Standard candles and the Cosmic distance ladder

---

1. Cosmic distance ladder

Figure light green box: techniques applicable to star-forming galaxies ([star-forming galaxies](https://en.wikipedia.org/wiki/Galaxy_formation_and_evolution)). Light blue boxes: techniques applicable to Population II galaxies ([Population II galaxies](https://en.wikipedia.org/wiki/Population_II)). Light purple box: geometric distance technique. Light red box: The planetary nebula luminosity function ([planetary nebula luminosity function](https://en.wikipedia.org/wiki/Planetary_nebula_luminosity_function)) technique is applicable to the Virgo Supercluster ([Virgo Supercluster](https://en .wikipedia.org/wiki/Virgo_Supercluster))). Solid black lines: well-calibrated steps. Black dotted line: Uncertain calibration ladder steps.

The cosmic distance ladder (also known as the extragalactic distance scale) is a series of methods used by astronomers to determine the distance of celestial objects. Only those objects that are "close enough" to Earth (within about a thousand parsecs) can make direct distance measurements to astronomical objects. Techniques for determining distances to more distant objects are all based on various measured correlations between methods that work at close range and those that work at greater distances. Several methods rely on the standard candle, an astronomical object with a known luminosity.

The staircase analogy arose because there is no single technique for measuring distances on all scales in astronomy. Instead, one method can be used to measure nearby distances, a second method can be used to measure nearby to intermediate distances, and so on. Each rung of the ladder provides information that can be used to determine the distance to the next higher rung.

2. Direct measurement

Fig. Astronomers demonstrate the use of parallax to measure distances. It was made from parts of the Yale-Columbia Refractor Telescope (1924), which was damaged when the Canberra bushfires destroyed the Mount Stromlo Observatory in 2003; the statue is in Questacon, Canberra.

At the bottom of the ladder are basic distance measurements, where distances are determined directly, without physical assumptions about the nature of the object in question. Precise measurements of the positions of stars are part of astrometry.

2.1 Astronomical unit

Direct distance measurements are based on the Astronomical Unit (AU), which is defined as the average distance between the Earth and the Sun. Kepler's laws provide a precise ratio of the size of the orbits of objects orbiting the sun, but not a measure of the overall size of the orbiting system. Radar is used to measure the distance between Earth's orbit and the orbit of a second object. From this measurement and the ratio of the two orbital sizes, the size of Earth's orbit can be calculated. It is known that the absolute accuracy of the earth's orbit is a few meters, and the relative accuracy is one hundred billionth ( $1\times 10^{-11}$ )。

Historically, observations of transits of Venus have been critical to determining the AU. Asteroid observations were also important in the first half of the 20th century. Earth's orbit is currently determined with high precision using radar to measure the distance to Venus and other nearby planets and asteroids, and to track the orbits of interplanetary spacecraft around the sun through the solar system.

2.2 Parallax

Figure from the Stellar Parallax Campaign for Annual Parallax. Half of the top angle is the parallax angle.

The most important basic distance measurement comes from trigonometric parallax. As the Earth orbits the sun, the positions of nearby stars appear to shift slightly against the more distant background. These displacements are the angles in an isosceles triangle, $2\text{AU}$ (the distance between the extreme positions of the Earth's orbit around the Sun) form the base of the triangle, with the distance to the star being the long side of equal length. Even for the nearest stars, the offset is very small, measured as 1 arcsecond for an object at a distance of 1 parsec (3.26 light-years), and then decreases in angular magnitude with increasing distance. Astronomers usually express distances in terms of parsecs (parallax arc seconds); light years are used in popular media.

Because the farther the star is, the smaller the parallax is, so only the stars that are close enough and the parallax is several times larger than the measurement accuracy can be measured. For example, in the 1990s, the Hipparcos mission obtained parallaxes for over 100,000 stars with an accuracy of about 1 millisecond, providing useful distances for stars within hundreds of parsecs. Hubble WFC3 now has the potential to provide 20 to 40 microarcseconds of precision, up to 5000 parsecs for a small number of stars ( $16,000ly$ ) for reliable distance measurement. In 2018, Data Release 2 from the Gaia space mission provided similar accurate distances to most stars brighter than 15th magnitude.

The velocity of the star relative to the sun results in proper motion (across the sky) and radial velocity (motion toward or away from the sun). The former is determined by mapping the change in the star's position over the years, while the latter comes from measuring the Doppler shift caused by the movement of the star's spectrum along the line of sight. For a group of stars of the same spectral class and similar magnitude range, the average parallax can be derived from a statistical analysis of proper motion with respect to their radial velocities. This statistical parallax method can be used to measure the distances of bright and giant variable stars beyond 50 parsecs, including Cepheids and RR Lyrae variables.

Figure parallax measurements could hold important clues to understanding three of the most elusive components of the universe: dark matter, dark energy and neutrinos. Tu Hubble's precise stellar distance measurements have been extended to 10 times the size of the Milky Way.

The motion of the sun through space provides a longer baseline, which will improve the accuracy of parallax measurements, known as long-term parallax. This corresponds to an average baseline of 4 AU per year for stars in the Milky Way and 40 AU per year for halo stars. Decades from now, the baseline could be orders of magnitude larger than the Earth-Sun baseline used for traditional parallax. However, long-term parallax introduces a higher level of uncertainty because the relative speed of the observed stars is an additional unknown. When applied to a multi-star sample, the uncertainty can be reduced; the uncertainty is inversely proportional to the square root of the sample size.

Moving cluster parallax is a technique in which the motion of individual stars in nearby star clusters is used to find the distance to the cluster. Only open clusters are close enough for this technique to be useful. The distance the Hyades get, in particular, has historically been a big step in the distance ladder.

Other individual objects can have basic distance estimates for them in special cases. If the expansion of a gas cloud (such as a supernova remnant or a planetary nebula) can be observed over time, the expansion parallax distance to that cloud can be estimated. However, these measurements are subject to uncertainties in the deviation of the object from sphericity. Binaries that are both visual and spectroscopic can also have their distances estimated in a similar way, and are not subject to the geometric uncertainties described above. A common feature of these methods is that the measurement of angular motion is combined with the measurement of absolute motion. Velocity (usually obtained by the Doppler effect). Distance estimation comes from calculating how far an object must travel in order for its observed absolute velocity to appear alongside its observed angular motion.

In particular expansion parallax can provide basic distance estimates for very distant objects because of the large expansion velocity and large size (compared to stars) of supernova ejecta. Additionally, they can be observed using radio interferometers, which can measure very small angular motions. These combine to provide basic distance estimates for supernovae in other galaxies. While valuable, such cases are very rare, so they serve as important consistency checks on the distance ladder, rather than individual workhorse steps.

3. Standard candle

Almost all astronomical objects used as physical distance indicators fall into the category of having a known brightness. By comparing this known luminosity to the object's observed brightness, the distance to the object can be calculated using the inverse square law. These objects of known brightness are called Standard candles, coined by Henrietta Swan Leavitt.

The brightness of an object can be expressed in terms of its absolute size. This quantity is obtained by looking at the logarithm of its luminosity at a distance of 10 parsecs ( units of astronomical distance). Apparent magnitude , which is the magnitude seen by an observer (using an instrument called a bolometer ) , can be measured and used along with absolute magnitude to calculate the magnitude of an object's distance $d$ , the unit is parsecs, as follows:

$$5\cdot {\log }_{10}d=m-M+55\cdot {\log }_{10}d=m-M+5$$

or

$$d=10^{(m-M+5)/5}$$

where $m$ is the apparent magnitude,$M$ is the absolute magnitude. To be accurate, the two magnitudes must be in the same frequency band and there must be no relative motion in the radial direction. Some way of correcting for interstellar extinction, which also makes objects appear dimmer and redder, is needed, especially if the object is located within a dusty or gaseous region. The difference between an object's absolute and apparent magnitude is called its distance modulus,and astronomical distances, especially between galaxies, are sometimes tabulated in this way.

3.1 Type

Standard candles are of these types:

• RR Lyrae variable star - belongs to the state of red giant star, used to measure the distance of globular clusters in the Milky Way and nearby.
• Eclipse Binary Stars - Within the last decade, it has become possible to measure fundamental parameters of eclipsing binary stars using 8-meter class telescopes, so they can be used to measure distances. In recent years, it has been successfully used to measure the distances of the Large Magellanic Cloud, Small Magellanic Cloud, Andromeda and Triangulum. Eclipsed binaries provide a direct way to measure distances. For galaxies at distances around 3 megaparsecs, the accuracy can be improved to within 5%.
• Cepheid variables — the first choice for galaxy astronomy, measuring distances in the tens of millions of parsecs.
  Red giant branch technology (TRGB) distance metrics.
• Type Ia supernova—the absolute magnitude of maximum brightness has a clear functional relationship with the luminosity curve, which can be used to confirm the distance of galaxies beyond hundreds of millions of parsecs.

In galaxy astronomy, X-ray bursts (thermonuclear flashes on the surface of neutron stars) also serve as standard candles. Sometimes observed X-ray bursts can show spectral lines that reveal the radius of the burst source. Therefore, the peak value of the X-ray burst flux should correspond to the Eddington luminosity, and the mass of the neutron star can be calculated accordingly (usually 1.5 solar masses can be assumed first). This method can measure the distance of some low-mass X-ray binaries. Low-mass X-ray binaries are very faint in visible light, making distance measurements extremely difficult.

The main difficulty with standard candles is how standard they are, e.g. all observations show Ia supernovae at the same distance to have the same brightness (after correction for photometric curves), but it is not known why they have the same , and the probability that a Type Ia supernova at a distant distance is qualitatively different from a nearby Type Ia supernova.

In the history of using Cepheids to measure distances, this is more than a purely philosophical debate. In the 1950s, Walter Budd discovered that the Cepheids used to calibrate standard candles at closer distances were of a different type than the Cepheids used to measure distances to nearby galaxies. Neighboring Cepheids are Population I stars that contain more metals (heavy elements) than Population II stars in neighboring galaxies. As a result, the diameter of the Milky Way and the distances of globular clusters and neighboring galaxies must all be doubled because Population II Cepheids are actually brighter.

3.2 Application issues

There are two problems with standard candles of any kind. The main thing is calibration, i.e. determining the absolute size of the candle. This involves defining the class well enough that members can be identified, and finding enough members of the class with known distances that their true absolute size can be determined with sufficient accuracy. The second problem lies in identifying members of a class, rather than incorrectly using standard candle calibration on objects that do not belong to that class. At extreme distances, where one would most expect a distance indicator, this recognition problem can be severe.

A big problem with standard candles is that they come in many different standards. For example, all observations seem to indicate that Type Ia supernovae at known distances have the same brightness (corrected by the shape of the light curve). The basis for brightness approximation is discussed below. However, distant Type Ia supernovae may have different properties than nearby Type Ia supernovae. The use of Type Ia supernovae is crucial for determining correct cosmological models. If the properties of Type Ia supernovae do differ over very large distances, i.e. if extrapolating their calibration to arbitrary distances is not valid, ignoring this variation could dangerously bias the reconstruction of cosmological parameters, especially matter density Parameter reconstruction.

As can be seen from the history of distance measurements using Cepheid variables , this is not just a philosophical question. In the 1950s, Walter Baade discovered that the nearby Cepheids used to calibrate standard candles were of a different type than the Cepheids used to measure distances to nearby galaxies. Nearby Cepheids are Group I stars with much higher metal contents than distant Group II stars. As a result, Group II stars are actually much brighter than thought, and after correction, this has the effect of doubling the distance to globular clusters, nearby galaxies, and the diameter of the Milky Way.

3.3 Related history

Let's start with a story.
In 1920, the U.S. government conducted a census in Cambridgeshire, Massachusetts. A census taker goes into a poorer neighborhood and goes door-to-door to check how many people live there and what their occupations are.
He knocked on the door of a family in the community and saw a mother and daughter who depended on each other. The daughter, a deaf woman, had a hard time figuring out what the census taker was up to. When asked about her profession, she answered "scientist".
The census taker just laughed. In the United States at the beginning of the 20th century, scientists were the exclusive domain of men, and few women could earn a Ph.D. So he couldn't believe that a deaf woman living in a poor neighborhood could be a scientist.

Picture Henrietta LeWitt.

Her name is Henrietta LeWitt, the mother of modern cosmology. She is the only person in history who can be called the mother of a certain great subject.

In 1868, LeWitt was born in a pastor's family in Massachusetts, USA. At the age of 20, she passed the rigorous examinations and was admitted to Radcliffe Ladies College (one of the famous Seven Sisters College, which was later absorbed by Harvard University). In 1892, LeWitt graduated with his bachelor's degree. Then, according to the tradition at the time, she took a boat to Europe and started her graduation trip.

But the sky is unpredictable. During this trip, a sudden and serious illness damaged her eyesight and hearing. Although her eyesight improved, her hearing deteriorated until she eventually became deaf. For nearly 30 years since then, she has been in a state of serious illness.

After returning from his trip, LeWitt decided to pursue a master's degree in astronomy. She joined Edward Pickering's Harvard University Observatory in 1893 as a "Harvard Calculator".

Tulewit became a "Harvard computer".

Unfortunately, LeWitt's health was a serious drag on her studies. Frail and sick, LeWitt had to call in sick every now and then, which made her scientific work fragmented. Of course, this also made her mentor Pickering quite dissatisfied.

In 1896, LeWitt realized that it was impossible for her to complete her studies. In desperation, she chose to give up and left the Harvard University Observatory, which lasted 6 years.

Six years later, in 1902, LeWitt wrote Pickering a letter. In the letter, LeWitt mentioned that due to her hearing impairment, she was no longer competent for other jobs, so she wanted to apply to return to the Harvard Observatory. Pickering agreed. But this time, Pickering was smart enough not to let LeWitt participate in the observatory's most important stellar classification work, and sent her to study Cepheid variables alone.

At the beginning of the 20th century, human beings couldn't even figure out the types of the simplest stars in the sky, let alone the extremely complicated problem of variable stars. In this case, sending someone to study Cepheid variables alone is tantamount to exile to the frontier.

Let’s pause this film about LeWitt for a moment and pay attention to her return to the Harvard Observatory. Ill, deaf, forced to make a living, had to return to the sad place where I gave up my master's degree, and was sent to a scientific wasteland where no one had ever set foot before. I am afraid that very few people can get out of such a desperate situation.

But it's our last look with the ordinary Henrietta LeWitt. What happened next is as legendary as Moses parted the Red Sea with his staff.

Since 1904, LeWitt has been finding new Cepheids in the Magellanic Clouds at an astonishing rate. She found it so quickly that an astronomer wrote to Pickering specifically: "Miss LeWitt is a master at finding variable stars. We didn't even have time to record her new discovery."

In 1908, LeWitt published a paper in the "Harvard Observatory Annals", announcing that he had found a total of 1777 Cepheid variables in the Magellanic Clouds (the total number of Cepheid variables found in the previous 100 years Only a few dozen). This astonishing number immediately caused a sensation in the astronomy community, and even got a report in the famous "Washington Post".

Figure Magellanic clouds.

But the number of sensational Cepheid variable star discoveries is nothing compared to the most valuable part of this paper.

At the end of this paper, LeWitt selected 16 Cepheid variable stars located in the Small Magellanic Cloud, and listed their light change periods (the time to complete a round of light and dark alternation) and apparent magnitudes in a table. She left this comment on the table: "It is worth noting that the brighter the variable star, the longer its photovarying period." Four years later, in
1912, LeWitt perfected this conclusion. She selected 25 Cepheid variables located in the Small Magellanic Cloud and plotted them on a graph with brightness on the X-axis and photoperiod on the Y-axis. As a result, these 25 Cepheid variable stars just lined up in a straight line. Based on this, LeWitt asserted that "the brightness of a Cepheid variable star is proportional to its light variation period".

In order to understand the weight of this seemingly ordinary sentence in the history of astronomy, you can imagine a wasteland that has been frozen for an unknown number of years. meter of beautiful flowers.
This statement came to be known as LeWitt's law. It was this earth-shattering LeWitt's law that opened the door to modern cosmology.

You may feel a little confused: "Why can such a simple law start a whole new discipline?" The answer is that it provides a new method of distance measurement, which is the famous standard candle.

In order to introduce the basic principle of measuring distance with standard candles, let us start with a phenomenon that is quite common in daily life. A candle is bright when viewed close up, but dim when viewed from a distance. This is because the brightness of the candle we see depends on the number of photons emitted by the candle and hitting our eyes. The more photons that are injected, the brighter the candle will appear; conversely, the dimmer it will appear.

Schematic diagram of measuring distance with standard candles.

As shown in the figure, the total number of photons emitted by a candle whose absolute brightness remains constant also remains constant. These photons spread outward in a spherical shape. So at a certain place, the number of photons received per unit area is inversely proportional to the square of the distance from the candle here. This means that the apparent brightness of a candle we see at a certain location is inversely proportional to the square of the distance from the candle there. For example, if the distance is increased by 4 times, the apparent brightness of the candle will be reduced to 1/16 of the original.

In this way, we can use candles to measure distance: first, put a candle in a relatively close place, and measure its distance and apparent brightness. Then, place another candle of the same absolute brightness at a particularly distant place, and measure its apparent brightness. Finally, using the inverse relationship between the apparent brightness and the square of the distance, the particularly far distance can be calculated.

Figure measuring the universe with candles.

The principle of using candles to measure distances, the principle of measuring distances with candles, is also applicable in the sky. To this end, it is necessary to find a special celestial body in the sky that can satisfy the following two conditions at the same time:

①It is very bright and can be seen even if it is far away;
②Its optical properties are stable, and its absolute brightness is fixed. If such a celestial body can be found, we can use it as a candle to measure distances on a cosmological scale. This special celestial body that can be used as a candle is the so-called standard candle.

Knowing the concept of standard candlelight, let's talk about the significance of LeWitt's law. Since the Cepheids selected by LeWitt are all located in the Small Magellanic Cloud, it can be approximated that they are all at the same distance from the Earth. Therefore, as long as their apparent brightness is equal, their absolute brightness must be equal.
LeWitt's law says that the absolute brightness of a Cepheid variable star is proportional to its light variation period. This means that as long as Cepheid variable stars with exactly the same light change period are selected, a batch of celestial bodies with exactly the same absolute brightness can be obtained.

So Lewitt's law means that the Cepheid variable star meets the two conditions of the standard candle, which is a true standard candle. This is also the first standard candle found in human history.

The discovery of standard candles provides a new method for measuring distant cosmological distances. Maybe you still have questions: "Why can the discovery of a new distance measurement method create a new discipline of modern cosmology?" In fact, it was this discovery that shook Copernicus' heliocentric theory.

Let's say a few more words about LeWitt. Very sadly, LeWitt's story does not have a happy ending.

Not long after discovering that Cepheids were standard candles, LeWitt left again due to stomach surgery. By the time she returned, Pickering had assigned her a new job: measuring the Polaris sequence, that is, analyzing the spectra of 96 stars near Polaris. This is the subject that Pickering likes most and wants to complete over the years.

It makes perfect sense for a manager to assign his best employees to the challenges he finds toughest. But for an astronomer of LeWitt's caliber, the arrangement was absurd, tantamount to forcing a mid-year Michael Jordan to give up his basketball career to play in a low-level baseball league. What's more cruel is that LeWitt, who is under the roof, has no choice at all. Since then, she has never been able to return to the study of standard candles.

And Pickering's selfish decision also set the world's research on variable stars back for decades.

Ironically, despite single-handedly creating a new discipline that has since fed thousands of Ph.D.s, LeWitt himself was never able to earn a Ph.D. Many years later, she is still a Harvard computer with half the salary of a man.
In 1921, LeWitt, who had always been dependent on his mother, fell ill again. This time it was incurable cancer. On December 12 of that year, she passed away on a rainy night. In her will, she left all her property to her mother. The estates are worth a total of $315, just enough to buy eight rugs.

After his death, LeWitt was buried in his family cemetery. She couldn't even have a tombstone of her own, and was forced to squeeze in with a dozen relatives. This tombstone is very small, with only enough space to write her name, birthday and death date.

Tulewit's Tomb.

This is the discoverer of the standard candle, the gravedigger of Copernican's heliocentric theory, the mother of modern cosmology, and the final outcome of a great female scientist. More than 100 years have passed, and now the name Henrietta LeWitt has been almost forgotten in the dust of history. But I still wanted to write an essay to commemorate the suffering and glory of this extraordinary woman. Despite illness, deafness, poverty, loneliness, being manipulated, despised, and forgotten, she is still an eternal candle that illuminates the entire universe.

4. Standard siren

Gravitational waves originating from the sucking phase of a compact binary star system such as a neutron star or a black hole have the useful property that the energy emitted in the form of gravitational radiation is derived entirely from the orbital energy of the pair, so that their orbital contraction is directly observable As the frequency of gravitational waves emitted increases. For leading order, $f$ frequency change rate:

$$\frac{\mathrm{d}f}{\mathrm{d}t}=\frac{96{BC}^{8/3}(G\mathcal{M}{)}^{\frac{5}{3}}{f}^{\frac{11}{3}}}{5c^5}$$

where $G$ is the gravitational constant,$c$ is the speed of light, and$M$ is a single (and therefore computable) number called the chirp mass of the system,the combination of masses$(m_1,m_2)$ two objects:

$$\mathcal{M}=\frac{(m_1{m}_2{)}^{3/5}}{(m_1+m_2{)}^{1/5}}$$

By observing the waveform, the quality of the chirp can be calculated, and then the power (energy emission rate) of the gravitational wave can be calculated. So this source of gravitational waves is a standard siren of known loudness.

Just like standard candles, the inverse square law determines the distance to the source given the magnitude of the emission and reception. However, there are some differences in standard candles. Gravitational waves are not emitted isotropically, but measuring the polarization of the waves provides enough information to determine the angle of emission. Gravitational wave detectors also have anisotropic antenna patterns, so the position of the source on the sky relative to the detector is needed to determine the angle of reception. Typically, if a wave is detected by a network of three detectors in different locations, the network will measure enough information to make these corrections and obtain the distance. Also unlike standard candles, gravitational waves do not require calibration for other distance measurements. Measuring the distance of course requires calibration of the gravitational-wave detectors, but the distance is basically a multiple of the wavelength of the laser used in the gravitational-wave interferometer.

There are other factors besides detector calibration that can limit the accuracy of this distance. Fortunately, gravitational waves do not disappear because of the intervening absorbing medium. But they are affected by gravitational lensing, just like light. If a signal is strongly lensed, then it may be received as multiple events, separated in time (e.g. simulation of multiple images of a quasar). Less easy to discern and control is weak lensing, where the path of a signal through space is affected by many small zooming in and out events. This is important for signals originating from cosmological redshifts greater than 1. Finally, it is difficult for a detector network to accurately measure the polarization of a signal if the binary system is viewed almost head-on; such a signal suffers from significantly larger errors in distance measurements. Unfortunately, binary stars radiate most strongly in directions perpendicular to the orbital plane, so the frontal signal is inherently stronger and most commonly observed.

If the binary consists of a pair of neutron stars, their merger will be accompanied by a kilonova/supernova explosion, which may allow electromagnetic telescopes to pinpoint the location. In this case, the redshift of the host galaxy can determine the Hubble constant $H_0$. This was the case with GW170817, which was used to make the first such measurement. Even if no electromagnetic counterpart can be identified for a set of signals, statistical methods can be used to infer $H_0$。

5. Standard ruler

Another type of physical distance indicator is the gauge ruler. In 2008, galaxy diameters were proposed as possible yardsticks for the determination of cosmological parameters. More recently, physical scales of the imprints of baryon acoustic oscillations (BAOs) in the early universe have been used. In the early universe (before recombination) baryons and photons scattered each other, forming a tightly coupled fluid that could support sound waves. These waves originate from primordial density perturbations and propagate at velocities that can be predicted from baryon density and other cosmological parameters. The total distance these sound waves can travel before recombination determines a fixed scale, which after recombination simply expands with the expansion of the universe. The BAO thus provides a standard yardstick that can be measured in terms of baryon effects on galaxy clusters in galaxy surveys. The method required extensive surveys of galaxies to make this scale visible, but has been measured with percent-level accuracy (see baryon acoustic oscillations). The scale does depend on cosmological parameters such as baryons and matter density, as well as the number of neutrinos, so BAO-based distances depend more on cosmological models than distances based on local measurements.

Optical echoes can also be used as a standard ruler, although measuring source geometry correctly is challenging.

6. Galactic distance indicators

With few exceptions, distances based on direct measurements are only up to about a thousand parsecs, which is only a small fraction of our galaxy. For distances beyond this range, measurements depend on physical assumptions that assert that a person recognizes the object in question, and that the object's class is sufficiently homogeneous that its membership can be used for meaningful distance estimates.

Physical distance metrics for progressively larger distance scales, including:

• Dynamic Parallax, which uses the orbital parameters of the visual binary to measure the mass of the system and therefore uses the mass-luminosity relationship to determine the luminosity
  • Eclipsed Binaries - Over the past decade, 8-meter class telescopes have been able to measure fundamental parameters of eclipsing binaries. This makes it feasible to use them as distance metrics. More recently, they have been used for direct distance estimates to the Large Magellanic Cloud (LMC), Small Magellanic Cloud (SMC), Andromeda and Triangulum galaxies. Eclipse binaries provide a direct way to measure distances to galaxies at a new and improved level of 5% accuracy, which is feasible with current technology at distances of about 3 Mpc (3 million parsecs).
• RR Lyrae Variables - Used to measure distances to globular clusters within galaxies and nearby.
• The following four indicators all use stars in the old star population (family II):
  • Tip of the red giant branch (TRGB) distance indicator.
  • Planetary Nebula Luminosity Function (PNLF)
  • Globular Cluster Luminosity Function (GCLF)
  • Surface Brightness Fluctuation (SBF)
• In galactic astronomy, X-ray bursts (thermonuclear flashes on the surface of neutron stars) are used as standard candles. Observations of X-ray bursts sometimes reveal X-ray spectra that indicate an enlarged radius. Therefore, the X-ray flux at the peak of the burst should correspond to the Eddington luminosity, which can be calculated once the mass of the neutron star is known (1.5 solar masses is a common assumption). This method can determine the distances of some low-mass X-ray binaries. Low-mass X-ray binaries are optically very faint, making their distances extremely difficult to determine.
• Interstellar masers can be used to derive distances to the Milky Way and some extragalactic objects with maser emissions.
• Cepheids and Novae
• Tully-Fisher relationship
• Faber-Jackson relationship
• Type Ia supernovae have a very well-defined maximum absolute magnitude as a function of the shape of their light curve, which can be used to determine up to a few hundred M pc \mathrm{Mpc}Extragalactic distance of$Mpc\; .$A notable exception is SN 2003fg, the "Champagne Supernova", a Type Ia supernova with unusual properties.
• Redshift and Hubble's Law

6.1 Main sequence fitting

When the absolute magnitudes of a set of stars are plotted against their spectral classification, in a Hertzsprung-Russell diagram, evolutionary patterns can be found that relate to the mass, age, and composition of the stars. In particular, during their hydrogen burn, stars follow a curved line in the diagram, known as the main-sequence stars. By measuring these properties in the stellar spectrum, it is possible to determine the position of the main sequence star on the HR diagram and thus estimate the absolute magnitude of the star. After correcting for interstellar extinction due to gas and dust, comparison of this value with the apparent magnitude allows the approximate distance to be determined.

In a gravitationally bound star cluster, such as the Hyades, stars form at about the same age and at the same distances. This allows relatively accurate master sequence fitting, providing age and distance determination.

7. Extragalactic distance scale

Diagram of the distance indicator outside the river.

Extragalactic distance scales are a series of techniques used by astronomers today to determine the distances of cosmic objects beyond our galaxy, which are not readily available by traditional methods. Some programs take advantage of the properties of these objects, such as stars, globular clusters, nebulae, and entire galaxies. Other methods are more based on statistics and probabilities of things like entire galaxy clusters.

7.1 Wilson–Bappu effect

Discovered by Olin Wilson and MK Vainu Bappu in 1956, the Wilson-Bappu effect exploits an effect known as spectral parallax. Many stars have features in their spectra, such as calcium K lines, that indicate their absolute size. The distance to a star can then be calculated from its apparent magnitude using the distance modulus.

This method of finding distances to stars has significant limitations. The calibration accuracy of spectral line intensities is limited, requiring correction for interstellar extinction. While this approach is theoretically capable of providing reliable distance calculations for stars up to 7 megaparsecs (Mpc), it is typically only used for stars down to hundreds of kiloparsecs (kpc).

7.2 Classical Cepheids

Beyond the Wilson-Bapp effect, the next approach relies on the period-luminosity relationship of classical Cepheids. The following relations can be used to calculate the distances to the Milky Way and extragalactic classical Cepheids:

$$5{\log }_{10}d=V+(3.34){\log }_{10}P-(2.45)(VI)+7.52$$
$$5{\log }_{10}d=V+(3.37){\log }_{10}P-(2.55)(VI)+7.48$$

Several issues complicate the use of Cepheids as standard candles and give rise to intense debate, chief among them: the nature and linearity of the period-luminosity relationship in the various passbands, and the contribution of metallicity to these relationships Null and slope effects, as well as photometric pollution (mixing) and changing (often unknown) extinction laws on Cepheid distances.

These unresolved issues lead to quoted values ​​for the Hubble constant between $60km/s/Mpc$ 和 $Between\; 80km/s/Mpc$ . Resolving this discrepancy is one of the most important problems in astronomy, because by providing a precise value for the Hubble constant, some cosmological parameters of the universe can be better constrained.

Cepheid variables were a key tool in Edwin Hubble's conclusion in 1923 that M31 (Andromeda) was an outer galaxy rather than a smaller nebula within the Milky Way. He was able to calculate M31 to $285Kpc$ distance, today's value is 770 Kpc.

So far, we found that a spiral galaxy NGC 3370 in the constellation Leo contains the most distant Cepheid variable star found so far, at a distance of $29Mpc$ . Cepheids are by no means perfect distance markers: they are off by about 7% in nearby galaxies and by as much as$15\%$。

7.3 Supernova

SN 1994D (bright spot at lower left) in galaxy NGC 4526. Image courtesy of NASA, ESA, Hubble Key Projects Team, and High Z Supernova Search Team
There are several different ways supernovae can be used to measure extragalactic distances.

7.4 Measuring the photosphere of a supernova

We can assume that supernovae expand in a spherically symmetric fashion. If the supernova is close enough that we can measure the angular range θ ( t ) \theta{(t)} of its photosphere$\theta \left(t\right)$ , we can use the equation

$$oh=\frac{D}{\Delta t}$$

where ω is the angular velocity and θ is the angular extent. In order to obtain accurate measurement results, it is necessary to make two observations separated by time Δt. Then, we can use

$$d=\frac{V_{ej}}{oh},d=\frac{V_{ej}}{oh},$$

where $d$ is the distance to the supernova,$V_{ej}$is the radial velocity of the supernova ejecta (if spherically symmetric, one can assume $V_{ej}$is equal to $V_{}$ ）。

This method only works if the supernova is close enough to be able to accurately measure the photosphere. Likewise, an expanding shell of gas is not actually a perfect sphere, nor is it a perfect black body. Interstellar extinction also hinders accurate measurements of the photosphere. Core-collapse supernovae further exacerbate the problem. All of these factors can lead to distance errors of up to 25%.

7.5 Type Ia light curve

Type Ia supernovae are one of the best ways to determine extragalactic distances. Ia occurs when a binary white dwarf starts accreting material from its companion star. As the white dwarf gains matter, eventually it reaches its Chandrasekhar limit of $1.4M{\_}_{\odot }$。

Once there, the star becomes unstable and undergoes runaway nuclear fusion reactions. Because all Type Ia supernova explosions have roughly the same mass, they all have the same absolute magnitude. This makes them very useful as standard candles. All Type Ia supernovae have standard blue and visual magnitudes

$$M_B\approx M_V\approx -19.3\pm 0.3$$

Therefore, when observing a Type Ia supernova, if its peak size can be determined, its distance can be calculated. It is not inherently necessary to directly capture a peak-sized supernova; using the multicolor light curve shape method (MLCS), the shape of the light curve (taken at any reasonable time after the initial explosion) is compared to a set of parametric curves that will Determines the absolute magnitude at maximum brightness. This approach also enables interstellar extinction/darkening from dust and gas.

Similarly, the stretching method fits a specific supernova intensity light curve to a template light curve. This template is not the same as multiple light curves for different wavelengths (MLCS), it is just one light curve that has been stretched (or compressed) in time. By using this Stretch Factor, the peak amplitude can be determined.

Using Type Ia supernovae is one of the most accurate methods, especially since supernova explosions can be seen at great distances (their luminosity is comparable to the galaxies in which they are located), farther away than Cepheids (500 times farther). Much time has been spent improving this method. The current uncertainty is only close to $5\%$ , the corresponding uncertainty is only$0.1$ magnitude.

7.6 Nova in distance determination

Novae can be used to derive extragalactic distances in much the same way as supernovae. There is a direct relationship between the maximum magnitude of a nova and the time it takes for it to drop two magnitudes in visible light. This relationship is shown as:

$$M_V^{\max }=-9.96-2.31{\log }_{10}\dot{x}$$

where $\dot{x}$ is the time derivative of nova mag, which describes the average rate of decline of the first two magnitudes.

After novas fade, they are as bright as the brightest Cepheids, so the maximum distance for both techniques is about the same: $\sim 20Mpc$ . The error in this method produces an uncertainty of about$\pm 0.4$

8. Globular Cluster Luminosity Function

According to the method of comparing the luminosity of globular clusters (in the halo of galaxies) of distant galaxies with that of the Virgo cluster, the globular cluster luminosity function has about $20\%$ (or$0.4$ etc.) for distance uncertainties.

American astronomer William Alvin Baum first attempted to use globular clusters to measure distant elliptical galaxies. He compared the brightest globular clusters in the Virgo A galaxy and Andromeda, assuming that the two clusters have the same luminosity. Knowing the distance to Andromeda, Baum assumed a direct correlation and estimated the distance to Virgo A.

Baum only used a single globular cluster, but individual formations are usually poor standard candles. Canadian astronomer René Racine hypothesized that using the globular cluster luminosity function (GCLF) would yield a better approximation. The number of globular clusters as a function of magnitude is given by:

$$\Phi \left(m\right)=A{e}^{(m-m_0{)}^2/2p^2}$$

where $m_0$is the turnover amplitude, $M_0$is the magnitude of the Virgo cluster, $\sigma$ is the dispersion$\sim 1.4$ etc.

Assume that globular clusters all have roughly the same luminosity in the universe. There is no general globular cluster luminosity function that works for all galaxies.

9. Planetary nebula luminosity function

As with the GCLF method, similar numerical analyzes can be used for planetary nebulae in distant galaxies. The planetary nebula luminosity function (PNLF) was first proposed by Holland Kerr and David Jenner in the late 1970s. They think that all planetary nebulae may have a similar maximum intrinsic brightness, which is now calculated as $M=-4.53$ . So this would make them potential standard candles for determining extragalactic distances.

Astronomer George Howard Jacobi and his colleagues later proposed that the PNLF function is equal to:

$$N(M)\propto e^{0.307M}(1-e^{3(M^{\ast }-M)})$$

where N(M) is the number of planetary nebulae and M is the absolute magnitude. M* equals the magnitude of the brightest nebula.

10. Surface Brightness Fluctuation Method

10.1 Galaxy clusters

The following methods deal with the global intrinsic properties of galaxies. These methods, while having different error percentages, are capable of performing over $100Mpc$ for distance estimation, although it is usually applied more locally.

The Surface Brightness Fluctuation (SBF) method exploits the use of a CCD camera on the telescope. Due to spatial fluctuations in the brightness of galaxies' surfaces, some pixels on these cameras will capture more stars than others. However, as the distance increases, the picture becomes smoother and smoother. Analysis of this describes the magnitude of the pixel-to-pixel variation, which is directly related to the galaxy's distance.

10.2 Sigma-D relationship

The Sigma-D relationship (or Σ-D relationship) for elliptical galaxies relates the angular diameter (D) of a galaxy to its velocity dispersion. In order to understand this approach, accurately describe $What\; the\; D$ stands for matters. More precisely, it is the angular diameter of the galaxy with an apparent brightness level of 20.75 B-mag arcsec -2. This apparent brightness is independent of the galaxy's actual distance from us. Instead,$D$ is inversely proportional to the distance of the galaxy, expressed as$d$ . Therefore, this relationship does not use standard candles. Instead,$D$ provides a standard scale. $D$ andΣ\SigmaThe relationship between$\Sigma \; is$

$$\log (D)=1.333\log \left(\Sigma \right)\_+C$$

where $C$ is a constant that depends on the distance to the galaxy cluster.

This method has the potential to be one of the strongest methods in galactic distance calculators, perhaps even surpassing the scope of the Tully-Fisher method. However, as of today, elliptical galaxies are not bright enough to provide calibration for this method by using techniques such as Cepheid variables. Instead, calibration is done using a coarser method.

11. Overlap and scaling

A series of distance indicators, known as distance ladders, are required to determine distances to other galaxies. The reason is that objects bright enough to be identified and measured at such distances are so rare that they have little or no proximity, so examples close enough with reliable triangular parallax to calibrate the pointer are too rare. For example, Cepheid variables, one of the best indicators of nearby spiral galaxies, cannot yet be satisfactorily calibrated by parallax alone, although the Gaia space mission can now weigh this specific issue. The situation is further complicated by the fact that different stellar populations often do not have all types of stars. Cepheids, in particular, are massive stars with short lifetimes, so they are only found where stars have recently formed. Therefore, since elliptical galaxies usually cease to have massive star formation long ago, they will not have Cepheid variables. Instead, distance metrics originating from older stellar populations (such as the Nova and RR Lyrae variables) must be used. However, RR Lyrae variables are not as bright as Cepheids, and novae are unpredictable, requiring an intensive monitoring program—and luck during that program—to collect enough novae in the target galaxy for a good distance estimate.

Because the farther steps in the cosmic distance ladder depend on the closer steps, and the farther steps include the effects of errors in the closer steps, including the effects of systematic and statistical errors. The consequences of these propagation errors mean that distances in astronomy are rarely measured with the same level of precision as measurements in other sciences, and for more distant object types the precision is necessarily worse.

Another problem, especially with the brightest standard candles, is their "standard": how uniform an object is in its true absolute size. For some of these different standard candles, the homogeneity is based on theories about the formation and evolution of stars and galaxies, so there are uncertainties in these as well. For the brightest distance indicators, Type Ia supernovae, this uniformity is poor; however, no other class of objects is bright enough to be detected at such distances, so this class is useful only because there are no real replacement of.

An observational consequence of Hubble's Law, the proportional relationship between distance and the speed at which galaxies move away from us (commonly known as redshift) is a product of the cosmic distance ladder. Edwin Hubble observed that dimmer galaxies are more redshifted. Finding the value of Hubble's constant is the result of decades of work by many astronomers, who both accumulated measurements of the redshift of galaxies and calibrated the steps of the distance ladder. Hubble's law is the primary means we use to estimate the distances of quasars and distant galaxies, where no single distance indicator can be seen.

---

• references

Wang Shuang. A Cosmic Odyssey: Across the Milky Way.

Wang Shuang. The Mother of Modern Cosmology: A Deaf Girl Born in Poverty, Illuminating the Universe with Candlelight

Cosmic distance ladder

Wikipedia: Standard candle

wiki: Distance measure