Astronomers estimate the distance of nearby objects in space by using a method called stellar parallax, or trigonometric parallax. Simply put, they measure a star's apparent movement against the background of more distant stars as Earth revolves around the sun.

Parallax is “the best way to get distance in astronomy,” said Mark Reid, an astronomer at the Harvard Smithsonian Center for Astrophysics. He described parallax as the “gold standard” for measuring stellar distances because it does not involve physics; rather, it relies solely on geometry. The method is based on measuring two angles and the included side of a triangle formed by the star, Earth on one side of its orbit and Earth six months later on the other side of its orbit, according to Edward L. Wright, a professor at UCLA.

If we do not know the distance to an object, it is not possible to measure an apparent sideways change in its position against the background in length units. However, we can measure it in angular units, i.e., the portion of a complete circle the object appears to move against the background.

Once we have determined the apparent angular displacement of the object caused by viewing it from a different angle, or its parallax, we need to measure the distance between the observation points, or the baseline. We then take the baseline measurement and divide that by the tangent of the parallax angle to obtain the distance to the object.

To measure the distance of a star, astronomers use a baseline of 1 astronomical unit (AU), which is the average distance between Earth and the sun, about 93 million miles (150 million kilometers). They also measure small angles in arcseconds. One arcsecond is a tiny angle. It is equivalent to a lateral shift in position of 0.017 inches seen from 100 yards (0.48 millimeters at 100 meters).

If we divide the baseline of one AU by the tangent of one arcsecond, it comes out to about 19.2 trillion miles (30.9 trillion kilometers), or about 3.26 light years. This unit of distance is called a parallax second, or parsec (pc). There is a reciprocal relationship between distance and parallax. For example, a star with a parallax of half an arcsecond would have a distance of 2 pc, and a star with a parallax of one-tenth of an arcsecond would have a distance of 10 pc.

However, since no stars are close enough to have a parallax of 1 arcsecond, measurements have to be even more precise than that. This made measurement of parallax quite difficult for early astronomers.

The first known astronomical measurement using parallax is thought to have occurred in 189 B.C., when a Greek astronomer, Hipparchus, used observations of a solar eclipse from two different locations to measure the distance to the moon, Reid said.

Hipparchus noted that on March 14 of that year there was a total solar eclipse in Hellespont, Turkey, while at the same time farther south in Alexandria, Egypt, the moon covered only four-fifths of the sun. Knowing the baseline distance between Hellespont and Alexandria — 9 degrees of latitude or about 600 miles (965 km), along with the angular displacement of the edge of the moon against the sun (about one-tenth of a degree), he calculated the distance to the moon to be about 350,000 miles (563,300 km), which was nearly 50 percent too far. His mistake was in assuming that the moon was directly overhead, thus miscalculating the angle difference between Hellespont and Alexandria.

In 1672, Italian astronomer Giovanni Cassini and a colleague, Jean Richer, made simultaneous observations of Mars, with Cassini in Paris and Richer in French Guiana. Cassini computed the parallax, determined Mars' distance from Earth. This allowed for the first estimation of the dimensions of the solar system.

The first person to succeed at measuring the distance to a star using parallax was F.W. Bessel, who in 1838 measured the parallax angle of 61 Cygni as 0.28 arcseconds, which gives a distance of 3.57 pc. The nearest star, Proxima Centauri, has a parallax of 0.77 arcseconds, giving a distance of 1.30 pc.

Parallax is an important rung in the cosmic distance ladder. By measuring the distances to a number of nearby stars, astronomers have been able to establish relationships between a star’s color and its intrinsic brightness, i.e., the brightness it would appear to be if viewed from a standard distance. These stars then become “standard candles.”

Then if a star is too far away to measure its parallax, astronomers can match its color and spectrum to one of the standard candles and determine its intrinsic brightness, Reid said. Comparing this to its apparent brightness, we can get a good measure of its distance by applying the 1/r2 rule. This rule states that the apparent brightness of a light source is proportional to the square of its distance. If you project of a one-foot square image onto a screen, and then move the projector twice as far away, the new image will be 2 feet by 2 feet, or 4 square feet. The light is spread over an area four times larger, and it will be only one-fourth as bright as when the projector was half as far away. If you move the projector three times farther away, the light will cover 9 square feet and appear only one-ninth as bright.

If a star measured in this manner happens to be part of a distant cluster, we can assume that all of those stars are the same distance, and we can add them to the library of standard candles.

In 1989, the European Space Agency (ESA) launched an orbiting telescope called Hipparcos (named after Hipparchus). Its main purpose was to measure stellar distances using parallax with an accuracy of 2–4 milliarcseconds (mas). According to their website, “ESA’s Hipparcos satellite pinpointed more than 100,000 stars, 200 times more accurately than ever before.” They were able to determine distances of stars out to 100 pc with an accuracy of plus-or-minus 10 percent. Their results are available in an online searchable catalog

The ESA’s successor mission to Hipparcos is Gaia, which was launched into Earth orbit in 2013. ESA describes it as “an ambitious mission to chart a three-dimensional map of our galaxy, the Milky Way, in the process revealing the composition, formation and evolution of the galaxy.” Among its tasks will be to obtain distances of 1 billion stars, about 1 percent of all the stars in the Milky Way, using parallax measurements with an accuracy of 24 microarcseconds (μas).

While parallax measurements at optical wavelengths have improved greatly over time, according to Reid, we can obtain even more accurate results from radio astronomy with a technique called very long baseline interferometry using the Very Long Baseline Array (VLBA). He describes the VLBA as a global network of radio telescopes, synchronized by atomic clocks that produce an effective aperture of thousands of miles. This is compared to our largest optical telescopes, which are only about 400 inches (10 m). [Related: Milky Way's Structure Mapped in Unprecedented Detail]

The theoretical maximum angular resolution of a telescope can be calculated as the wavelength being observed divided by the size of the aperture, according to Reid. However, atmospheric effects reduce this considerably. A ground-based optical telescope with an effective aperture of 26 feet (8 m) has a resolution of about 50 milliarcseconds (mas). This is matched by the much smaller 2.4-meter Hubble Space Telescope because it operates above the atmosphere. The VLBA generally observes wavelengths of a few centimeters, but because the effective aperture is several thousand kilometers, and the longer wavelengths are little affected by the atmosphere, it can resolve parallax motions of 10 μas allowing distance measurements of objects at up to 100,000 parsecs (100 kpc).

The objects that Reid and his team generally observe for these measurements include young active stars that emit synchrotron radiation.  This is caused by the star ejecting charged particles, typically electrons, which tend to spiral around intense magnetic field lines. Because the particles are being forced to move in a curved path, this causes them to emit photons at radio wavelengths. Another source of radio emissions is gas clouds around even younger and more active stars that act as natural masers

Another application of parallax is the reproduction and display of 3D images. The key is to capture 2D images of the subject from two slightly different angles, similar to the way human eyes do, and present them in such a way that each eye sees only one of the two images.

For example, a stereopticon, or stereoscope, which was a popular device in the 19th century, uses parallax to display photographs in 3D. Two pictures mounted next to each other are viewed through a set of lenses. Each picture is taken from a slightly different viewpoint that corresponds closely to the spacing of the eyes. The left picture represents what the left eye would see, and the right picture shows what the right eye would see. Through a special viewer, the pair of 2D pictures merge into a single 3D photograph. The modern View-Master toy uses the same principle. [Video: Queen's Brian May Assembles First Stereoscopic Pluto Image]

Another method for capturing and viewing 3D images, Anaglyph 3D, separates images by photographing them through colored filters. The images are then viewed using special colored glasses. One lens is usually red and the other cyan (blue-green). This effect works for movies and printed images, but most or all of the color information from the original scene is lost.

Some movies achieve a 3D effect using polarized light. The two images are polarized in orthogonal directions, or at right angles to each other, typically in an X pattern, and projected together on the screen. The special 3D glasses worn by audience members block one of the two overlaid images to each eye.

Some 3D televisions and computer monitors also use polarization to separate the images seen by the left and right eye. However, most of today’s 3D televisions use an active-shutter scheme to display images for each eye that alternate at 240 Hz. Special glasses are synchronized with the TV so they alternately block the left and right images to each eye.

Other 3D display devices use “parallax blocking” whereby two vertically interlaced images are viewed through a “picket fence” of opaque vertical lines. The lines block one of the images when viewed by one eye and the other image when viewed by the other eye. The disadvantages of this system are that it only works for a specific viewing angle, and it cuts the horizontal resolution and the total light by half.

Virtual reality gaming headsets, such as the Oculus Rift and the HTC Vive, produce 3D virtual environments by projecting an image from a different viewing angle to each eye to simulate a parallax effect.

There are also many uses for 3D imaging in science and medicine. For example, CT scans — which are actual 3D images of regions inside the body, not just a pair of 2D projections — can be displayed so each eye sees the image from a slightly different angle to produce a parallax effect. The image can then be rotated and tilted as it is being viewed. Scientists can also use 3D images to visualize molecules, viruses, crystals, thin film surfaces, nanostructures, and other objects that cannot be seen directly with optical microscopes because they are too small or are imbedded in opaque materials.