Micro / Macro numbers in Science

Jai Ganesh · 2015-11-16 05:25:18

In astrophysics, the Eddington number,

, is the number of protons in the observable universe. The term honors the British astrophysicist Arthur Eddington, who in 1938 was the first to propose a value of and to explain why this number might be important for cosmology and the foundations of physics.

Eddington argued that the value of the fine-structure constant, α, could be obtained by pure deduction. He related α to the Eddington number, which was his estimate of the number of protons in the universe. This led him in 1929 to conjecture that α was exactly 1/137. Other physicists did not adopt this conjecture and did not accept his argument.
In the late 1930s, the best experimental value of the fine-structure constant,

, was approximately 1/136. Eddington then argued, from aesthetic and numerological considerations, that should be exactly 1/136. He devised a "proof" that , or about
.

Some estimates of

point to a value of about . These estimates assume that all matter can be taken to be hydrogen and require assumed values for the number and size of galaxies and stars in the universe.

Attempts to find a mathematical basis for this dimensionless constant have continued up to the present time.

Last edited by Jai Ganesh (2015-11-16 07:23:17)

Jai Ganesh · 2015-11-16 07:31:35

Continued....

In the 1938 Tarner Lecture at Trinity College, Cambridge, Eddington averred that:

I believe there are 15 747 724 136 275 002 577 605 653 961 181 555 468 044 717 914 527 116 709 366 231 425 076 185 631 031 296 protons in the universe and the same number of electrons.

This large number was soon named the "Eddington number."

Shortly thereafter, improved measurements of α yielded values closer to 1/137, whereupon Eddington changed his "proof" to show that α had to be exactly 1/137.
Recent theory

The most precise value of α (obtained experimentally in 2012) is:

Consequently, no one maintains any longer that α is the reciprocal of an integer. Nor does anyone take seriously a mathematical relationship between α and

.

Agnishom · 2015-11-16 11:59:32

Why'd Eddington state that?

Jai Ganesh · 2015-11-16 14:13:03

Hi,

Fundamental theory and the Eddington number

During the 1920s until Sir Arthur Stanley Eddington's death, he increasingly concentrated on what he called "fundamental theory" which was intended to be a unification of quantum theory, relativity, cosmology, and gravitation. At first he progressed along "traditional" lines, but turned increasingly to an almost numerological analysis of the dimensionless ratios of fundamental constants.

His basic approach was to combine several fundamental constants in order to produce a dimensionless number. In many cases these would result in numbers close to

, its square, or its square root. He was convinced that the mass of the proton and the charge of the electron were a natural and complete specification for constructing a Universe and that their values were not accidental. One of the discoverers of quantum mechanics, Paul Dirac, also pursued this line of investigation, which has become known as the Dirac large numbers hypothesis, and some scientists even today believe it has something to it.

A somewhat damaging statement in his defence of these concepts involved the fine structure constant, α. At the time it was measured to be very close to 1/136, and he argued that the value should in fact be exactly 1/136 for epistemological reasons. Later measurements placed the value much closer to 1/137, at which point he switched his line of reasoning to argue that one more should be added to the degrees of freedom, so that the value should in fact be exactly 1/137, the Eddington number. Wags at the time started calling him "Arthur Adding-one". This change of stance detracted from Eddington's credibility in the physics community. The current measured value is estimated at
1/137.035 999 074(44).

Eddington believed he had identified an algebraic basis for fundamental physics, which he termed "E-numbers" (representing a certain group – a Clifford algebra). These in effect incorporated spacetime into a higher-dimensional structure. While his theory has long been neglected by the general physics community, similar algebraic notions underlie many modern attempts at a grand unified theory. Moreover, Eddington's emphasis on the values of the fundamental constants, and specifically upon dimensionless numbers derived from them, is nowadays a central concern of physics. In particular, he predicted a number of hydrogen atoms in the Universe

, or equivalently the half of the total number of particles protons + electrons. When equalized with the non-dark energy equivalent number of hydrogen atoms , this corresponds to a Universe radius R = 13.8 Giga light year, a value predicted for years from universal constants using an atomic-cosmic symmetry, and compatible with c-times the so-called Universe age 13.80(4) Gyr, as determined by the recent mission Planck (March 2003).

He did not complete this line of research before his death in 1944; his book Fundamental Theory was published posthumously in 1948.

Jai Ganesh · 2015-11-17 17:39:47

The Milky Way is the second-largest galaxy in the Local Group, with its stellar disk approximately 100,000 ly (30 kpc) in diameter, and, on average, approximately 1,000 ly (0.3 kpc) thick. As a guide to the relative physical scale of the Milky Way, if the Solar System out to Neptune were the size of a US quarter (25mm), the Milky Way would be approximately the size of the United States. A ring-like filament of stars wrapping around the Milky Way may actually belong to the Milky Way itself, rippling above and below the relatively flat galactic plane. If so, that would mean a diameter of 150,000–180,000 light-years (46–55 kpc).

Estimates of the mass of the Milky Way vary, depending upon the method and data used. At the low end of the estimate range, the mass of the Milky Way is

solar masses (M☉), somewhat less than that of the Andromeda Galaxy. Measurements using the Very Long Baseline Array in 2009 found velocities as large as 254 km/s for stars at the outer edge of the Milky Way. Because the orbital velocity depends on the total mass inside the orbital radius, this suggests that the Milky Way is more massive, roughly equaling the mass of Andromeda Galaxy at M☉ within 160,000 ly (49 kpc) of its center. In 2010, a measurement of the radial velocity of halo stars finds that the mass enclosed within 80 kiloparsecs is M☉. According to a study published in 2014, the mass of the entire Milky Way is estimated to be M☉, which is about half the mass of the Andromeda Galaxy.

Most of the mass of the Milky Way appears to be dark matter, an unknown and invisible form of matter that interacts gravitationally with ordinary matter. A dark matter halo is spread out relatively uniformly to a distance beyond one hundred kiloparsecs from the Galactic Center. Mathematical models of the Milky Way suggest that its total mass is

M☉. More-recent studies indicate a mass as large as M☉ and as small as M☉.

The total mass of all the stars in the Milky Way is estimated to be between

M☉ and M☉. In addition to that mass in stars, there's between 10% and 15%[54] of it in the form of gas (90% of hydrogen and 10% of helium by mass, with two thirds of the former in atomic form and the remaining one third in molecular form), as well as an 1% of the total gas mass in the form of interstellar dust.

Jai Ganesh · 2015-11-17 23:47:09

Andromeda Galaxy

Distance : 2.54 ± 0.11 Mly : (778 ± 33 kpc)
Mass :

M☉
Size (ly) : ~220 kly (diameter)
Number of stars : 1 trillion
Apparent dimensions (V) : 190′ × 60′
Apparent magnitude (V) : 3.44
Absolute magnitude (V) : -21.5
Other designations
M31, NGC 224

The Andromeda Galaxy, also known as Messier 31, M31, or NGC 224, is a spiral galaxy approximately 780 kiloparsecs (2.5 million light-years) from Earth. It is the nearest major galaxy to the Milky Way and was often referred to as the Great Andromeda Nebula in older texts. It received its name from the area of the sky in which it appears, the constellation of Andromeda, which was named after the mythological princess Andromeda. Being approximately 220,000 light years across, it is the largest galaxy of the Local Group, which also contains the Milky Way, the Triangulum Galaxy, and about 44 other smaller galaxies.

The Andromeda Galaxy is the most massive galaxy in the Local Group as well. Despite earlier findings that suggested that the Milky Way contains more dark matter and could be the most massive in the grouping, the 2006 observations by the Spitzer Space Telescope revealed that Andromeda contains one trillion

stars at least twice the number of stars in the Milky Way, which is estimated to be 200–400 billion.

The Andromeda Galaxy is estimated to be

solar masses, while the mass of the Milky Way is estimated to be solar masses. In comparison, a 2009 study estimated that the Milky Way and M31 are about equal in mass, while a 2006 study put the mass of the Milky Way at ~80% of the mass of the Andromeda Galaxy. The Milky Way and Andromeda are expected to collide in 3.75 billion years, eventually merging to form a giant elliptical galaxy or perhaps a large disk galaxy.

At 3.4, the apparent magnitude of the Andromeda Galaxy is one of the brightest of any of the Messier objects, making it visible to the naked eye on moonless nights even when viewed from areas with moderate light pollution. Although it appears more than six times as wide as the full Moon when photographed through a larger telescope, only the brighter central region is visible to the naked eye or when viewed using binoculars or a small telescope, making it appear similar to a star.

Last edited by Jai Ganesh (2015-11-18 00:02:58)

Jai Ganesh · 2015-11-18 20:05:16

Sirius is the brightest star (in fact, a star system) in the Earth's night sky. With a visual apparent magnitude of -1.46, it is almost twice as bright as Canopus, the next brightest star. The name "Sirius" is derived from the Ancient Greek (Seirios), meaning "glowing" or "scorcher". The system has the Bayer designation Alpha Canis Majoris (α CMa). What the naked eye perceives as a single star is actually a binary star system, consisting of a white main-sequence star of spectral type A1V, termed Sirius A, and a faint white dwarf companion of spectral type DA2, called Sirius B. The distance separating Sirius A from its companion varies between 8.2 and 31.5 AU.

Sirius appears bright because of both its intrinsic luminosity and its proximity to Earth. At a distance of 2.6 parsecs (8.6 ly), as determined by the Hipparcos astrometry satellite, the Sirius system is one of Earth's near neighbors. Sirius is gradually moving closer to the Solar System, so it will slightly increase in brightness over the next 60,000 years. After that time its distance will begin to increase, but it will continue to be the brightest star in the Earth's sky for the next 210,000 years.

Sirius A is about twice as massive as the Sun (M☉) and has an absolute visual magnitude of 1.42. It is 25 times more luminous than the Sun but has a significantly lower luminosity than other bright stars such as Canopus or Rigel. The system is between 200 and 300 million years old. It was originally composed of two bright bluish stars. The more massive of these, Sirius B, consumed its resources and became a red giant before shedding its outer layers and collapsing into its current state as a white dwarf around 120 million years ago.

Sirius is also known colloquially as the "Dog Star", reflecting its prominence in its constellation, Canis Major (Greater Dog). The heliacal rising of Sirius marked the flooding of the Nile in Ancient Egypt and the "dog days" of summer for the ancient Greeks, while to the Polynesians in the Southern Hemisphere the star marked winter and was an important reference for their navigation around the Pacific Ocean.

Jai Ganesh · 2015-11-20 01:34:02

Proxima Centauri is a red dwarf about 4.24 light-years from the Sun, inside the G-cloud, in the constellation of Centaurus. It was discovered in 1915 by the Scottish astronomer Robert Innes, the Director of the Union Observatory in South Africa, and is the nearest known star to the Sun, although it is too faint to be seen with the naked eye, with an apparent magnitude of 11.05. Its distance to the second- and third-nearest stars, which form the bright binary Alpha Centauri, is 0.237 ± 0.011 ly (15,000 ± 700 AU). Proxima Centauri is very likely part of a triple star system with Alpha Centauri A and B, but its orbital period may be greater than 500,000 years.

Because of Proxima Centauri's proximity, its distance from Earth and angular diameter can be measured directly, from which it can be determined that its diameter is about one-seventh of that of the Sun. Proxima Centauri's mass is about an eighth of the Sun's (M☉), and its average density is about 40 times that of the Sun. Although it has a very low average luminosity, Proxima is a flare star that undergoes random dramatic increases in brightness because of magnetic activity. The star's magnetic field is created by convection throughout the stellar body, and the resulting flare activity generates a total X-ray emission similar to that produced by the Sun. The mixing of the fuel at Proxima Centauri's core through convection and its relatively low energy-production rate mean that it will be a main-sequence star for another four trillion years, or nearly 300 times the current age of the universe.

Searches for companions orbiting Proxima Centauri have been unsuccessful, ruling out the presence of brown dwarfs and supermassive planets. Precision radial velocity surveys have also ruled out the presence of super-Earths within the star's habitable zone. The detection of smaller objects will require the use of new instruments, such as the James Webb Space Telescope, which is scheduled for deployment in 2018. Because Proxima Centauri is a red dwarf and a flare star, whether a planet orbiting it could support life is disputed. Nevertheless, because of the star's proximity to Earth, it has been proposed as a destination for interstellar travel.

Observation

In 1915, the Scottish astronomer Robert Innes, Director of the Union Observatory in Johannesburg, South Africa, discovered a star that had the same proper motion as Alpha Centauri. He suggested it be named Proxima Centauri (actually Proxima Centaurus). In 1917, at the Royal Observatory at the Cape of Good Hope, the Dutch astronomer Joan Voûte measured the star's trigonometric parallax at 0.755 ± 0.028″ and determined that Proxima Centauri was approximately the same distance from the Sun as Alpha Centauri. It was also found to be the lowest-luminosity star known at the time. An equally accurate parallax determination of Proxima Centauri was made by American astronomer Harold L. Alden in 1928, who confirmed Innes's view that it is closer, with a parallax of 0.783 ± 0.005″.
Stars closest to the Sun, including Proxima Centauri (April 25, 2014).

In 1951, American astronomer Harlow Shapley announced that Proxima Centauri is a flare star. Examination of past photographic records showed that the star displayed a measurable increase in magnitude on about 8% of the images, making it the most active flare star then known. The proximity of the star allows for detailed observation of its flare activity. In 1980, the Einstein Observatory produced a detailed X-ray energy curve of a stellar flare on Proxima Centauri. Further observations of flare activity were made with the EXOSAT and ROSAT satellites, and the X-ray emissions of smaller, solar-like flares were observed by the Japanese ASCA satellite in 1995. Proxima Centauri has since been the subject of study by most X-ray observatories, including XMM-Newton and Chandra.

Because of Proxima Centauri's southern declination, it can only be viewed south of latitude 27° N. Red dwarfs such as Proxima Centauri are far too faint to be seen with the naked eye. Even from Alpha Centauri A or B, Proxima would only be seen as a fifth magnitude star. It has an apparent visual magnitude of 11, so a telescope with an aperture of at least 8 cm (3.1 in.) is needed to observe it, even under ideal viewing conditions - under clear, dark skies with Proxima Centauri well above the horizon.

Characteristics

Proxima Centauri is a red dwarf, because it belongs to the main sequence on the Hertzsprung–Russell diagram and is of spectral class M6. M6 means that it falls in the low-mass end of M-type stars. Its absolute visual magnitude, or its visual magnitude as viewed from a distance of 10 parsecs, is 15.5. Its total luminosity over all wavelengths is 0.17% that of the Sun, although when observed in the wavelengths of visible light the eye is most sensitive to, it is only 0.0056% as luminous as the Sun. More than 85% of its radiated power is at infrared wavelengths.
This illustration shows the comparative sizes of (from left to right) the Sun, α Centauri A, α Centauri B, and Proxima Centauri
The two bright stars are (left) Alpha Centauri and (right) Beta Centauri. The faint red star in the center of the red circle is Proxima Centauri.

In 2002, optical interferometry with the Very Large Telescope (VLTI) found that the angular diameter of Proxima Centauri was 1.02 ± 0.08 milliarcsec. Because its distance is known, the actual diameter of Proxima Centauri can be calculated to be about 1/7 that of the Sun, or 1.5 times that of Jupiter. The star's estimated mass is 12.3% M☉, or 129 Jupiter masses (MJ). The mean density of a main-sequence star increases with decreasing mass, and Proxima Centauri is no exception: it has a mean density of

kilograms per meter cube (56.8 grams per cubic centimeters), compared with the Sun's mean density of kilograms per cubic meters (1.411 grams per cubic centimeters).

Because of its low mass, the interior of the star is completely convective, causing energy to be transferred to the exterior by the physical movement of plasma rather than through radiative processes. This convection means that the helium ash left over from the thermonuclear fusion of hydrogen does not accumulate at the core, but is instead circulated throughout the star. Unlike the Sun, which will only burn through about 10% of its total hydrogen supply before leaving the main sequence, Proxima Centauri will consume nearly all of its fuel before the fusion of hydrogen comes to an end.

Convection is associated with the generation and persistence of a magnetic field. The magnetic energy from this field is released at the surface through stellar flares that briefly increase the overall luminosity of the star. These flares can grow as large as the star and reach temperatures measured as high as 27 million K - hot enough to radiate X-rays. Indeed, Proxima Centauri's quiescent X-ray luminosity, approximately

erg/s W), is roughly equal to that of the much larger Sun. The peak X-ray luminosity of the largest flares can reach erg/s ( W.)

Proxima Centauri's chromosphere is active, and its spectrum displays a strong emission line of singly ionized magnesium at a wavelength of 280 nm. About 88% of the surface of Proxima Centauri may be active, a percentage that is much higher than that of the Sun even at the peak of the solar cycle. Even during quiescent periods with few or no flares, this activity increases the corona temperature of Proxima Centauri to 3.5 million K, compared to the 2 million K of the Sun's corona. However, Proxima Centauri's overall activity level is considered low compared to other red dwarfs, which is consistent with the star's estimated age of

years, since the activity level of a red dwarf is expected to steadily wane over billions of years as its stellar rotation rate decreases. The activity level also appears to vary with a period of roughly 442 days, which is shorter than the solar cycle of 11 years.

Proxima Centauri has a relatively weak stellar wind, no more than 20% of mass loss rate of the solar wind. Because the star is much smaller than the Sun, however, the mass loss per unit surface area from Proxima Centauri may be eight times that from the solar surface.

A red dwarf with the mass of Proxima Centauri will remain on the main sequence for about four trillion years. As the proportion of helium increases because of hydrogen fusion, the star will become smaller and hotter, gradually transforming from red to blue. Near the end of this period it will become significantly more luminous, reaching 2.5% of the Sun's luminosity (L☉) and warming up any orbiting bodies for a period of several billion years. Once the hydrogen fuel is exhausted, Proxima Centauri will then evolve into a white dwarf (without passing through the red giant phase) and steadily lose any remaining heat energy.

Distance and motion

Based on the parallax of 768.7 ± 0.3 milliarcseconds, measured using the Hipparcos astrometry satellite, and more precisely with the Fine Guidance Sensors on the Hubble Space Telescope, Proxima Centauri is about 4.24 light-years (ly) from the Sun, or 270,000 times more distant than Earth is from the Sun. From Earth's vantage point, Proxima is separated by 2.18° from Alpha Centauri, or four times the angular diameter of the full Moon. Proxima also has a relatively large proper motion - moving 3.85 arcseconds per year across the sky. It has a radial velocity toward the Sun of 22.4 km/s.

Among the known stars, Proxima Centauri has been the closest star to the Sun for about 32,000 years and will be so for about another 33,000 years, after which the closest star to the Sun will be Ross 248. In 2001, J. García-Sánchez et al. predicted that Proxima will make its closest approach to the Sun, coming within 3.11 ly of the latter, in approximately 26,700 years. A 2010 study by V. V. Bobylev predicted a closest approach distance of 2.90 ly in about 27,400 years. Proxima Centauri is orbiting through the Milky Way at a distance from the Galactic Center that varies from 8.3 to 9.5 kpc, with an orbital eccentricity of 0.07.

Ever since the discovery of Proxima it has been suspected to be a true companion of the Alpha Centauri binary star system. At a distance to Alpha Centauri of just 0.21 ly (15,000 ± 700 AU), Proxima Centauri may be in orbit around Alpha Centauri, with an orbital period of the order of 500,000 years or more. For this reason, Proxima is sometimes referred to as Alpha Centauri C. Modern estimates, taking into account the small separation between and relative velocity of the stars, suggest that the chance of the observed alignment being a coincidence is roughly one in a million. Data from the Hipparcos satellite, combined with ground-based observations, is consistent with the hypothesis that the three stars are truly a bound system. If so, Proxima would currently be near apastron, the farthest point in its orbit from the Alpha Centauri system. Such a triple system can form naturally through a low-mass star being dynamically captured by a more massive binary of 1.5 - 2 M☉ within their embedded star cluster before the cluster disperses. More accurate measurement of the radial velocity is needed to confirm this hypothesis.

If Proxima was bound to the Alpha Centauri system during its formation, the stars would be likely to share the same elemental composition. The gravitational influence of Proxima may also have stirred up the Alpha Centauri protoplanetary disks. This would have increased the delivery of volatiles such as water to the dry inner regions. Any terrestrial planets in the system may have been enriched by this material.

Six single stars, two binary star systems, and a triple star share a common motion through space with Proxima Centauri and the Alpha Centauri system. The space velocities of these stars are all within 10 km/s of Alpha Centauri's peculiar motion. Thus, they may form a moving group of stars, which would indicate a common point of origin, such as in a star cluster. If it is determined that Proxima Centauri is not gravitationally bound to Alpha Centauri, then such a moving group would help explain their relatively close proximity.

Though Proxima Centauri is the nearest bona fide star, it is still possible that one or more as-yet undetected sub-stellar brown dwarfs may lie closer.

If a massive planet is orbiting Proxima Centauri, it would cause some displacement of Proxima Centauri over the course of the planet's orbit. If the orbital plane of the planet is not perpendicular to the line of sight from Earth, then this displacement would cause periodic changes in the radial velocity of Proxima Centauri. The fact that multiple measurements of the star's radial velocity have detected no such shifts has lowered the maximum mass that a possible companion to Proxima Centauri could possess. The activity level of the star adds noise to the radial velocity measurements, limiting future prospects for detection of a companion using this method.

In 1998, an examination of Proxima Centauri using the Faint Object Spectrograph on board the Hubble Space Telescope appeared to show evidence of a companion orbiting at a distance of about 0.5 AU.[68] However, a subsequent search using the Wide Field Planetary Camera 2 failed to locate any companions. Astrometric measurements at the Cerro Tololo Inter-American Observatory appear to rule out a Jovian companion with an orbital period of 2 - 12 years.

Proxima Centauri, along with Alpha Centauri A and B, was among the "Tier 1" target stars for NASA's now-canceled Space Interferometry Mission (SIM), which would theoretically have been able to detect planets as small as three Earth masses (M⊕) within two AU of a "Tier 1" target star.

Habitable zone

The TV documentary Alien Worlds hypothesized that a life-sustaining planet could exist in orbit around Proxima Centauri or other red dwarfs. Such a planet would lie within the habitable zone of Proxima Centauri, about 0.023–0.054 AU from the star, and would have an orbital period of 3.6–14 days. A planet orbiting within this zone will experience tidal locking to the star, so that Proxima Centauri moves little in the planet's sky, and most of the surface experiences either day or night perpetually. However, the presence of an atmosphere could serve to redistribute the energy from the star-lit side to the far side of the planet.

Proxima Centauri's flare outbursts could erode the atmosphere of any planet in its habitable zone, but the documentary's scientists thought that this obstacle could be overcome (see continued theories). Gibor Basri of the University of California, Berkeley, even mentioned that "no one [has] found any showstoppers to habitability." For example, one concern was that the torrents of charged particles from the star's flares could strip the atmosphere off any nearby planet. However, if the planet had a strong magnetic field, the field would deflect the particles from the atmosphere; even the slow rotation of a tidally locked dwarf planet that spins once for every time it orbits its star would be enough to generate a magnetic field, as long as part of the planet's interior remained molten.

Other scientists, especially proponents of the Rare Earth hypothesis, disagree that red dwarfs can sustain life. The tide-locked rotation may result in a relatively weak planetary magnetic moment, leading to strong atmospheric erosion by coronal mass ejections from Proxima Centauri.

Interstellar travel

Proxima Centauri has been suggested as a possible first destination for interstellar travel. The star is in motion toward Earth at a rate of 22.4 km/s. Ηowever, after 26,700 years, when it will come as close as 3.11 light-years, it will begin to move farther away. If non-nuclear propulsion were used, a voyage of a spacecraft to a planet orbiting Proxima Centauri would probably require thousands of years. For example, Voyager 1, which is now travelling 17.043 km/s (38,120 mph) relative to the Sun, would reach Proxima in 73,775 years, were the spacecraft traveling in the direction of that star. A slow-moving probe would have only several tens of thousands of years to catch Proxima Centauri near its closest approach, and could end up watching it recede into the distance. Nuclear pulse propulsion might enable such interstellar travel with a trip timescale of a century, beginning within the next century, inspiring several studies such as Project Orion, Project Daedalus, and Project Longshot.

From Proxima Centauri, the Sun would appear as a bright 0.4-magnitude star in the constellation Cassiopeia.

Last edited by Jai Ganesh (2015-11-20 02:07:37)

Jai Ganesh · 2015-11-23 18:34:15

The Boltzmann constant

or k, named after Ludwig Boltzmann, is a physical constant relating energy at the individual particle level with temperature. It is the gas constant R divided by the Avogadro constant :

The Boltzmann constant has the dimension energy divided by temperature, the same as entropy. The accepted value in SI units is J/K.
Boltzmann's constant, k, is a bridge between macroscopic and microscopic physics. Macroscopically, the ideal gas law states that, for an ideal gas, the product of pressure p and volume V is proportional to the product of amount of substance n (in moles) and absolute temperature T:

where R is the gas constant . Introducing the Boltzmann constant transforms the ideal gas law into an alternative form:
,
where N is the number of molecules of gas. For n = 1 mol, N is equal to the number of particles in one mole (Avogadro's number).

Given a thermodynamic system at an absolute temperature T, the average thermal energy carried by each microscopic degree of freedom in the system is on the order of magnitude of kT/2 (i.e., about

, or 0.013 eV, at room temperature).

Last edited by Jai Ganesh (2015-11-23 18:46:57)

Jai Ganesh · 2015-11-25 01:09:11

Venus is the second planet from the Sun, orbiting it every 224.7 Earth days. It has no natural satellite. It is named after the Roman goddess of love and beauty. After the Moon, it is the brightest natural object in the night sky, reaching an apparent magnitude of -4.6, bright enough to cast shadows. Because Venus is an inferior planet from Earth, it never appears to venture far from the Sun: its elongation reaches a maximum of 47.8°.

Venus is a terrestrial planet and is sometimes called Earth's "sister planet" because of their similar size, mass, proximity to the Sun and bulk composition. It is radically different from Earth in other respects. It has the densest atmosphere of the four terrestrial planets, consisting of more than 96% carbon dioxide. The atmospheric pressure at the planet's surface is 92 times that of Earth's. With a mean surface temperature of 735 K (462 °C; 863 °F), Venus is by far the hottest planet in the Solar System, even though Mercury is closer to the Sun. Venus is shrouded by an opaque layer of highly reflective clouds of sulfuric acid, preventing its surface from being seen from space in visible light. It may have had oceans in the past, but these would have vaporized as the temperature rose due to a runaway greenhouse effect. The water has most probably photodissociated, and, because of the lack of a planetary magnetic field, the free hydrogen has been swept into interplanetary space by the solar wind. Venus's surface is a dry desertscape interspersed with slab-like rocks and periodically refreshed by volcanism.

Orbital characteristics -

Aphelion - 0.728213 AU - 108939000 km
Perihelion - 0.718440 AU - 107477000 km

Semi-major axis - 0.723332 AU - 108208000 km
Eccentricity - 0.006772
Orbital period - 224.701 d - 0.615198 yr
1.92 Venus solar day
Synodic period - 583.92 days
Average orbital speed - 35.02 km/s
Mean anomaly - 50.115°
Inclination - 3.39458° to ecliptic - 3.86° to Sun's equator
2.19° to invariable plane
Longitude of ascending node - 76.680°
Argument of perihelion - 54.884°
Satellites - None

Physical characteristics
Mean radius - 6051.8±1.0 km - 0.9499 Earths

Flattening 0
Surface area -

square kilometers - 0.902 Earths.
Volume - cubic kilometers - 0.866 Earths.
Mass - kilograms - 0.815 Earths
Mean density - 5.243 grams per cubic centimeters.
Surface gravity - 8.87 meters per second square - 0.904 g.
Escape velocity - 10.36 km/s (6.44 mi/s)
Sidereal rotation period - (-)243.025 d (retrograde)
Equatorial rotation velocity - 6.52 km/h (1.81 m/s)
Axial tilt - 2.64° (for retrograde rotation)
177.36° (to orbit)
North pole right ascension - 18h 11m 2s - 272.76°
North pole declination - 67.16°
Albedo - 0.67 (geometric) - 0.90 (Bond)
Surface temp. min mean max
Kelvin 737 K
Celsius 462 °C
Apparent magnitude - brightest (-4.9) (crescent) - (-3.8) (full)
Angular diameter - 9.7″ to 66.0″
Atmosphere - Surface pressure - 92 bar (9.2 MPa)
Composition by volume
≈ 96.5% carbon dioxide
≈ 3.5% nitrogen
0.015% sulfur dioxide
0.007% argon
0.002% water vapour
0.0017% carbon monoxide
0.0012% helium
0.0007% neon
trace carbonyl sulfide
trace hydrogen chloride
trace hydrogen fluoride

Last edited by Jai Ganesh (2015-11-25 01:10:54)

Jai Ganesh · 2015-11-26 16:44:49

Mars is the fourth planet from the Sun and the second smallest planet in the Solar System, after Mercury. Named after the Roman god of war, it is often referred to as the "Red Planet" because the iron oxide prevalent on its surface gives it a reddish appearance. Mars is a terrestrial planet with a thin atmosphere, having surface features reminiscent both of the impact craters of the Moon and the volcanoes, valleys, deserts, and polar ice caps of Earth.

The rotational period and seasonal cycles of Mars are likewise similar to those of Earth, as is the tilt that produces the seasons. Mars is the site of Olympus Mons, the largest volcano and second-highest known mountain in the Solar System, and of Valles Marineris, one of the largest canyons in the Solar System. The smooth Borealis basin in the northern hemisphere covers 40% of the planet and may be a giant impact feature. Mars has two moons, Phobos and Deimos, which are small and irregularly shaped. These may be captured asteroids, similar to 5261 Eureka, a Mars trojan.

On 28 September 2015, NASA announced the presence of briny flowing salt water on the Martian surface.

Aphelion - 1.6660 AU - 249.2 million km
Perihelion - 1.3814 AU - 206.7 million km
Semi-major axis - 1.523679 AU - 227,939,100 km
Eccentricity - 0.0935±0.0001
Orbital period - 1.8808 Julian years - 686.971 d
668.5991 sols
Synodic period - 779.96 days - 2.135 Julian years
Average orbital speed - 24.077 km/s
Mean anomaly - 19.3564°
Inclination - 1.850° to ecliptic - 5.65° to Sun's equator - 1.67° to invariable plane
Longitude of ascending node - 49.562°
Argument of perihelion - 286.537°
Satellites - 2

Physical characteristics
Mean radius - 3389.5±0.2 km
Equatorial radius - 3396.2±0.1 km - 0.533 Earths
Polar radius - 3,376.2±0.1 km - 0.531 Earths
Flattening - 0.00589±0.00015
Surface area - 144,798,500 square kilometers - 0.284 Earths
Volume -

cubic kilometers - 0.151 Earths
Mass - kilograms per meter cube - 0.107 Earths
Mean density - 3.9335±0.0004 g/cm³
Surface gravity - 3.711 m/s² - 0.376 g
Moment of inertia factor - 0.3662±0.0017
Escape velocity - 5.027 km/s
Sidereal rotation period - 1.025957 d - 24h 37m 22s
Equatorial rotation velocity - 868.22 km/h (241.17 m/s)
Axial tilt - 25.19° to its orbital plane
North pole right ascension - 21h 10m 44s - 317.68143°
North pole declination - 52.88650°
Albedo - 0.170 (geometric) - 0.25 (Bond)
Surface temp. min mean max
Kelvin 130 K 210 K 308 K
Celsius −143 °C −63 °C 35 °C
Apparent magnitude - +1.6 to −3.0
Angular diameter - 3.5–25.1″

Atmosphere :

Surface pressure - 0.636 (0.4–0.87) kPa

Composition by volume

95.97% carbon dioxide
1.93% argon
1.89% nitrogen
0.146% oxygen
0.0557% carbon monoxide
210 ppm water vapor
100 ppm nitric oxide
15 ppm molecular hydrogen
2.5 ppm neon
850 ppb HDO
300 ppb krypton
130 ppb formaldehyde
80 ppb xenon
18 ppb hydrogen peroxide
10 ppb methane

Last edited by Jai Ganesh (2015-11-29 17:38:20)

Jai Ganesh · 2015-11-29 18:43:41

Earth is the third planet from the Sun, the densest planet in the Solar System, the largest of the Solar System's four terrestrial planets, and the only astronomical object known to harbor life. The earliest life on Earth arose at least 3.5 billion years ago. Earlier physical evidences of life include graphite, a biogenic substance, in 3.7 billion-year-old metasedimentary rocks discovered in southwestern Greenland, as well as, "remains of biotic life" found in 4.1 billion-year-old rocks in Western Australia. Earth's biodiversity has expanded continually except when interrupted by mass extinctions. Although scholars estimate that over 99 percent of all species of life (over five billion) that ever lived on Earth are extinct, there are still an estimated 10–14 million extant species, of which about 1.2 million have been documented and over 86 percent have not yet been described. Over 7.3 billion humans live on Earth and depend on its biosphere and minerals for their survival. Earth's human population is divided among about two hundred sovereign states which interact through diplomacy, conflict, travel, trade and communication media.

According to evidence from radiometric dating and other sources, Earth was formed about 4.54 billion years ago. Within its first billion years, life appeared in its oceans and began to affect its atmosphere and surface, promoting the proliferation of aerobic as well as anaerobic organisms. Since then, the combination of Earth's distance from the Sun, its physical properties and its geological history have allowed life to thrive and evolve.

Earth's lithosphere is divided into several rigid tectonic plates that migrate across the surface over periods of many millions of years. Seventy-one percent of Earth's surface is covered with water, with the remainder consisting of continents and islands that together have many lakes and other sources of water that contribute to the hydrosphere. Earth's polar regions are mostly covered with ice, including the Antarctic ice sheet and the sea ice of the polar ice packs. Earth's interior remains active with a solid iron inner core, a liquid outer core that generates the magnetic field, and a convecting mantle that drives plate tectonics.

Earth gravitationally interacts with other objects in space, especially the Sun and the Moon. During one orbit around the Sun, Earth rotates about its own axis 366.26 times, creating 365.26 solar days or one sidereal year. Earth's axis of rotation is tilted 23.4° away from the perpendicular of its orbital plane, producing seasonal variations on the planet's surface with a period of one tropical year (365.24 solar days). The Moon is Earth's only permanent natural satellite. Its gravitational interaction with Earth causes ocean tides, stabilizes the orientation of Earth's rotational axis, and gradually slows Earth's rotational rate.

Aphelion - 152,100,000 km (94,500,000 mi) - (1.01673 AU)
Perihelion - 147,095,000 km (91,401,000 mi) - (0.9832687 AU)
Semi-major axis - 149,598,023 km (92,955,902 mi) - (1.000001018 AU)
Eccentricity - 0.0167086
Orbital period - 365.256363004 d - (1.00001742096 yr)
Average orbital speed - 29.78 km/s (18.50 mi/s) - (107,200 km/h (66,600 mph))
Mean anomaly - 358.617 deg
Inclination - 7.155 deg to Sun's equator; 1.57869 deg to invariable plane;
Longitude of ascending node (-) −11.26064 deg to J2000 ecliptic
Argument of perihelion - 114.20783 deg
Satellites - One natural satellite; 1305 operational artificial satellites

Physical characteristics
Mean radius - 6,371.0 km (3,958.8 mi)
Equatorial radius - 6,378.1 km (3,963.2 mi)
Polar radius - 6,356.8 km (3,949.9 mi)
Flattening - 0.0033528
1/298.257222101
Circumference - 40,075.017 km (24,901.461 mi) (equatorial); 40,007.86 km (24,859.73 mi) (meridional)

Surface area - 510,072,000 square kilometers (196,940,000 sq mi)
(148,940,000 square kilometers (57,510,000 sq mi) (29.2%) land
361,132,000 square kilometers (139,434,000 sq mi) (70.8%) water)

Volume -

cubic kilometers cu mi)
Mass - kg ( lb)
M☉)
Mean density - 5.514 grams per cubic centimeters (0.1992 lb/cu in)
Surface gravity - 9.807 meters per square second (32.18 feet per square second) - (1 g)
Moment of inertia factor - 0.3307
Escape velocity - 11.186 km/s (6.951 mi/s)
Sidereal rotation period - 0.99726968 d - (23h 56m 4.100s)
Equatorial rotation velocity - 1,674.4 km/h (1,040.4 mph)
Axial tilt - 23.4392811°
Albedo - 0.367 geometric, 0.306 Bond

Surface temp. : min : mean : max
Kelvin : 184 K : 288 K : 330 K
Celsius : −89.2 °C :15 °C : 56.7 °C
Fahrenheit : −128.5 °F : 59 °F : 134 °F
Atmosphere
Surface pressure : 101.325 kPa (at MSL)
Composition by volume

78.08% nitrogen (N2) (dry air)
20.95% oxygen (O2)
0.930% argon
0.039% carbon dioxide
~ 1% water vapor (climate-variable)

Last edited by Jai Ganesh (2015-11-29 18:47:53)

Jai Ganesh · 2015-11-29 21:21:16

Asankhyeya

The asankhyeya (also called asaṃkhyeya) is a number described in Buddhist texts that is equal to

, or 1 followed by 140 zeroes. It is pronounced Asougi in Japanese where it is equal to , and means "innumerable".

The Avatamsaka Sutra gives an alternate description of Asankhyeya as

, defining a series of numbers that are squares of each other starting with one koti equalling , one koti kotis making an ayuta (), one ayuta ayutas making a nayuta , and so on, with Asankhyeya being the 104th number in this chain.

Jai Ganesh · 2015-11-30 02:05:56

Tallakshana

The tallakshana is equal to

.

Dvajagravati

The dvajagravati is equal to

Mahakathana

The mahakathana is equal to

, or 1 followed by 133 zeroes.

Asankhyeya

The asankhyeya (also called asaṃkhyeya) is a number described in Buddhist texts that is equal to

, or 1 followed by 140 zeroes. It is pronounced Asougi in Japanese where it is equal to , and means "innumerable"

Dvajagranisamani

The dvajagranisamani is equal to

Vahanaprajnapti

The vahanaprajnapti is equal to

Inga

The inga is equal to

Kuruta

The kuruta is equal to

Sarvanikshepa

The sarvanikshepa is equal to

Agrasara

The agrasara is equal to

Uttaraparamanurajahpravesa

The uttaraparamanurajahpravesa is equal to

Avatamsaka Sutra

The Avatamsaka Sutra (sometimes called the Flower Garland Sutra) is one of the better-known Buddhist scriptures. In the 30th chapter, the Buddha teaches a king the concept of large numbers.

The Buddha begins by calculating:

...

He continues until

, at which he states, "that number squared is an incalculable." The Buddha then names the following:

The "Untold" number is between the first Skewes number and a Dakillion.

Last edited by Jai Ganesh (2015-11-30 02:09:17)

Jai Ganesh · 2015-12-14 19:39:34

-
Estimated mass (in kilograms) of the observable universe.

-
Estimated total mass-energy (in Joules) of the observable universe.

Estimate the total number of fundamental particles in the observable universe (other estimates go up to )

Planck density, the density (in kilograms per meter cube) of the universe at one unit of Planck time after the Big Bang.

Jai Ganesh · 2015-12-14 20:02:05

Approximate density (in kilograms per meter cube) of the universe as a whole.

Approximate mass (in kilograms) of a stationary electron.

Planck Length (in meters), the size of a hypothetical string. Lengths smaller than this are considered not make any physical sense in our current understanding of physics.

Planck Time (in seconds), the shortest meaningful interval of time, and the earliest time the known universe can be measured from.

Jai Ganesh · 2016-01-01 00:22:47

Titan (or Saturn VI) is the largest moon of Saturn. It is the only natural satellite known to have a dense atmosphere, and the only object other than Earth where clear evidence of stable bodies of surface liquid has been found.

Titan is the sixth ellipsoidal moon from Saturn. Frequently described as a planet-like moon, Titan's diameter is 50% larger than Earth's natural satellite, the Moon, and it is 80% more massive. It is the second-largest moon in the Solar System, after Jupiter's moon Ganymede, and is larger by volume than the smallest planet, Mercury, although only 40% as massive. Discovered in 1655 by the Dutch astronomer Christiaan Huygens, Titan was the first known moon of Saturn, and the sixth known planetary satellite.

Titan is primarily composed of water ice and rocky material. Much as with Venus before the Space Age, the dense opaque atmosphere prevented understanding of Titan's surface until new information accumulated when the Cassini–Huygens mission arrived in 2004, including the discovery of liquid hydrocarbon lakes in Titan's polar regions. The geologically young surface is generally smooth, with few impact craters, although mountains and several possible cryovolcanoes have been found.

The atmosphere of Titan is largely nitrogen; minor components lead to the formation of methane–ethane clouds and nitrogen-rich organic smog. The climate—including wind and rain—creates surface features similar to those of Earth, such as dunes, rivers, lakes, seas (probably of liquid methane–ethane), and deltas, and is dominated by seasonal weather patterns as on Earth. With its liquids (both surface and subsurface) and robust nitrogen atmosphere, Titan's methane cycle is analogous to Earth's water cycle, although at a much lower temperature.

Mean radius :

(0.404 Earths, 1.480 Moons)
Surface area
square kilometers
Volume cubic kilometers (0.066 Earths) (3.3 Moons)
Mass
(0.0225 Earths) (1.829 Moons)
Mean density
grams per cubic centimeters

mathaholic · 2016-02-12 01:39:37

I see.

Useful numbers. And the data on Titan.

Jai Ganesh · 2016-02-12 17:41:25

Thanks, mathaholic!

Vega (α Lyr, α Lyrae, Alpha Lyrae) is the brightest star in the constellation Lyra, the fifth brightest star in the night sky and the second brightest star in the northern celestial hemisphere, after Arcturus. It is a relatively close star at only 25 light-years from Earth, and, together with Arcturus and Sirius, one of the most luminous stars in the Sun's neighborhood.

Vega has been extensively studied by astronomers, leading it to be termed "arguably the next most important star in the sky after the Sun." Vega was the northern pole star around 12,000 BCE and will be so again around the year 13,727 when the declination will be +86°14'. Vega was the first star other than the Sun to be photographed and the first to have its spectrum recorded. It was one of the first stars whose distance was estimated through parallax measurements. Vega has served as the baseline for calibrating the photometric brightness scale, and was one of the stars used to define the mean values for the UBV photometric system.

Vega is only about a tenth of the age of the Sun, but since it is 2.1 times as massive its expected lifetime is also one tenth of that of the Sun; both stars are at present approaching the midpoint of their life expectancies. Vega has an unusually low abundance of the elements with a higher atomic number than that of helium. Vega is also a suspected variable star that may vary slightly in magnitude in a periodic manner. It is rotating rapidly with a velocity of 274 km/s at the equator. This is causing the equator to bulge outward because of centrifugal effects, and, as a result, there is a variation of temperature across the star's photosphere that reaches a maximum at the poles. From Earth, Vega is being observed from the direction of one of these poles.

Based on an observed excess emission of infrared radiation, Vega appears to have a circumstellar disk of dust. This dust is likely to be the result of collisions between objects in an orbiting debris disk, which is analogous to the Kuiper belt in the Solar System. Stars that display an infrared excess because of dust emission are termed Vega-like stars.

Jai Ganesh · 2022-02-23 22:05:14

Orders of magnitude (numbers):

Smaller than {10}^{-100} (one googolth)

Mathematics – random selections: Approximately {10}^{-183,800} is a rough first estimate of the probability that a typing "monkey", or an English-illiterate typing robot, when placed in front of a typewriter, will type out William Shakespeare's play Hamlet as its first set of inputs, on the precondition it typed the needed number of characters. However, demanding correct punctuation, capitalization, and spacing, the probability falls to around {10}^{-360,783}.

Computing: 2.2 × {10}^{-78913} is approximately equal to the smallest positive non-zero value that can be represented by an octuple-precision IEEE floating-point value.
1 × {10}^{-6176} is equal to the smallest positive non-zero value that can be represented by a quadruple-precision IEEE decimal floating-point value.
6.5 × {10}^{-4966} is approximately equal to the smallest positive non-zero value that can be represented by a quadruple-precision IEEE floating-point value.
3.6 × {10}^{-4951} is approximately equal to the smallest positive non-zero value that can be represented by an 80-bit x86 double-extended IEEE floating-point value.
1 × {10}^{-398} is equal to the smallest positive non-zero value that can be represented by a double-precision IEEE decimal floating-point value.
4.9 × {10}^{-324} is approximately equal to the smallest positive non-zero value that can be represented by a double-precision IEEE floating-point value.
1 × {10}^{-101} is equal to the smallest positive non-zero value that can be represented by a single-precision IEEE decimal floating-point value.

(The Institute of Electrical and Electronics Engineers (IEEE) is a 501(c)(3) professional association for electronic engineering and electrical engineering (and associated disciplines) with its corporate office in New York City and its operations center in Piscataway, New Jersey. It was formed in 1963 from the amalgamation of the American Institute of Electrical Engineers and the Institute of Radio Engineers.

Due to its expansion of scope into so many related fields, it is simply referred to by the letters I-E-E-E (pronounced I-triple-E), except on legal business documents. As of 2018, it is the world's largest association of technical professionals with more than 423,000 members in over 160 countries around the world. Its objectives are the educational and technical advancement of electrical and electronic engineering, telecommunications, computer engineering and similar disciplines.)

Math Is Fun Forum

#51 2015-11-16 05:25:18

Re: Micro / Macro numbers in Science

#52 2015-11-16 07:31:35

Re: Micro / Macro numbers in Science

#53 2015-11-16 11:59:32

Re: Micro / Macro numbers in Science

#54 2015-11-16 14:13:03

Re: Micro / Macro numbers in Science

#55 2015-11-17 17:39:47

Re: Micro / Macro numbers in Science

#56 2015-11-17 23:47:09

Re: Micro / Macro numbers in Science

#57 2015-11-18 20:05:16

Re: Micro / Macro numbers in Science

#58 2015-11-20 01:34:02

Re: Micro / Macro numbers in Science

#59 2015-11-23 18:34:15

Re: Micro / Macro numbers in Science

#60 2015-11-25 01:09:11

Re: Micro / Macro numbers in Science

#61 2015-11-26 16:44:49

Re: Micro / Macro numbers in Science

#62 2015-11-29 18:43:41

Re: Micro / Macro numbers in Science

#63 2015-11-29 21:21:16

Re: Micro / Macro numbers in Science

#64 2015-11-30 02:05:56

Re: Micro / Macro numbers in Science

#65 2015-12-14 19:39:34

Re: Micro / Macro numbers in Science

#66 2015-12-14 20:02:05

Re: Micro / Macro numbers in Science

#67 2016-01-01 00:22:47

Re: Micro / Macro numbers in Science

#68 2016-02-12 01:39:37

Re: Micro / Macro numbers in Science

#69 2016-02-12 17:41:25

Re: Micro / Macro numbers in Science

#70 2022-02-23 22:05:14

Re: Micro / Macro numbers in Science

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