cosmic distance scale, Kirkwood slots, orbital eccentricity

Cosmic distance scale

Hierarchy of methods for determining distances in the Universe (red color – direct methods, black – intermediate, blue – indirect).

The scale, or ladder of cosmic distances, is a series of methods by which astronomers determine the distances to celestial objects. Direct measurement of the distance to an astronomical object is possible only for those objects that are “close enough” (within a thousand parsecs) to the Earth. All methods for determining distances to more distant objects are based on various measured correlations between methods that work at close distances and methods that work at long distances. Some methods are based on standard candles, astronomical objects with a known luminosity.

The ladder analogy arises because no single method can measure distances at all distances encountered in astronomy. Instead, one method can be used to measure close distances, a second method can be used to measure close and intermediate distances, and so on. Each rung of the ladder provides information that can be used to determine distances on the next higher rung.

The distance to an object is one of the main characteristics that are determined from astronomical observations. Moreover, only knowledge of distances allows us to determine many characteristics of stars, such as luminosities, masses, and others.

The distance scale in astronomy is based on knowledge of the distance between the Earth and the Sun (astronomical unit, AU), which is determined in several ways – the oldest of them is the direct measurement of the daily parallax of the Sun. Due to a number of difficulties (solar brightness, thermal deformations of the instrument), this method is the least accurate. From Kepler’s third law, the mutual position of the planets in the solar system is easily determined. It remains only to scale it, which is achieved with high accuracy by determining the parallax of Venus from observations of its passage across the solar disk or by radar of the planet.

The distances to the stars were also originally determined only by the trigonometric method. The determination of the annual parallax relies precisely on reliable knowledge of 1 AU. The accuracy of ground-based measurements of parallaxes is limited by the Earth’s atmosphere and is approximately ±0.01″. Also for this purpose, statistical and group parallaxes (parallaxes of moving clusters) are used. The latter are highly accurate and form the basis of the astronomical distance scale, providing a link between trigonometric and photometric methods.

The rungs of the distance ladder provide different measurement methods. Distances between planets are now measured using radar. Interstellar distances up to 100 parsecs were previously measured using trigonometric parallaxes (the HIPPARCOS satellite moved this limit to 1000 parsecs). Up to distances of a million parsecs (1 Mpc), the Cepheid method is used. Even more distant objects – galaxies and quasars are hundreds and thousands of megaparsecs away from us. Such distances are measured by redshift and require knowledge of the Hubble constant.

Naturally, when passing the baton from one method to another, each of the “older” methods must be checked using the “junior” method. To do this, there must be areas of overlap in which at least two methods can be applied:

The most reliable annual parallaxes provide us with extra-atmospheric observations – the accuracy of the HIPPARCOS satellite reaches ±0.001″. In any case, direct methods (trigonometric) allow us to determine distances not exceeding 1 kpc.

More distant objects are distanced by indirect methods (photometric): the brightness decreases in proportion to the square of the distance, and if the luminosity of the star is known,
you can find the distance. Examples are “spectral parallaxes” – according to the statistical dependences of line intensity on luminosity, or “Cepheid parallaxes” – according to the “period/luminosity” dependence. The accuracy of photometric methods is limited by the dispersion of empirical dependences and, as a rule, does not exceed ±25%.

The main photometric method is the Cepheid method. Back in 1908, G. Leavitt, studying the variables in the Small Magellanic Cloud (SMC), noted that brighter Cepheids have a longer period. Since all variables are at the same distance (the dimensions of the IMO are much smaller than the distance to it), it is enough to know the luminosity of at least one such Cepheid in order to determine the distance for any star of this type from the “period-luminosity” dependence.

However, in the vicinity of the Sun, accessible for determining distances by the trigonometric method, there is not a single Cepheid. Therefore, H. Shapley considered that variables in globular clusters with similar light curves (subsequently – pulsating ones of the RRLyr type!), Have the same luminosity as Cepheids. As a result, the distance estimate was underestimated. Its use has led to a significant underestimation of the size of the Andromeda Nebula (M31) and the luminosity of objects in it.

The error persisted for almost 30 years until W. Baade and then D. Irwin solved this problem by creating a modern scale of distances in the Universe. This was helped by the discovery of several Cepheids in open clusters, the distances to which are confidently determined from the Hertzsprung-Russell diagram.

There is not a single Cepheid in a sphere with a radius of 100 parsecs (one of the Cepheids closest to us is the North Star, it is 122 parsecs away from us). Therefore, prior to the implementation of the HIPPARCOS project, the distance scale based on Cepheids in clusters was not consistent with the results of direct measurements to stars using trigonometric parallax methods. Now the situation has changed: the Cepheid scale has been refined, on the basis of which it was concluded that the currently accepted value of the Hubble constant should be reduced by 5-10 percent.

The gravitational waves generated by the gradual approach of objects in compact binary systems, such as neutron stars or black holes, have the useful property that the energy emitted in the form of gravitational radiation depends solely on the orbital energy of the pair, and the resulting contraction of their orbits is directly observed as an increase in the frequency of emitted gravitational waves. Therefore, by analogy with standard candles, the distance to such systems can be estimated without the presence of other measurement methods – only instead of the luminosity of a star, the characteristics of gravitational waves are used, and therefore such objects are called “standard sirens”, drawing an analogy between sound and gravitational waves.

As in the case of standard candles, taking into account the amplitudes of the emitted and received energy, the distance to the source is determined by the inverse square law. However, there are some differences from standard candles. Gravitational waves are not emitted isotropically – but measuring the polarization of the wave provides enough information to determine the angle of radiation. Gravitational wave detectors also have anisotropic antenna patterns, so to determine the reception angle it is necessary to know the position of the source in the sky relative to the detectors. As a rule, if a wave is detected by a network of three detectors in different places, this gives us enough information to make the necessary corrections and get the distance to the object. Also, unlike standard candles, gravitational waves do not need to be calibrated against other measures of distance.

Fortunately, gravitational waves are not subject to disappearance due to the presence of an intermediate absorbing medium. But they are subject to gravitational lensing, just like light. If the signal is strongly lensed, then it can be received as several events separated in time (analogous to several images of a quasar, for example). It is less easy to distinguish and control the effect of weak lensing when the signal path through space is affected by many small magnification and degaussing events. This will be important for signals occurring at cosmological redshifts greater than 1. Finally, it is difficult for detector networks to accurately measure the polarization of a signal if the binary system is observed almost face-to-face – such signals have much larger errors in measuring the distance. Unfortunately, binary systems radiate most strongly perpendicular to the plane of the orbit, so unidirectional signals are inherently stronger and more commonly observed.

crevices of kirkwood

Kirkwood crevices or Kirkwood trapdoors are specific areas in the asteroid belt that are created by the resonant influence of Jupiter. In these areas, asteroids are practically absent. These regions correspond to the places of orbital resonances of asteroids with Jupiter.

For example, there are very few asteroids with a semi-major axis of about 2.50 AU and a period of 3.95 years. Such asteroids go through three orbits for each orbit of Jupiter (hence their orbital resonance is called 3:1). Other orbital resonances correspond to orbital periods that are mere fractions of Jupiter’s period.

These tears were first noticed in 1866 by Daniel Kirkwood, a professor at Jefferson College in Canonsburg, Pennsylvania. He, moreover, correctly explained their origin by orbital resonances with Jupiter.

Most Kirkwood ruptures are not rich in asteroids, unlike Neptune’s mid-motion resonances (MMR) or Jupiter’s 3:2 resonance, which preserve objects captured during the migration of giant planets.

More recently, a relatively small number of asteroids with high eccentricity orbits have been discovered that do lie within the Kirkwood gaps. An example is the Alinda and Griqua groups. These orbits slowly increase their eccentricity over tens of millions of years and eventually fall out of resonance due to a close encounter with a large planet. That is why asteroids are rarely found in Kirkwood gaps.

Orbital eccentricity

In astrodynamics, the orbital eccentricity of an astronomical object is a dimensionless parameter that determines the amount of deviation of the orbit around another body from an ideal circle. An eccentricity value of 0 corresponds to a circular orbit, values ​​from 0 to 1 form an elliptical orbit, 1 – a parabolic escape (or capture) orbit of the object, and more than 1 – a hyperbola.

The term takes its name from the parameters of conic sections, since every Kepler orbit is a conic section. It is usually used for an isolated two-body problem, but there are extensions for objects following rosette orbit through the galaxy.

The word “eccentricity” comes from Medieval Latin eccentricus, derived from the Greek ἔκκεντρος “outside the center”, from ἐκ- ek-, “from” + κέντρον “center”. The word “eccentric” first appeared in English in 1551 with the definition “… a circle in which the earth, sun, etc. deviate from their center. In 1556, five years later, the adjectival form of the word appeared.

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