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Finding and Defining Real Astronomical Distances
By using observation and mathematics, the size of the earth and the disatnce to the sun, moon and stars can be meaured
Measuring the size of the earh and the distance to the sun and moon have a long history
There are a lot of people suggesting that the universe is much smaller than that given by the findings of astronomers. If this is true, then we never would have made it to the moon, Mars and the other planets based on findings and calculations that have their roots in ancient Greece and have been refined ever since based on observation, countless experiments, well defined measure and mathematics. Among the people who progressed our understanding are Galileo, Newton and all their successors. Where we can directly measure things, beginning with the size of the earth, we can gain some concrete understanding between the earth, the moon and sun. Getting a fix on the size of the Earth is crucial in obtaining a foundation in understanding the rest of the solar system. From this, we can measure the distance to the nearest stars based on something called parallax. For all of this, we need to define a measure and stick to it in order to establish a relationship with the rest. For our era, that measure has been until going metric, the foot, the yard and the mile. We begin our journey with Eratosthenes of Syene who was the first person to coin the word geography. He was also the inventor of latitude and longitude. But by far, his most important contribution was determining the size (circumference) of the earth. In his era of the 3^{rd} and 2^{nd} century BC, the measure in use was the stade.
Eratosthenes calculated the circumference of the earth without ever leaving Egypt where he was staying when he made it. He knew that on the summer solstice at local noon in the Ancient Egyptian city of Swenet, known in Greek as Syene, and today as Aswan, on the Tropic of Cancer, the sun would appear directly overhead at what is called the zenith. Eratosthenes also knew, from his measurement already established, that in his hometown of Alexandria, the angle of elevation of the sun would be 1/50^{th} of a full circle; 7°12' directly south of the zenith at the same time. As Alexandria was north of Syene he concluded that the distance from Alexandria to Syene must be 1/50^{th} of the total circumference of the earth. His estimated distance between the two cities was 5000 stadia; or about 500 geographical miles/800 kilometres using today's measures; by estimating the time that he had taken to travel from Syene to Alexandria by camel. He rounded off the result to a final value of 700 stadia per degree, which implies a circumference of 252,000 stadia for the earth. The exact size of the stadion he used is often disputed in argument. The common Attic stadium was about 185 meters, which would imply a circumference of 46,620 kilometres, i.e. 16.3% too large. However, if Eratosthenes used the "Egyptian stadium" (1) of about 157.5 meters, his measurement turns out to be 39,690 kilometres, an error of less than 1% (2). Modern sources say that his measure was out by only 400 miles. This was to go uncontested until the modern era. We can do a similar thing today with greater accuracy. In the offing, while most people thought the earth to be flat, Eratosthenes proved it to be a globe. With some simple math, we can then derive the diameter and use this as a base line for trigonometric measures of distances to the moon and sun.
From here, we have a base line (diameter of the earth = 12,756.2kilometres) where we can then use angular measurements simultaneously on opposite sides of the planet to find the distance to the moon and sun for a series of times during the moon's and earth's orbital periods to get a mean orbital distance. We get the fundamentals of trigonometry from Pythagoras.
Six trigonometric functions exist in terms of an (x, y) point located on the terminal side of the input angle This is based on the Cartesian system of plotting a quadratic relationship. You should be familiar with:

The (x, y) Cartesian coordinate plane.

How to find the distance from the origin to an (x, y) point.

The graphs of the trigonometric functions.

Angles in the standard position.

Radian measure for angles.
The six trigonometric functions, (sine, cosine, tangent, cotangent, secant and cosecant), are usually thought to accept an angle as input and give us an output in a pure number. For the purposes of the definitions this relative angle is to be placed in standard position (x,y) = (0,0). We will be concerned with any (x, y) point located on the terminal side of this angle or at the end furthest from the measured angle. These definitions are based on such an (x, y) point. These definitions also use the distance from the origin to the (x, y) = (0,0) point. This distance can be referred to as r and can be calculated using;
r = √(x^{2} + y^{2})
The six definitions are:
sin(angle) = y/r
cos(angle) = x/r
tan(angle) = y/x (x not equal to zero)
csc(angle) = r/y (y not equal to zero)
sec(angle) = r/x (x not equal to zero)
cot(angle) = x/y (y not equal to zero)
With this basic tool set and known distances and angles, we can find the distance to the moon using an anglesideangle calculation, or two such calculations as described above, done simultaneously. For this, we can be content with the use of the first two definitions. One of the best times to measure that distance, is during a solar eclipse, observed by two people at a synchronized moment that are located a known distance apart to get the parallax angle. In this case, the sun will represent a “fixed” background where we can see the movement of the moon and to take our angle of sight to one of the moon's “limbs” as described in astronomy. We also need to know for accuracies sake, the tracks of both the sun and moon. The sun represents the background against which we will take simultaneous sightings of the parallax difference from two separate locations on earth that witness the solar eclipse. Some caution should be taken when accounting for the earth's curvature. Ultimately what will be found are three angles and one distance relation that we know. There will be the angle for one observer that is the angle between the line to the moon's leading limb and the ground. There will also be the angle for the second observer that is also the angle between the line to the moon's leading limb and the ground. Now as the earth is curved, so these angles are deceptive. We do know the distance between the two observers over the curvature of the earth. As we already know the circumference of the earth, so we can determine the flat line distance between the two observers from the known curved distance. This is somewhat the reverse of the calculation that Eratosthenes used to find the size of the earth. We just need the straight line distance and connection to the two observers to get the correct angles. Then we have the knowledge to determine the angle of the two lines on sight at the moon's limb. Other have done this same calculation countless times by now and we know the mean distance to the moon as being 384,399 kilometres (238,857 miles) for what is called the semimajor axis.
Finding the distance to the sun is a little trickier, but it can be inferred a few ways using the same methods. Owing to the brightness of the sun, we cannot use parallax against the “fixed” stars. Instead, we can calculate the distance to Venus seen from opposite sides of the earth at the same synchronized moment by two observers. We can at the same time, take a sighting on the sun for both limbs to obtain an averaged line of sight. In this case, it would be good to have four separate observers doing all of this at the same time; two for Venus and two for the sun. With the two Venus sightings, the diameter line of the Earth, and the angles to Venus from the Earth base line, we can find the distance to Venus. Collect the averaged angle of sight for the sun in both cases. Do a little constructive geometry for the earth, Venus and the Sun and make the calculations. The angle that you find between the Earth, Venus to Sun can be found. From here we can derive all the information we want on the bodies in question. The Sun Earth mean distance will eventually be our base line for calculating parallax distance to the stars measured half a year apart. Again, there have been countless observations of this type done, so we know that the sun's mean distance from earth is 149,597,871 kilometres (92,955,820.5 miles), also known as the Astronomical Unit (AU). Our base line to measure distance to the stars is twice this at 299,195,742 kilometres (185,911,641 miles).
As we tend to observe the stars at night, we can see the nearby stars shift against the background of far more distant ones over the course of a year. This parallax can be used to sight the stars the same way we sight the moon, sun and Venus. These calculations are far more tricky due to distances and fine angular adjustment required. Once more, observation has been cone countless times and we can establish distance relations to stars out to about 1,000 light years. Using the base line mean diameter of the Earth's orbit, and drawing an isosceles triangle, we create one with an apex of one second of angle. This is the basic construction of the parsec that is defined as;
A unit of astronomical length based on the distance from Earth at which stellar parallax is one second of arc and equal to 3.258 lightyears, 3.086 × 10^{13} kilometres, or 1.918 × 10^{13} miles. With stars there is an additional problem insofar as they are themselves in motion. By watching them over several years, we can determine that difference year by year and calculate for this change. As far as we know, there are no stars within one parsec of the solar system, despite what some will say. The final proof in all of this, is to repeat the observations, which is part of what hard science is all about. We have to be sceptical of unverified claims, subject them to observation and rigorous testing. For stars outside of the parallax limit, we have to turn to other methods that as yet have to be solidly proven.
References:
1. Isaac Moreno Gallo (3–6 November 2004). "Roman Surveying" (PDF). Archived from the original on 20070205
2. There is a huge Eratosthenes got it right literature based upon attacking the applicability of the standard 185m stadium to his experiment. Among advocates: F. Hultsch, Griechische und Römische Metrologie, Berlin, 1882; E. LehmannHaupt, Stadion entry in Paulys RealEncyclopädie, Stuttgart, 1929; I. Fischer, Q. Jl. Royal Astronomical. Society. 16.2:152–167, 1975; Gulbekian (1987); Dutka (1993). The means employed include worrying various ratios of the stadium to the unstably defined "schoenus", or using a truncated passage from Pliny. (Gulbekian just computes the stadium from Eratosthenes' experiment instead of the reverse.) A disproportionality of literature exists because professional scholars of ancient science have generally retarded such speculation as special pleading and so have not bothered to write extensively on the issue. Skeptical works include E. Bunbury's classic History of Ancient Geography, 1883; D. Dicks, Geographical Fragments of Hipparchus, University of London, 1960; O. Neugebauer, History of Ancient Mathematical Astronomy, Springer, 1975; J. Berggren and A. Jones, Ptolemy's Geography, Princeton, 2000. Some difficulties with the several arguments for Eratosthenes' exact correctness are discussed by Rawlins in 1982b page 218 and in his Contributions and Distillate. See also, at [2], "The Shores of the Cosmic Ocean", chapter 1 of Cosmos: A Personal Voyage, a TV series by Carl Sagan, Ann Druyan and Steven Sotter (1978–1979), where a description of Eratosthenes' experiment is presented.
3. http://zonalandeducation.com/mmts/trigonometryRealms/TrigFuncPointDef/TrigFuncPointDefinitions.html