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Astronomy as a physical science is a cacophony of different practices and methods as the universe exists it is too massive and distant to prescribe the more archaic forms of research. Much of it is simply based on observation, and in this way only using the few eyes that have been educated formally limits the scope of the statistical research that can be taken. Instead, many astronomical organizations rely on amateur observers and hobbyists to join in on their missions. This is a great way to continue your interest in astronomy and to contribute to knowledge that can better help the field understand the nature of the universe, and the dynamics of our own planet though it.
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The American Association of Star Observers is an organization that brings together a collection of professionals and amateurs to study variable stars, which are ones that have an inconsistent pattern of brightness. In this situation there are a large number of amateurs that help accrue data that is then used in research by astronomers doing work. Long format observation as well as photo work is collected by amateurs and then sent down the line, as well as work done in conjunction with astronomy professionals. This non-profit organization is one of the premiere groups that observe this type of star phenomenon and use people from different locations where they then send their collected research to the headquarters. The AAOSO is open to almost all volunteers, and is also a great way for students to gain experience in the field.
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There are a number of other organizations based in the U.S. that are utilizing this pool of interested amateurs, mainly because it is a good way to staff their projects without having to pay professional wages. There are huge populations of interested people who do not use this as their primary focus but still have a lot to contribute. This goes into all areas of astronomy, from observation to classification work like that done by the Galaxy Zoo. This project was started because of the need to classify images of galaxies that were seen through the Sloan Digital Sky Survey, which was an eight year project that got deep space images through a massive telescope in the Apache Point Observatory in New Mexico. All of these images then needed to be put into classifications according to their design for research, which would take an unreasonable amount of time if working astronomers had to go through all of them. Now it takes volunteers who are given an image of a galaxy and are then asked to go through a series of questions, such if it is round or if it is a disk with a viewable edge, and then the answers of the cumulative questions allow you to classify it. This has been a very successful project, so much so that it continues to do so with new data be accumulated - Galaxy Zoo 2.
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Many programs really just want to harness the power of home computers all across the country. The Einstein at Home Project asks people to voluntarily give up idle time on their computer to contribute processing power for use in gravitational wave detection. This process is based on the idea that gravitational areas alter space and time, especially when it is from massively concentrated areas such as neutron stars and black holes. This comes from Einstein proposition that our universe is made up of gravitational waves and sources of massive waves create fundamental alterations, so this project uses that as inspiration to find these types of massive areas. This uses data coming from Laser Interferometer Gravitational Wave Observatory domestically and Germany's GEO 600. Einstein at home utilizes the home computer in a similar fashion as SETI at Home, which is currently the largest scientific use of home computer participants. SETI, known as the Search for Extraterrestrial Intelligence, is a project that many people are aware of because of its commitment to the search for alien life. Utilizing radio telescopes to search for possible signals from space that could be from a technologically advanced society looking to communicate intergalactically. Just like Einstein at Home, this uses home computers of volunteers to process data that is being received. The project, out of UC Berkeley, breaks up the data stream that is coming up and allows it to be processed in pieces by different computers around the country. This is done in the form of a screensaver that volunteers download, which then uses computer resources when inactive.
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These are only a few of the many programs that use amateurs as part of major operations. This trickles down to even the local observatory and science center where contributions of time from the community are as important as financial and professional support.