of the planets were easily calculated. The distance to one planet was needed. In 1671, a parallax measurement of Mars was made using telescopes (which made small shifts against the star background look bigger). The French astronomer Jean Richer measured at Cayenne in French Guiana, and the Italian-French astronomer Giovanni Cassini measured from Paris. Once that measurement was made, the scale of Kepler's model was known, and all other distances in the solar system could be calculated. The distance from the earth to the sun was calculated by Cassini to be 87,000,000 miles.

Parallax measurements of Venus can be made at times when Venus crosses the sun's disc. In 1835, the German astronomer Johann Encke used data from Venus transits, and calculated the distance to the sun to be 95,370,000 miles. In 1931, the asteroid Eros was parallax measured by a vast international project. Being small, and with no atmosphere, it could be measured more accurately. From this the distance to the sun was determined to be a little less than 93 million miles.

In 1961, microwave signals were reflected off Venus. The speed of light had been accurately measured long before. From this the average distance to each planet was accurately calculated.

The speed of light

In 1676, the Danish astronomer Ole Roemer (Olaus Rohmer?) determined the speed of light from observations of the moons of Jupiter. He noticed that the eclipses of Jupiter's moons appeared later when the earth was farther away from Jupiter. He reasoned that it was because it took longer for the light to reach him. His value was 140,000 miles per second. Armande Fizeau measured it in 1849 in vacuum, air, & water using a rotating wheel with slots. James Clerk Maxwell developed equations for electric and magnetic fields in 1862. From them came a formula for the speed of light from the permeability and permittivity of vacuum (which can be measured in the laboratory): c = 1/SQR(u0·e0). Albert Michelson measured it around 1900 using an apparatus with a rotating mirror and a slotted wheel. This approach was repeated several times in the USA, the most accurately about 1947 under the FDR administration. The value is 186,282 mps.

Force of gravity

The Brit Henry Cavendish demonstrated gravitational attraction between spheres. He balanced two small spheres on a long thread as thin as a cobweb inside a glass box to keep air currents from disturbing it. Outside the glass box he put two massive spheres that could be rotated around the central axis. After equilibrium, the massive spheres were rotated part way around the glass box. The small spheres on the bar followed for a certain angle. Measuring the angle, and knowing the resistance of the thread to a twist, Cavendish calculated the force of gravity between the spheres. He found the value of the coefficient G in Isaac Newton's formula for gravitational force between two bodies (F = G·M1·M2/d2). Using that value, the force between two apples close together was found to be the equivalent of the weight of one billionth of an ounce!

Mass of the earth

A modified form of the Cavendish experiment was performed by the British physicist C.V. Boys. After having balanced two equal weights on equal-balance scales, he placed a massive sphere under one of the plates, and observed a slight deflection from the gravitational attraction of the sphere. From this, Boys determined the ratio of the mass of the sphere to the mass of the earth. The earth, he found, weighs 6,100,000,000,000,000,000,000,000 kilograms.

Distance to nearest stars

It makes little difference whether it's from the sun or the earth. The brightness of a star varies inversely with the square of the distance. The distance can therefore be estimated from relative brightness to the sun. This method assumes that all stars have the same luminance (incorrect). Rigel and Deneb are 25,000 times as luminous as the sun, and S Doradus is 600,000 times as luminous!).

Proper motion (stars moving through space) can also be used to estimate distance. Closer stars should have more proper motion than farther ones, but stars are traveling at different speeds, and those traveling straight toward us (or straight away) show no proper motion. Parallax ellipses over the course of a year can theoretically be seen. It is very difficult to measure parallax to stars for several reasons. The ellipses are very small, and stars have proper motion. Another problem, the "aberration of light", caused by the earth's speed relative to the speed of light (about 18.5 miles per second), causes a larger ellipse than the parallax ellipse. The nutation of the earth is also involved. After these effects have been accounted for, parallax gives the best measurement. It is limited to about 150 light years.

Size of the Milky Way Galaxy

Certain stars such as Delta Cephei have varying luminosity with a period that is measured in days. These are called "Cepheid variables". The American astronomer Henrietta Swan Leavitt studied these in the Magellanic clouds. She noted that the brighter the cepheid variable, the longer it's period. The American astronomer Harlow Shapley used Cepheids to find the relative distance to globular clusters (from the relative period of the Cepheids, the relative luminosity was assumed). The clusters seemed to be in a spherical arrangement. This was assumed to be the center of the galaxy. He measured the average proper motion (transverse) of Cepheids in each cluster. The radial velocity (toward or away from us) should be the same as the transverse velocity. The radial velocity was determined from the spectral shift of the light (Doppler shift toward the red or violet). With transverse velocity and proper motion known, the distance can be calculated. Shapely came up with 50,000 light-years. He had not accounted for dust in the Galaxy which reddens and reduces the light. The figure was revised downward to 30,000 light-years. The diameter of the Galaxy is about 100,000 light-years. The thickness is about 16,000 light-years at the center, and about 3,000 light-years where the sun is. The globular clusters are distributed spherically about the center of the Galaxy in a 100,000 light-year sphere. Jan Oort determined the general nature of the rotation of the Galaxy, and from that calculated the direction and distance of the center without using globular clusters. He got 30,000 light-years in the same direction as the clusters. Oort's calculations gave an estimate of the sun orbiting the galaxy once every 230 million years, and of 100 billion stars in the Galaxy.

Diameters of giant stars

Even the nearest stars seem no more than a point of light in the best Modern telescopes. The German-American physicist Albert Michelson invented a device called a light interferometer. It produces interference patterns in light rays, and made it possible to measure the very small angle between the light coming from one side of a star and the light coming from the other side. From the angle, and the distance of the star, the diameter can be calculated. Betelgeuse was measured this way in 1920, and found to be as large as the orbit of Mars around the sun. Epsilon Aurigae, which is only visible in infrared, is as large as the orbit of Uranus!

Distance to nearest galaxy

Sometimes stars suddenly brighten to thousands of times their normal brightness. These are called Novae. They dim down again after a few days to a few months. A Nova (S Andromedae) appeared in Andromeda in 1885 (Andromeda was thought to be a nebula at the time). In 1901, a nova was seen in the Milky Way Galaxy (Nova Persei). It's distance was measured by parallax to be 100 light-years. On the assumption that all novae have the same luminosity, the distance to Andromeda was calculated at 1600 light-years. The American astronomer Heber Curtis found and studied many novae in Andromeda. They were all far dimmer than S Andromedae had been. In 1917, a 100 inch (mirror size) telescope was installed on Mt. Wilson. The American astronomer Edwin Hubble used it to look at Andromeda. He could make out individual stars on the outskirts. Eventually, he found Cepheid variables in it, and showed that it was a galaxy. He calculated its distance at 800,000 light-years. By 1950, it was known that this was wrong. The Milky Way appeared to be larger than any other, even though it had the shape of an intermediate sized galaxy. Also, Andromeda had globular clusters like the Milky Way, and they appeared to be smaller. The Cepheid variables were classified into two types, and Baade showed that one type in the Milky Way had been compared to the other type in Andromeda. The distance to Andromeda was revised to 2.3 million light-years. This is the closest one! It was decided that S Andromedae was in a different class than most Novae. These are called Supernovae. One Supernova can outshine an entire galaxy for a few days! They occur about 3 per galaxy per millennium.

Distance to farther galaxies

For galaxies too far away to see Cephieds, Hubble made use of any stars he could see, by assuming these were supergiants that were bright like S Doradus. If no stars could be seen, he used the relative brightness of the galaxy as a whole. Clusters of galaxies are compared to farther clusters by the relative brightness of the average of the galaxies in the cluster.

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#1

Re: Measurements of the Universe

04/22/2007 7:01 PM

Ref the man in the forest. Only if the trees grass him up to the woman. The rest is interesting. Good effort to impart some useful info.

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#2

Re: Measurements of the Universe

04/24/2007 12:07 AM

this is very interesting .

now i have a question .

if you are placed on no mans land and have no tools like cell etc, and are to describe your location ( only tool is the sky stars etc, may some stick or things that would naturaly be present . ) can you find out

your geographycal location.

can you find out the day , month and year ????

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#3

Re: Measurements of the Universe

04/24/2007 1:27 AM

Hi SG,

Interesting summary! Some of the cosmological figures are a bit old, e.g., parallax measurements were done to more than 750 light years by the Hipparcos space telescope.

I also wrote up something on the so-called "distance ladder" on my website. It's in a download from this page: The Expanding Universe: an Engineer's View (if you have the eBook, that's in section 16.3, page 207)

Sadly, things change so fast that my book is probably also outdated by now!

J

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#4

Re: Measurements of the Universe

04/24/2007 1:10 PM

Hi StandardsGuy,

Nice piece, I love stuff like this.

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#5

Re: Measurements of the Universe

04/25/2007 10:47 PM

Thanks guys, I really appreciate it.

Yes, I admitted that it was old, but it gives a flavor for the uncertainties involved in these kinds of measurements.

Jorrie, I didn't know you had that in your book. I'm going to have to read the whole thing someday, but you are still writing it? Getting parallax measurements to 750 light years seems phenomenal, but you mentioned 5,000? That would be very phenomenal! What has advanced so much to make that possible?

S

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#6
In reply to #5

Re: Measurements of the Universe

04/26/2007 3:59 AM

Hi SG, you wrote: "I'm going to have to read the whole thing someday, but you are still writing it?"

The book's 1st issue (Sept 06) is a completed work. I just re-issued it with few errata in Nov 06. In order to keep it reasonably up to date, I plan to issue a minor update every year or so. I plan a major extension in about 18 months from now, which will include new topics like rotating black holes and some others.

"Getting parallax measurements to 750 light years seems phenomenal, but you mentioned 5,000? That would be very phenomenal! What has advanced so much to make that possible?"

I believe it's VLBA astronomy. The very long baseline is still tiny compared to the diameter of Earth's orbit, but it helps to measure the instantaneous angle to a star more accurately than what Hipparcos could have done. I do not think 5000 ly has been achieved yet, but I think that's about the theoretical limit of parallax.

Will see if I can find the latest results.

-J

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#7
In reply to #5

Re: Measurements of the Universe

04/26/2007 7:33 AM

Hi SG, I found some articles on the VLBA project reporting "model-dependant distances" using parallax of up to 7500 ly at ~10% accuracy (2.37±0.23 kpc), where 1 kpc = 3.26 ly.

I'm not too sure what this "model-dependence" means, but it may be irrelevant for the "distance ladder" anyway, because the measurements were all for pulsars. For the distance ladder one needs the distance to Cepheid variable stars or other "standard candles".

-J

CR4 Admin: removed broken link

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#8
In reply to #7

Re: Measurements of the Universe

04/26/2007 8:39 PM

Thanks for the astrometric update.

I dig my graduate work in astrometry under George Gatewood at Allegheny Observatory (University of Pittsburgh). That was all optical (visual wavelength) and ground based work. The accuracy of a single digitized image was 0.05-0.1 arc seconds. Optical ground based parallaxes were determined by taking thousands of images over 10-70 years.

In space Hipparcos can do a lot better than ground based telescopes (no atmospheric distortion).

The VLBA is comprised of 10 radio telescopes, each is a 25 m diameter dish. They are mostly in the USA: Arizona, New Mexico, Texas, Washington, Iowa, New Hampshire. But include Mauna Kea in Hawaii and St. Croix, Virgin Islands. This interferometry gives the equivalent resolution of a radio telescope approaching the diameter of the earth.

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#9

Re: Measurements of the Universe

04/26/2007 9:28 PM

"The American astronomer Harlow Shapley used Cepheids to find the relative distance to globular cluster"

There are many types of variable stars. Cepheid variables are bright stars which pulsate. The longer the pulsation period, the brighter the star. Typical luminosities are 800 to 10,000 times our sun. These can be observed in nearby galaxies. Type I Cepheids are younger and brighter than Type II Cepheids.

RR Lyrae stars are pulsating variable stars, similar to Cepheids, but different. They are about the mass of our sun (0.8-1.0 solar masses) and about 40 times brighter. Their periods are shorter (typically 1 day or less). They are stars on their way to becoming white dwarfs. These are the variable stars, which are observable in globular clusters, are the ones Shapley used to measure distances to globular clusters. As their scatter compared with theory is smaller, they yield better distances than the classic Cepheids. Also averaging many RR Lyrae stars in each globular cluster means a more accurate distance measurement to the cluster.

Interstellar dust dims stars so they appear farther away. Dust lies in the galactic plane. Basic astrometry (parallax) was used to determine the distances to close stars and other methods were used to extend the distance measurements with so called standard candles as Cepheid variables. The distance equation was developed by observing stars in the galactic plane, hence the wrong distance relationship emerged when dust was ignored. But this does not directly affect the distances to globular clusters as they lie above or below the galactic plane where there is little dust. But using the wrong equation led to the wrong globular cluster distances.

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#10
In reply to #9

Re: Measurements of the Universe

04/26/2007 11:53 PM

Hi GWJ

Thanks for the valuable inputs - the standard candles make for fascinating astronomy!

BTW, I spotted a "bad" in my previous post when I wrote: "... parallax of up to 7500 ly at ~10% accuracy (2.37±0.23 kpc), where 1 kpc = 3.26 ly."

The k in kpc actually (and obviously) means 1 kpc = 3260 ly, or more precisely, 3261.6 ly.

-J

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