When assessing the comparative sizes of the planets of the solar system, it is customary to operate with such a concept as astronomical unit. What is it and what is it equal to?

History of the introduction of the astronomical unit

Thanks to centuries-old efforts of scientists (in particular, Keplerian celestial mechanics), we have learned that the planets, each in its own orbit, revolve around the Sun. And the stars, which look like sparkles in the sky, are at such distances from us that it is impossible to even imagine. Constantly expanding after new discoveries by scientists, the Universe has expanded so much that now no one even knows how big it is. Astronomy, rapidly developing, has become one of the most advanced sciences.

Concept of astronomical unit

Scientists came to the conclusion that the Earth is not and has never been the center of the Universe 300 years ago. After numerous observations and endless repeated checks, astronomers began to discover the true dimensions of our cosmic home - the Solar System. As it turned out, they turned out to be so huge that earthly units of measurement were clearly not suitable here. Kilometers to the nearest planets were indicated by numbers with many zeros. And no one, except the scientists themselves, even knew what to call these numbers. That is why astronomers introduced a special unit to measure the distances from the Sun to the planets and between the planets in the solar system. That's what it's called - astronomical unit(symbol a.e.) and is equal to the average distance from the Earth to the Sun. This is approximately 150 million kilometers (more precisely, 149,597,870.691 km). In ordinary astronomical calculations, the rounded value of 149,600,000 km is used..

Not so little, considering that the earth's equator is the longest path that can be laid on our globe - approximately 40,000 kilometers long. And the Moon, the Earth’s satellite and the closest celestial body, revolves around the Earth at a distance of more than 380,000 kilometers.

Why is the distance from the Earth to the Sun taken as a measurement measure? Yes, because the Sun is the central body of the Solar System, and the Earth is the location of observers and revolves in an almost circular (elliptical) orbit. For this reason, the radius of this orbit was adopted as the unit of measurement.

The above is illustrated by the following schematic diagram:

So, astronomical unit is a measure of distances to space objects, equal to the semi-major axis of the Earth's elliptical orbit and, according to the properties of the ellipse, the average distance of the Earth from the Sun. This definition satisfies not only amateurs, but also most professional astronomers.

Examples of distances in astronomical units

Thus, an astronomical unit is almost 400 times greater than the distance from the Earth to the Moon. It is also quite suitable for measuring distances between planets. For example, the distance from Earth to Mars is on average 0.3 astronomical units. Mars is further from the Sun than Earth. Thus, it is easy to calculate that the distance from the Sun to Mars is 1.52 astronomical units. Even to distant Jupiter from the Sun it is a little more than 5 astronomical units. The distance from Earth to Uranus is about 20 astronomical units. The orbital radius of Neptune, one of the most distant objects in the solar system, is equal to 30 astronomical units. Sirius is a double star. Companion stars Sirius A and Sirius B rotate among themselves at a distance of 20 astronomical units.

Light travels the distance from the Earth to the Sun in approximately 500 seconds (8 minutes 20 seconds). Interestingly, this distance has a steady tendency to slowly increase at a rate of approximately 15 meters per 100 years. This may be due to the loss of solar mass due to the solar wind. However, this effect of increasing the astronomical unit is so slow that it can be completely neglected, since it is an order of magnitude greater than the calculated values.

Several generations of scientists successfully used the astronomical unit. The distances within the solar system, expressed in this measure of measurement, were relatively small, and they were easy and convenient to work with. And most importantly, everyone understood them. Any schoolchild, looking at distances in astronomical units, could say that Venus is located closer to the Sun than the Earth. And Jupiter is approximately halfway from the Sun to Saturn.

But it turned out that they rejoiced too early. As soon as it was possible to determine the distance to the nearest stars, it became clear that in the stellar world the astronomical unit was too small and therefore unsuitable for measurements.

Distance in AU

Astronomical unit (Russian designation: A. e.; international: au) is a historical unit of measurement of distances in astronomy, approximately equal to the average distance from to.

Light travels this distance in approximately 500 seconds (8 minutes 20 seconds).

It is mainly used to measure distances between objects, extrasolar systems, and between components of binaries.

In September 2012, the 28th General Assembly of the International Astronomical Union in Beijing decided to link the astronomical unit to the International System of Units (SI). An astronomical unit by definition is exactly 149,597,870,700 meters. In addition, the IAU decided to standardize the international designation of the astronomical unit: “au”. Sometimes the notation “a. u." or "AU". There is also an international standard ISO 80000-3, which recommends the use of the designation “ua”.

In the Russian Federation, an astronomical unit is approved for use as a non-systemic unit without a time limit with the field of application “astronomy”. In accordance with GOST 8.417-2002, the name and designation of the astronomical unit is not allowed to be used with submultiple and multiple SI prefixes.

Previous definitions

In accordance with the decision of the 10th General Assembly of the IAU in 1976, the astronomical unit was defined as the radius of the circular orbit of a test body in isotropic coordinates, the angular velocity of which, neglecting all bodies of the Solar system except the Sun, would be exactly equal to 0.017 202 098 95 radians on ephemeris days. In the IERS 2003 constant system, the astronomical unit was assumed to be equal to 149,597,870.691 km.

Story

Since the advent of the heliocentric system, and especially Keplerian celestial mechanics, relative distances in the Solar system (excluding the too close one) have become known with good accuracy. Since the Sun is the central body of the system, and the Earth, revolving in an almost circular orbit, is the location of observers, it was natural to take the radius of this orbit as a unit of measurement. However, there was no way to reliably measure the value of this unit, that is, to compare it with earthly scales. The Sun is too far away to reliably measure parallax from Earth. The distance to the Moon was known, but based on the data known in the 17th century, it was not possible to estimate the ratio of the distances to the Sun and the Moon - observation of the Moon does not provide the required accuracy, and the ratio of the masses of the Earth and the Sun was also not known.

In 1672, Giovanni Cassini, together with his collaborator Jean Richet, measured parallax. Since the orbital parameters of the Earth and Mars were measured with high accuracy, it became possible to estimate the size of the astronomical unit - in modern units they turned out to be approximately 140 million km. Subsequently, refined measurements of the astronomical unit were carried out using passages across the solar disk. The approach of Eros to the Earth in 1901 and the measurement of its parallax made it possible to obtain an even more accurate estimate.

The astronomical unit was also refined using radar. The location of Venus in 1961 established that the astronomical unit is equal to 149,599,300 km. The possible error did not exceed 2000 km. Repeated radar detection of Venus in 1962 made it possible to reduce this uncertainty and clarify the value of the astronomical unit: it turned out to be equal to 149,598,100 ± 750 km. It turned out that before the 1961 location, the value of a. e. was known with an accuracy of 0.1%.

Long-term measurements of the distance from the Earth to the Sun have recorded its slow increase at a rate of about 15 meters per hundred years (which is an order of magnitude greater than the accuracy of modern measurements). One of the reasons may be the loss of mass by the Sun (due to ), but the observed effect significantly exceeds the calculated values.

Some distances

• The orbital radius of the most distant planet in the Solar System is about 30 AU. e.
• As of April 23, 2016, it was located at a distance of 134.75 a. e. from the Sun, moving away from it at a speed of 3.6 a. e./year It is the furthest from Earth and the fastest moving object created by man.
• The distance to our closest star, Proxima Centauri, is about 270,000 AU. e.

The task of measuring cosmic distances has faced astronomers since ancient times. In one of the problems we already discussed modern methods for measuring distances to distant galaxies. But this whole epic with measuring distances began with the objects of the solar system closest to us.

Here we apply the parallax method, which is based on the fact that a specific celestial object is located not too much far away, and its position in the sky depends on where you look at it from. By the way, the stereoscopic perception of our eyes works in a similar way, with the help of which the brain determines the approximate distance to objects: the left and right eyes see the object from different (albeit close) angles. Knowing the angles and distances between the eyes - the so-called base length - you can quite accurately estimate the distance to the object (Fig. 1).

In geodesy, this method of measuring distances is called triangulation. Well, in astronomy, parallaxes are the most accurate way to calculate the distances to the stars closest to us. In this case, the semi-axis of the Earth's orbit is taken as the base and the angular position of the star is determined twice with an interval of six months. But where did it all start? How do we know the size of the Earth's orbit?

The astronomical unit (the average distance from the Earth to the Sun) - one of the main standards of distances in space - was adopted after Kepler proposed and justified the heliocentric system, in which the Earth revolves around the Sun in an (almost) circular orbit. The natural solution was to take the radius of this orbit as the unit of measurement.

Now the parameters of the earth's orbit are measured with great accuracy, but then, in the 18th century, astronomy hit a dead end. Scientists by that time were able to determine the distances to many planets in the solar system, expressing them in astronomical units. But the very value of the astronomical unit in units familiar to humans (for example, kilometers) was not precisely known.

At the same time, the radius of the Earth has already been measured quite accurately. Thus, the value of the base was reliably known, and all that was required was to measure the parallactic angle to any of the solar system objects to which the relative distance in astronomical units was known.

Therefore, astronomers around the world had great hopes for the passage of Venus across the disk of the Sun in 1761 and 1769. A properly organized observation of this phenomenon would potentially make it possible to measure the parallax of Venus relative to the parallax of the Sun (more precisely, their difference), and, knowing the radius of the Earth (the length of the base), to find out the astronomical unit.

The fact is that from different points on the Earth, the passage of Venus across the disk of the Sun looks different (Fig. 2). If it were possible to measure these trajectories at different points, then the problem would be solved, because then you can either directly find the angular dimensions of these trajectories, or the travel time, and from that find the required one. And so it happened: as a result of observations that took place in different parts of the globe, scientists were able to determine the value of the astronomical unit with fairly high accuracy.

In particular, Thomas Hornsby obtained a value for the distance from the Earth to the Sun of approximately 93,726,900 English miles (150,838,449 km), which is very close to the truth.

This problem proposes to make similar measurements of the parallax of Venus.

Task

Two photographs of the passage of Venus are given, taken simultaneously at 22:25:52 UTC on June 5, 2012 (Fig. 4). On the left is a photograph taken in Princeton, New Jersey. On the right is a photograph taken from the summit of Haleakala Volcano on the island of Maui, Hawaii.

Differences in the location of the disk of Venus are associated with parallax. It is known that the distance from Earth to Venus at the time of the photograph was 0.2887 AU. e., the distance to the Sun is 1.0147 a. i.e. the angular size of the Sun is 31.57 arc minutes, and the effective radius of the Earth can be taken as 6378.1 km. Venus was almost exactly at its zenith in Hawaii when the photographs were taken. Define According to these data and photographs, the distance from the Earth to the Sun.

Hint 1

Determining the length of the base in the general case is a rather complex issue. However, at the time of the photograph, the Sun on the island of Maui was almost exactly at its zenith. You can verify this using the Stellarium program by setting the current position in Hawaii and the time of 12 hours 25 minutes on June 5, 2012.

In this case, the length of the base is easily determined (Fig. 5).

Hint 2

Before you measure anything, you need to consider that the photographs were taken with a random camera orientation, so you need to match them correctly to measure the real displacement of Venus. This can be done using the Sun, or rather, sunspots, as a background. True, then the measured parallax will be relative, since the Sun also has its own parallax.

Solution

After fiddling around, you can compare the two proposed images of Venus on the disk of the Sun in a graphics editor. Since the boundaries of the Sun are quite blurred due to clouds and darkening towards the edge, you can focus on sunspots. It is enough to combine three pairs of spots. This is what you get as a result (photos slightly processed to highlight the edges):

Then we find the centers of two silhouettes of Venus (Fig. 7). Since we are still working with images, we can measure distances in pixels, but then, naturally, we will have to convert everything into “normal” units of length. The coordinates of the centers are as follows: C 1 (red center in Fig. 7) - X: 624.5 px, Y: 317 px, C 2 - X: 631.5 px, Y: 324.5 px.

Now we calculate the relative parallax of Venus (also in pixels):

\[ p=\sqrt((624(,)5-631(,)5)^2+(317-324(,)5)^2)=10(,)3\pm0(,)25~\text (px). \]

You might get a different number, but that's okay, because these values ​​are relative, and their specific values ​​depend on the size and resolution of the photos.

The diameter of the Sun can also be measured in pixels (Fig. 8), and this will give a conversion scale. From our pictures it turns out that D s= 936±1 px, which corresponds to a value of 31.57±0.005 arc minutes or 1894.2±0.3 arc seconds. Hence 1 px = 2.024±0.002 arcseconds.

We find that the parallax of Venus (relative to the Sun) is equal to

p vs= 10.3·2.024 = 20.9±0.5 arcseconds.

Since we want to find the absolute value of the astronomical unit, we are interested in the absolute parallax of Venus. Pay attention to fig. 9. On it pv And ps- these are the real parallaxes of Venus and the Sun, and p vs- parallax of Venus relative to the Sun (what we calculated above). It is clear from the figure that p vs = pv − ps.

Since the angles are small, we will use approximate equalities for small angles: sin φ ≈ tan φ ≈ φ in radians. Then in the notation of Fig. 9: d ⊥ /EV ≈ pv, d ⊥ /ES ≈ ps, Where EV And ES- distances from the Earth to Venus and the Sun, respectively. From here we find the real parallax:

\[ p_v=\frac(p_(vs))(1-\frac(EV)(ES))=29(,)2\pm 0(,)7~\text(arcseconds). \]

Using any mapping service with the function of measuring distances on the Earth’s surface (or some other method), we determine that the shortest distance between two observation points is 7834 km (Fig. 10). This is the length of arc AB in Fig. 9. Then α ≈ 1.2282 radians, and the length of the base can be found: d⊥ ≈ 6007.6 km.

The simplest thing remains. Knowing the base length and parallax, you can find the distance to Venus: d v = d ⊥ /pv=42±1 million km. And since it is known that the relative distance to Venus in astronomical units is 0.2887 a. e., then we get that 1 a. e. = 147±3 million km. The accuracy of these calculations could be greatly improved with higher resolution imagery.

Afterword

It is not surprising that the first more or less accurate measurements of the value of the astronomical unit were made precisely with the help of the transit of Venus. The Sun itself was a rather poor candidate for such observations, since it is not a point object, and, in addition, measurements of angles in the 18th century were quite inaccurate. For the same reason, it was quite difficult to measure the parallax of Mars.

Venus itself, which at inferior conjunction is located closer to the Earth than Mars, is also not very convenient. The fact is that in this position Venus is located directly between the Earth and the Sun and therefore represents a thin strip of a halo. And the Sun itself in this case makes it very difficult to measure the angular position of Venus relative to the background stars. Therefore, the paired passage of Venus across the disk of the Sun in 1761 and 1769 became a truly grandiose event in the world of science at that time.

Associated with parallax and the astronomical unit is another measure of length, often found in astrophysics and cosmology. As noted above, using the parallax method, astronomers today measure distances to the nearest objects outside the solar system (Fig. 11)

Due to the Earth's revolution around the Sun, the image of a star against the background of distant stars, which are not subject to (or much less subject to) the parallax effect, will shift slightly (by a parallax angle). By definition, if the parallax of a star is 1 arcsecond, then the star is at a distance of 1 parsec (abbreviated pc), which is approximately 3.26 light years. In other words, 1 parsec is the distance from which the Earth-Sun system has an angular size of only 1 arcsecond.

The distance to our closest star, Proxima Centauri, is 1.301 parsecs. The center of our Galaxy is 8000 parsecs (8 kiloparsecs). The nearest large galaxy, Andromeda, is 778 kpc.

In astrophysics and cosmology, it is this unit of measurement of distances that is used, and not light years, as many people think. In particular, for example, the Hubble constant, according to the Planck telescope, is approximately equal to 68 km/s/Mpc, that is, after every megaparsec (million parsecs), the speed of “escaping” galaxies due to the expansion of the Universe increases by 68 km/s.

Measuring distances in cosmology, as mentioned above, is the most important problem that astronomers have been facing for many decades.

Basically, the parallax method measures distances up to several hundred parsecs. However, there is also a kind of record here. It was delivered by the Hubble Telescope, which was able to measure the precise parallax of stars up to 5000 parsecs away! To do this, the telescope required a resolution of 20 microarcseconds (using an observation accumulation technique that improved measurement accuracy with limited resolution). It's like reading from Earth the writing on a piece of paper held by an astronaut on the Moon.

Farther distances are measured in other ways, for example using standard candles (such as supernovae, RR Lyrae stars, Cepheids, etc.). The problem is that all these measurements depend on specific models, and therefore are not independent. To do this, they need to be calibrated using model-independent methods such as parallax.

However, these models also have their limits of applicability, beyond which new methods are needed, which, again, need to be calibrated on old ones. This system of methods, each of which works on more distant objects, but is calibrated on nearby objects using previous methods, is called the cosmological “ladder” of distances (see also the article by M. Musin “Star speaks to star”). And this ladder originates precisely in the method studied in this problem.

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1 kilometer [km] = 6.6845871226706E-09 astronomical unit [a. e.]

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Convert feet and inches to meters and vice versa

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More about length and distance

General information

Length is the largest measurement of the body. In three-dimensional space, length is usually measured horizontally.

Distance is a quantity that determines how far two bodies are from each other.

Measuring distance and length

Units of distance and length

In the SI system, length is measured in meters. Derived units such as kilometer (1000 meters) and centimeter (1/100 meter) are also commonly used in the metric system. Countries that do not use the metric system, such as the US and UK, use units such as inches, feet and miles.

Distance in physics and biology

In biology and physics, lengths are often measured at much less than one millimeter. For this purpose, a special value has been adopted, the micrometer. One micrometer is equal to 1×10⁻⁶ meters. In biology, the size of microorganisms and cells is measured in micrometers, and in physics, the length of infrared electromagnetic radiation is measured. A micrometer is also called a micron and is sometimes, especially in English literature, denoted by the Greek letter µ. Other derivatives of the meter are also widely used: nanometers (1 × 10⁻⁹ meters), picometers (1 × 10⁻¹² meters), femtometers (1 × 10⁻¹⁵ meters and attometers (1 × 10⁻¹⁸ meters).

Navigation distance

Shipping uses nautical miles. One nautical mile is equal to 1852 meters. It was originally measured as an arc of one minute along the meridian, that is, 1/(60x180) of the meridian. This made latitude calculations easier, since 60 nautical miles equaled one degree of latitude. When distance is measured in nautical miles, speed is often measured in knots. One sea knot equals a speed of one nautical mile per hour.

Distance in astronomy

In astronomy, large distances are measured, so special quantities are adopted to facilitate calculations.

Astronomical unit(au, au) is equal to 149,597,870,700 meters. The value of one astronomical unit is a constant, that is, a constant value. It is generally accepted that the Earth is located at a distance of one astronomical unit from the Sun.

Light year equal to 10,000,000,000,000 or 10¹³ kilometers. This is the distance that light travels in a vacuum in one Julian year. This quantity is used in popular science literature more often than in physics and astronomy.

Parsec approximately equal to 30,856,775,814,671,900 meters or approximately 3.09 × 10¹³ kilometers. One parsec is the distance from the Sun to another astronomical object, such as a planet, star, moon, or asteroid, with an angle of one arcsecond. One arcsecond is 1/3600 of a degree, or approximately 4.8481368 microrads in radians. Parsec can be calculated using parallax - the effect of a visible change in body position, depending on the observation point. When making measurements, lay a segment E1A2 (in the illustration) from the Earth (point E1) to a star or other astronomical object (point A2). Six months later, when the Sun is on the other side of the Earth, a new segment E2A1 is laid from the new position of the Earth (point E2) to the new position in space of the same astronomical object (point A1). In this case, the Sun will be at the intersection of these two segments, at point S. The length of each of the segments E1S and E2S is equal to one astronomical unit. If we plot a segment through point S, perpendicular to E1E2, it will pass through the intersection point of segments E1A2 and E2A1, I. The distance from the Sun to point I is segment SI, it is equal to one parsec, when the angle between segments A1I and A2I is two arcseconds.

In the picture:

• A1, A2: apparent star position
• E1, E2: Earth position
• S: Sun position
• I: point of intersection
• IS = 1 parsec
• ∠P or ∠XIA2: parallax angle
• ∠P = 1 arcsecond

Other units

League- an obsolete unit of length previously used in many countries. It is still used in some places, such as the Yucatan Peninsula and rural areas of Mexico. This is the distance a person travels in an hour. Sea League - three nautical miles, approximately 5.6 kilometers. Lieu is a unit approximately equal to a league. In English, both leagues and leagues are called the same, league. In literature, league is sometimes found in the title of books, such as “20,000 Leagues Under the Sea” - the famous novel by Jules Verne.

Elbow- an ancient value equal to the distance from the tip of the middle finger to the elbow. This value was widespread in the ancient world, in the Middle Ages, and until modern times.

Yard used in the British Imperial system and is equal to three feet or 0.9144 meters. In some countries, such as Canada, where the metric system is adopted, yards are used to measure fabric and the length of swimming pools and sports fields and fields, such as golf and football courses.

Definition of meter

The definition of meter has changed several times. The meter was originally defined as 1/10,000,000 of the distance from the North Pole to the equator. Later, the meter was equal to the length of the platinum-iridium standard. The meter was later equated to the wavelength of the orange line of the electromagnetic spectrum of the krypton ⁸⁶Kr atom in a vacuum, multiplied by 1,650,763.73. Today, a meter is defined as the distance traveled by light in a vacuum in 1/299,792,458 of a second.

Computations

In geometry, the distance between two points, A and B, with coordinates A(x₁, y₁) and B(x₂, y₂) is calculated by the formula:

Calculations for converting units in the converter " Length and distance converter" are performed using unitconversion.org functions.