question: how astronomers can observe for a prolonged period of time a tiny part of the sky, with the earth rotating around its axis and the sun
comment: They mostly use a device called a heliostat. This is a device which you add to your telescope mount and configure it so that it aligns with the plane of the rotation of the planet. There is a motor attached to it, or you can turn it manually, that will rotate your telescope once every 24 hours. So the end result is that your telescope will be pointed towards the same point in the sky even if the Earth is rotating underneath it since the heliostat will counter the movement of the Earth. More fancy setups will have a fully motorized mount for the telescope which includes a heliostat function.
passages: BULLET::::- Nobeyama Radioheliograph (NoRH) is an interferometer installed at the Nobeyama Radio Observatory, Japan, formed by 84 small (80 cm) antennas, with receivers at 17 GHz (left and right polarization) and 34 GHz operating simultaneously. It continuously observes the Sun, producing daily snapshots.
   Helioseismology benefits most from continuous monitoring of the Sun, which began first with uninterrupted observations from near the South Pole over the southern summer. In addition, observations over multiple solar cycles have allowed helioseismologists to study changes in the Sun's structure over decades. These studies are made possible by global telescope networks like the Global Oscillations Network Group (GONG) and the Birmingham Solar Oscillations Network (BiSON), which have been operating for over 20 years.
  
  Section::::Types of solar oscillation.
   Historically, optical astronomy, also called visible light astronomy, is the oldest form of astronomy. Images of observations were originally drawn by hand. In the late 19th century and most of the 20th century, images were made using photographic equipment. Modern images are made using digital detectors, particularly using charge-coupled devices (CCDs) and recorded on modern medium. Although visible light itself extends from approximately 4000 Å to 7000 Å (400 nm to 700 nm), that same equipment can be used to observe some near-ultraviolet and near-infrared radiation.
  
  Section::::Observational astronomy.:Ultraviolet astronomy.
   A fix is called a "running fix" when one or more of the LOPs used to obtain it is an LOP advanced or retrieved over time. In order to get a fix the LOP must cross at an angle, the closer to 90° the better. This means the observations must have different azimuths. During the day, if only the Sun is visible, it is possible to get an LOP from the observation but not a fix as another LOP is needed. What may be done is take a first sight which yields one LOP and, some hours later, when the Sun's azimuth has changed substantially, take a second sight which yields a second LOP. Knowing the distance and course sailed in the interval, the first LOP can be advanced to its new position and the intersection with the second LOP yields a "running fix".
   Dipleidoscope
  
  A dipleidoscope is an instrument used to determine true noon, its name comes from the Greek for "double image viewer". It consists of a small telescope and a prism that creates a double image of the sun. When the two images overlap, it is local true noon. The instrument is capable of determining true noon to within ten seconds.
  
  The dipleidoscope was invented by Giovanni Battista Amici in the first half of the 19th century.
   Gravitational lensing occurs when a distant object and a massive intermediate object form a straight line as seen from Earth. Then the gravitational field of the intermediate object bends the light from distant object, magnifying it. When the two objects are stars, as opposed to galaxies, it is called gravitational microlensing. The alignment must be very precise, in fact so precise that Albert Einstein concluded "there is no great chance of observing this phenomenon". However, modern surveys such as the Optical Gravitational Lensing Experiment (OGLE) observe millions of stars each night, and see microlensing many times each year.
   When the solar time is exactly noon, or known from another clock, the instrument can be used to determine the time of year.
   BULLET::::- Nobeyama Radio Polarimeters are a set of radio telescopes installed at the Nobeyama Radio Observatory that continuously observes the full Sun (no images) at the frequencies of 1, 2, 3.75, 9.4, 17, 35, and 80 GHz, at left and right circular polarization.
   Contrary to the classical image of an old astronomer peering through a telescope through the dark hours of the night, it is far more common to use a charge-coupled device (CCD) camera to record a long, deep exposure, allowing a more sensitive image to be created because the light is added over time. Before CCDs, photographic plates were a common method of observation. Modern astronomers spend relatively little time at telescopes usually just a few weeks per year. Analysis of observed phenomena, along with making predictions as to the causes of what they observe, takes the majority of observational astronomers' time.
   BULLET::::1. It causes the images of point sources (such as stars), which in the absence of atmospheric turbulence would be steady Airy patterns produced by diffraction, to break up into speckle patterns, which change very rapidly with time (the resulting speckled images can be processed using speckle imaging)
  
  BULLET::::2. Long exposure images of these changing speckle patterns result in a blurred image of the point source, called a "seeing disc"
  
  BULLET::::3. The brightness of stars appears to fluctuate in a process known as scintillation or twinkling
  
  BULLET::::4. Atmospheric seeing causes the fringes in an astronomical interferometer to move rapidly
   BULLET::::- Optical Gravitational Lensing Experiment (OGLE) – large-scale variability sky survey (in I and V bands), 1992-present
  
  BULLET::::- Infrared
  
  BULLET::::- Infrared Astronomical Satellite did an all sky survey at 12, 25, 60, and 100 μm, 1983
  
  BULLET::::- The 2-micron All-Sky Survey (2MASS), a ground-based all sky survey at J, H, and Ks bands (1.25, 1.65, and 2.17 μm) 1997–2001
  
  BULLET::::- Akari (Astro-F) a Japanese mid and far infrared all-sky survey satellite, 2006–2008
   A modern-day example of this type of telescope is the 8 inch (~0.2m) Flagstaff Astrometric Scanning Transit Telescope (FASTT) at the USNO Flagstaff Station Observatory. Modern meridian circles are usually automated. The observer is replaced with a CCD camera. As the sky drifts across the field of view, the image built up in the CCD is clocked across (and out of) the chip at the same rate. This allows some improvements: 
  
  BULLET::::- The CCD can collect light for as long as the image is crossing it, allowing a dimmer limiting magnitude to be reached.
   Sunspots are observed with land-based and Earth-orbiting solar telescopes. These telescopes use filtration and projection techniques for direct observation, in addition to various types of filtered cameras. Specialized tools such as spectroscopes and spectrohelioscopes are used to examine sunspots and sunspot areas. Artificial eclipses allow viewing of the circumference of the Sun as sunspots rotate through the horizon.
   BULLET::::- Parallax. For events lasting for months, the motion of the Earth around the Sun can cause the alignment to change slightly, affecting the light curve.
  
  Most focus is currently on the more unusual microlensing events, especially those that might lead to the discovery of extrasolar planets. Although it has not yet been observed, another way to get more information from microlensing events that may soon be feasible involves measuring the astrometric shifts in the source position during the course of the event and even resolving the separate images with interferometry.
  
  Section::::Observing microlensing.
   Section::::Common techniques.:Imaging techniques.
  
  Amateur astronomers engage in many imaging techniques including film, DSLR, LRGB, and CCD astrophotography. Because CCD imagers are linear, image processing may be used to subtract away the effects of light pollution, which has increased the popularity of astrophotography in urban areas. Narrowband filters may also be used to minimize light pollution.
  
  Section::::Scientific research.
   TAUVEX was mounted to the GSAT-4 spacecraft on a plate that could rotate around its axis (the MDP), enabling to point the telescopes' line-of-sight to any desired declination. Being on a geostationary satellite, the observation would therefore have been of a scanning type. A 'ribbon' of a constant declination, 0.9 degree wide, would have been scanned as time advanced, completing an entire 360 degree circuit during one sidereal day. In this mode of operation, the dwell time of a source within the detector field of view is a function of the pointing declination and of the exact location in the FOV relative to the detector diameter. The closer a source is to one of the celestial poles, the longer it resides in the TAUVEX field of view during a single scan. The longest theoretically-possible exposure is for sources at |δ|89°30'; these could be observed all day.
   Section::::Observation tools.:Photography.:Advantages.
  
  Prior to the invention of photography, all astronomy was done with the naked eye. However, even before films became sensitive enough, scientific astronomy moved entirely to film, because of the overwhelming advantages:
  
  BULLET::::- The human eye discards what it sees from split-second to split-second, but photographic film gathers more and more light for as long as the shutter is open.
  
  BULLET::::- The resulting image is permanent, so many astronomers can use the same data.
  
  BULLET::::- It is possible to see objects as they change over time (SN 1987A is a spectacular example).
  
  Section::::Observation tools.:Photography.:Blink comparator.
   Gravitational microlensing occurs when the gravitational field of a star acts like a lens, magnifying the light of a distant background star. This effect occurs only when the two stars are almost exactly aligned. Lensing events are brief, lasting for weeks or days, as the two stars and Earth are all moving relative to each other. More than a thousand such events have been observed over the past ten years.
   PROMPT consists of six 0.41-m Ritchey-Chrétien telescopes by RC Optical Systems on rapidly slewing (9°/sec) Paramount ME mounts by Software Bisque, each under a clamshell dome by Astro Haven. Five of these telescopes use rapid-readout (2 sec) Alta U47+ cameras by Apogee, which make use of E2V CCDs. The sixth is being outfitted with an LN2-cooled Micro-Cam by Rockwell Scientific for NIR imaging. Each mirror and camera coating combination has been optimized for a different wavelength range, including a u-band optimized telescope. Although other filters are available, PROMPT automatically observes GRB localizations in ugriz simultaneously. The R-band telescope will additionally measure polarizations. The polarimeter was designed and built at UNC's Goodman Laboratory for Astronomical Instrumentation.
   BULLET::::- video – the observer uses a small video camera, usually mounted in the focuser of the telescope. A Video Time Inserter (VTI) is commonly used to insert a UTC time-stamp onto each frame of the recording. Either a camcorder, DVR or a laptop computer is used to record the video stream. The video recording is later analysed to extract event times.
  
  All known lunar occultations, (both total lunar occultations as well as grazing lunar occultations) are archived at the VizieR service.
  
  Such observations are useful for;
  
  BULLET::::- refining knowledge of the positions and motions of stars
   Stable features of the solar wind can be particularly well studied. At a given time, observers on Earth have a fixed line of sight through the solar wind, but as the Sun rotates over an approximately month-long period, the perspective on Earth changes. It is then possible to do "tomographic reconstruction of the distribution of the solar wind" for the features of the solar wind which remain static.
  
  Section::::Applications.:Compact sources.
   By taking two exposures of the same section of the sky days or weeks apart, it is possible to find objects such as asteroids, meteors, comets, variable stars, novae, and even unknown planets. By comparing the pair of images, using a device such as a blink comparator, astronomers are able to find objects that moved or changed brightness between the two exposures or simply appear in one image only, as in the case of a nova or meteor. Sometimes objects can even be found in one exposure since a fast moving object will appear as a "line" in a long exposure.
   During any given 24-hour period there are approximately 16 orbits. Almost the entire globe is observed in either daylight or nighttime mode, many in both. Polar regions are observed nearly every 100 minutes.
  
  Section::::Instrument characteristics.
   In contrast to radio arrays, the largest optical arrays currently have only 6 telescopes, giving poorer image quality from the 15 baselines between the telescopes. 
  
  Most aperture synthesis interferometers use the rotation of the Earth to increase the number of different baselines included in an observation (see diagram on right). Taking data at different times provides measurements with different telescope separations and angles without the need for buying additional telescopes or moving the telescopes manually, as the rotation of the Earth moves the telescopes to new baselines.
   Accurate telescope pointing and tracking is critical for obtaining good astronomical images and astrophotographs. However, because the Earth rotates, the sky appears to be in a constant state of motion relative to the Earth. Although this movement appears to be relatively slow when viewed with the naked eye, with the high magnification and consequently smaller field of view provided by even a small telescope this motion becomes apparent on timescales of the order of seconds.
  
  Though space telescopes are not mounted on a spinning planet, they still use guide stars including those listed in the HST Guide Star Catalog.
  