This leads to Quantization of θ because there are only specific values that become possible for θ when we impose that after n rotations it has to return back to its same starting point. **
And the following are those values:
The problem of finding the values of θ for given values of n is more generally known as the N roots of unity.
We will leave it as an exercise, but the following animation plots all the possible the values of θ for integer values of n=2,3,4,.. from the above equation that we found:
And those are the roots of the equation z^n = z i.e if you start at these points on the unit circle and make n rotations you will get back to the same point that you started with.
Kapustin Yar (known today as Znamensk) is one of the Soviet Union’s first rocket launch and missile development sites. The test ground was established on May 13, 1946, and to mark its 70th anniversary Russia’s Defense Ministry has declassified revealing photographs of the site that offer a peek inside the top secret military complex.
These are some incredible photos! Up until the activation of Vostochny last month, Kapustin Yar, Plestek, and Baikonur were the main launch sites of the Soviet union/Russia.
Skylab Sketch (Archive: NASA, Marshall, 08/19/66) by NASA’s Marshall Space Flight Center Via Flickr: At a meeting at the Marshall Space Flight Center in Huntsville, Alabama, on August 19, 1966, George E. Mueller, NASA Associate Administrator for Manned Space Flight, used a felt pen and poster paper to pin down the final conceptual layout for the budding space station’s (established as the Skylab in 1970) major elements. General Davy Jones, first program director, added his initials and those of Dr. Mueller in the lower right corner. The goals of the Skylab were to enrich our scientific knowledge of the Earth, the Sun, the stars, and cosmic space; to study the effects of weightlessness on living organisms, including man; to study the effects of the processing and manufacturing of materials utilizing the absence of gravity; and to conduct Earth resource observations. The Skylab also conducted 19 selected experiments submitted by high school students. Skylab’s 3 different 3-man crews spent up to 84 days in Earth orbit. The Marshall Center developed and integrated many of the major components of the Skylab: the Orbital Workshop (OWS), Airlock Module (AM), Multiple Docking Adapter (MDA), Apollo Telescope Mount (ATM), Payload Shroud (PS), and most of the experiments. Marshall provided the Saturn IB launch vehicles for three Apollo spacecraft and crews and a Saturn V launch vehicle for the Skylab.
Image credit: NASA
Original image:
www.nasa.gov/centers/marshall/history/skylab_140819.html#…
More about Marshall Center history:
www.nasa.gov/centers/marshall/history/index.html
Marshall History Album on Flickr:
www.flickr.com/photos/nasamarshall/sets/72157636868630444/
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These official NASA photographs are being made available for publication by news organizations and/or for personal use printing by the subject(s) of the photographs. The photographs may not be used in materials, advertisements, products, or promotions that in any way suggest approval or endorsement by NASA. All Images used must be credited. For information on usage rights please visit: www.nasa.gov/audience/formedia/features/MP_Photo_Guidelin…
Nerds workplace 1 .jpg by Matt Via Flickr: I was shooting some photos at a nearby bunker from the Second World War. The different rooms, which are for rent, were shown to me by the owner. They are now used for other purposes. In addition to a recording studio, there was also the “Commodore 64 Club”. I could take a look of it…
Triple Jupiter Eclipse by NASA on The Commons Via Flickr: Description: At first glance, Jupiter looks like it has a mild case of the measles. Five spots - one colored white, one blue, and three black are scattered across the upper half of the planet. Closer inspection by NASA’s Hubble Space Telescope reveals that these spots are actually a rare alignment of three of Jupiter’s largest moons - Io, Ganymede, and Callisto - across the planet’s face. In this image, the telltale signatures of this alignment are the shadows [the three black circles] cast by the moons. Io’s shadow is located just above center and to the left; Ganymede’s on the planet’s left edge; and Callisto’s near the right edge. Only two of the moons, however, are visible in this image. Io is the white circle in the center of the image, and Ganymede is the blue circle at upper right. Callisto is out of the image and to the right. On Earth, we witness a solar eclipse when our Moon’s shadow sweeps across our planet’s face as it passes in front of our Sun. Jupiter, however, has four moons roughly the same size as Earth’s Moon. The shadows of three of them occasionally sweep simultaneously across Jupiter. The image was taken March 28, 2004, with Hubble’s Near Infrared Camera and Multi-Object Spectrometer. Seeing three shadows on Jupiter happens only about once or twice a decade. Why is this triple eclipse so unique? Io, Ganymede, and Callisto orbit Jupiter at different rates. Their shadows likewise cross Jupiter’s face at different rates. For example, the outermost moon Callisto orbits the slowest of the three satellites. Callisto’s shadow moves across the planet once for every 20 shadow crossings of Io. Add the crossing rate of Ganymede’s shadow and the possibility of a triple eclipse becomes even more rare. Viewing the triple shadows in 2004 was even more special, because two of the moons were crossing Jupiter’s face at the same time as the three shadows. Jupiter appears in pastel colors in this photo because the observation was taken in near-infrared light. Astronomers combined images taken in three near-infrared wavelengths to make this color image. The photo shows sunlight reflected from Jupiter’s clouds. In the near infrared, methane gas in Jupiter’s atmosphere limits the penetration of sunlight, which causes clouds to appear in different colors depending on their altitude. Studying clouds in near-infrared light is very useful for scientists studying the layers of clouds that make up Jupiter’s atmosphere. Yellow colors indicate high clouds; red colors lower clouds; and blue colors even lower clouds in Jupiter’s atmosphere. The green color near the poles comes from a thin haze very high in the atmosphere. Ganymede’s blue color comes from the absorption of water ice on its surface at longer wavelengths. Io’s white color is from light reflected off bright sulfur compounds on the satellite’s surface. In viewing this rare alignment, astronomers also tested a new imaging technique. To increase the sharpness of the near-infrared camera images, astronomers speeded up Hubble’s tracking system so that Jupiter traveled through the telescope’s field of view much faster than normal. This technique allowed scientists to take rapid-fire snapshots of the planet and its moons. They then combined the images into one single picture to show more details of the planet and its moons.
Image # SPD-SLRSY-3083
Date: March 28, 2004
Explorer XVII Satellite by NASA on The Commons Via Flickr: Weighing 405 lbs. (184 kg), this 35-inch (89-cm) pressurized stainless steel sphere measured the density, composition, pressure and temperature of Earth’s atmosphere after its launch from Cape Canaveral on April 3, 1963. The mission was one of three that Goddard Space Flight Center specifically conducted to learn more about the atmosphere’s physical properties, knowledge that they ultimately used for scientific and meteorological purposes. Explorer XVII carried two spectrometers, four vacuum pressure gauges and two electrostatic probes. Before it reached its intended orbit that ranged from 158 to 570 miles (254-917 km) above Earth, the satellite was spun up to about 90 rpm.
Image # : G-63-4001
Date: January 1, 1963