Astrophysics

Tidal acceleration

 * See also:

Image shows an approximation of the shape of a rapidly spinning planet. North pole is at the top. South pole is at the bottom. The equator reaches orbital velocity.



Orbital velocity:


 * $$v_o = \sqrt{\frac{GM}{r}}$$

Orbital period:


 * $$T = 2\pi\sqrt{\frac{r^3}{GM}}$$

Orbital angular momentum:


 * $$mvr \quad = \quad m \Bigg ( \sqrt{\frac{GM}{r}} \Bigg ) r \quad = \quad m \sqrt{GMr}$$

Rotational angular momentum of solid sphere:


 * $$L=I \omega = \frac{2}{5}mr^2 \frac{v}{r} = \frac{2}{5}mvr$$

where:
 * r is the orbit's
 * G is the ,
 * M is the mass of the more massive body.
 * m is the mass of the less massive body.

Moons orbital angular momentum is

Earths rotational angular momentum is


 * Correcting for Earths uneven mass distribution: +  +  +  = 4.6 * 10^33 Js

The total amount of angular momentum for the Earth-Moon system is 28.73 + 4.6 = 33.33 * 10^33 Js

Moons current orbit is 384,399 km. Its orbital period is 2.372 * 106 seconds. (27 days, 10 hours, 50 minutes). Its orbital velocity is 1.022 km/s.

for the moon is


 * Fluid: 18,381 km fluid


 * 384,399 / 18,381 = 20.9


 * Orbital momentum of moon at fluid Roche limit = 28.73 * 10^33 Js / sqrt(20.9) = 6.3 * 10^33


 * Earth would spin (28.73-6.3+4.6)/4.6 = 5.876 times faster


 * Rigid: 9,492 km


 * 384,399 / 9,492 = 40.5


 * Orbital momentum of moon at rigid Roche limit = 28.73 * 10^33 Js / sqrt(40.5) = 4.5 * 10^33


 * Earth would spin (28.73-4.5+4.6)/4.6 = 6.27 times faster

Orbital radius with period = 4 hours:


 * $$\sqrt[3]{G \cdot 1 \text{ Earth mass} \cdot \Bigg ( \frac{4\text{ hours}}{2\pi} \Bigg )^2 }$$ = 12,800 km

Alternately we can ask what the orbital period would be if Earth had a moon (not necessarily the moon) at 18,381 km.


 * $$T = 2\pi\sqrt{\frac{(18,381 \text{ km})^3}{G*1 \text{ Earth mass} }}$$ =


 * Earth would spin 24/7.554 = 3.177 times faster


 * Earths angular momentum would be 3.177 * 4.6 * 10^33 Js = 14.6142 * 10^33 Js


 * Our current Moons angular momentum would be 28.73 - (14.6142 - 4.6) * 10^33 Js = 18.7158 * 10^33 Js


 * Thats 18.7158 / 28.73 = 0.65


 * So the current moons orbit would have been 0.65^2 * 384,399 km = 0.424 * 384,399 km = 162985 km

Tidal rhythmites are alternating layers of sand and silt laid down offshore from estuaries having great tidal flows. Daily, monthly and seasonal cycles can be found in the deposits. This geological record indicates that 620 million years ago there were 400±7 solar days/year

The motion of the Moon can be followed with an accuracy of a few centimeters by lunar laser ranging. Laser pulses are bounced off mirrors on the surface of the moon. The results are:


 * +38.08±0.04 mm/yr (384,399 km / 63.4 billion years)
 * (33.33 * 10^33 Js / 23 billion years)
 * 1.42*10^26 Js/century

The corresponding change in the length of the day can be computed:


 * (1.42*10^26)/(4.6 * 10^33) * 24 hours = 3.087*10^-8  * 24 hours =

620 million yrs ago the Moon had 1.42*10^24 * 620*10^6 = less angular momentum. The moons orbit was therefore 384,399 km * ((28.73-0.88)/28.73)^2 = 361,211 km. One month lasted 2.161 * 106 seconds. (25 days, 16 minutes, 40 seconds)

The Earth spun (4.6+0.88)/4.6 = 1.19 times faster so the day was 24 hours / 1.19 = 20.1680672 hours

The year was 400 "days" * 20.1680672 hours per "day" = 336.135 24-hour periods

Earths orbit was therefore


 * $$\sqrt[3]{G \cdot 1 \text{ Solar mass} \Bigg ( \frac{336.134453 \text{ days}}{2\pi} \Bigg )^2}$$ =

Therefore Earth must be receding from the sun at


 * Thats

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Planets


From 16 Psyche:

16 Psyche is one of the ten most massive asteroids in the asteroid belt. It is over 200 km (120 mi) in diameter and contains a little less than 1% of the mass of the entire asteroid belt. It is thought to be the exposed iron core of a protoplanet.











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Brown dwarfs


As can be seen in the image to the right, all planets (Brown dwarfs) from 1 to 100 Jupiter masses are about 1 Jupiter radius which is 69,911 km. The largest "puffy" planets are 2 Jupiter radii. 1 Jupiter volume = 1.431×1015 km3

This suggests that the pressure an electron shell (in degenerate matter) can withstand without again becoming degenerate is inversely proportional to the sixth power of its radius:


 * $$P=\frac{1}{r^6}$$

(This formula only applies to degenerate matter like metallic hydrogen. Non-degenerate matter can withstand far more pressure).

If so then the maximum size (radius) that a planet composed entirely of one (degenerate) element could grow would depend only on, and be inversely proportional to, the atomic mass of its atoms. (Use 2 for the atomic mass of diatomic hydrogen).

Simplified calculation of radius of brown dwarf as core grows from zero to 1 Jupiter radius:




 * r is radius of core with 2.83 (sqrt(2)3) times the density of overlying material

Rock floats on top of the metallic hydrogen but iron sinks to the Core. 0.1% of the mass of the brown dwarf is iron. Assuming iron density of 231.85 g/cm3 (as in Earths core), the gravity of the iron core will cause the brown dwarf to be about smaller then it would be otherwise.

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Dark matter
From Dark matter

Dark matter is a type of unidentified matter that may constitute about 80% of the total matter in the universe. It has not been directly observed, but its gravitational effects are evident in a variety of astrophysical measurements. The primary evidence for dark matter is that calculations show that many galaxies would fly apart instead of rotating if they did not contain a large amount of matter beyond what can be observed.

From Gravitational microlensing





Microlensing allows the study of objects that emit little or no light. With microlensing, the lens mass is too low for the displacement of light to be observed easily, but the apparent brightening of the source may still be detected. In such a situation, the lens will pass by the source in seconds to years instead of millions of years.

The Einstein radius, also called the Einstein angle, is the angular radius of the Einstein ring in the event of perfect alignment. It depends on the lens mass M, the distance of the lens dL, and the distance of the source dS:


 * $$\theta_E = \sqrt{\frac{4GM}{c^2} \frac{d_S - d_L}{d_S d_L}}$$ (in radians).

For M equal to 60 Jupiter masses, dL = 4000 parsecs, and dS = 8000 parsecs (typical for a Bulge microlensing event), the Einstein radius is (angle subtended by ). By comparison, ideal Earth-based observations have angular resolution around 0.4 arcseconds, 1660 times greater. One parsec is equal to about 3.26 light-years (30 trillion km).

Any brown dwarf surrounded by a circumstellar disk larger and thicker than 1 au would therefore be virtually completely undetectable.

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Stars

 * See also:, , ,





Fusion of diatomic hydrogen begins around 60 Jupiter masses. Fusion of monatomic helium requires significantly more pressure.

Fusion releases energy that heats the star causing it to expand. The expansion reduces the pressure in the core which reduces the rate of fusion. So the rate of fusion is self limiting. A low mass star has a lifetime of billions of years. A high mass star has a lifetime of only a few tens of millions of years despite starting with more hydrogen.

Low mass stars are far more common than high mass stars. The masses of the two component stars of NGC 3603-A1, A1a and A1b, determined from the orbital parameters are 116 ± 31 M☉ and 89 ± 16 M☉respectively. This makes them the two most massive stars directly measured, i.e. not estimated from models.

The luminousity of a star is:


 * $$L = 4 \pi R^2 \sigma T^4$$


 * where σ is the :


 * $$\sigma = \frac{2\pi^5k_{\rm B}^4}{15h^3c^2} = \frac{\pi^2k_{\rm B}^4}{60\hbar^3c^2} = 5.670373(21) \, \times 10^{-8}\ \textrm{J}\,\textrm{m}^{-2}\,\textrm{s}^{-1}\,\textrm{K}^{-4}

$$

The luminosity of the sun at 5772 K and 695,700 km is


 * Thats

The brightness of sunlight at the surface of the Earth is 1400 watt/meter2

The plasma inside a star is non-relativistic. A relativistic plasma with a thermal has temperatures greater than around 260 keV, or. Those sorts of temperatures are only created in a supernova. The core of the sun is about 15 * 106 K.

Plasmas, which are normally opaque to light, are transparent to light with frequency higher than the. The plasma literally cant vibrate fast enough to keep up with the light. Plasma frequency is proportional to the square root of the electron density.


 * $$\omega = \sqrt{\frac{n_\mathrm{e} q_e^{2}}{m_e \varepsilon_0}}$$


 * where


 * ne = number of electrons / volume.

See also:

From Radiative zone

From 0.3 to 1.2 solar masses, the region around the stellar core is a radiative zone. (The light frequency is higher than the plasma frequency). The radius of the radiative zone increases monotonically with mass, with stars around 1.2 solar masses being almost entirely radiative.

From Convective zone

In main sequence stars of less than about 1.3 solar masses, the outer envelope of the star contains a region of relatively low temperature which causes the frequency of the light to be lower than the plasma frequency which causes the opacity to be high enough to produce a steep temperature gradient. This produces an outer convection zone. The Sun's convection zone extends from 0.7 solar radii (500,000 km) to near the surface.

From Cepheid variable

A Cepheid variable is a type of star that pulsates radially, varying in both diameter and temperature and producing changes in brightness with a well-defined stable period and amplitude.

A strong direct relationship between a Cepheid variable's luminosity and pulsation period allows one to know the true luminosity of a Cepheid by simply observing its pulsation period. This in turn allows one to determine the distance to the star, by comparing its known luminosity to its observed brightness.

From Variable star

The pulsation of cepheids is known to be driven by oscillations in the ionization of helium. From fully ionized (more opaque) He++ to partially ionized (more transparent) He+ and back to He++. See.

In the swelling phase. Its outer layers expand, causing them to cool. Because of the decreasing temperature the degree of ionization also decreases. This makes the gas more transparent, and thus makes it easier for the star to radiate its energy. This in turn will make the star start to contract. As the gas is thereby compressed, it is heated and the degree of ionization again increases. This makes the gas more opaque, and radiation temporarily becomes captured in the gas. This heats the gas further, leading it to expand once again. Thus a cycle of expansion and compression (swelling and shrinking) is maintained.

From Instability strip

In normal A-F-G stars He is neutral in the stellar photosphere. Deeper below the photosphere, at about 25,000–30,000K, begins the He II layer (first He ionization). Second ionization (He III) starts at about 35,000–50,000K.

The Sun's photosphere has a temperature between 4,500 and 6,000 K. Negative hydrogen ions (H-) are the primary reason for the highly opaque nature of the photosphere.

As the star fuses hydrogen into heavier elements the heavier elements build up in the core. Eventually the outer layers of the star are blown away and all thats left is the core. We call whats left a white dwarf.

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White dwarfs
A white dwarf is about the same size as the Earth but is far denser and far more massive. A typical temperature for a white dwarf is 25,000 K. That would make its surface brightness 350 times the surface brightness of the sun.

Simplified calculation of radius of White dwarf as core grows from zero to half the original radius:




 * r is radius of core. The core has 16 times the density (twice the atomic number) of the overlying material. The final state has half the radius and twice the mass of the original white dwarf.

A 0.6 solar mass White dwarf is 8900 km in radius which Is 8.03 times smaller than Jupiter which suggests a composition of oxygen. It has a surface gravity of


 * $$\frac{G \cdot 0.6 \text{ solar mass}}{(8900 \text{ km})^2}$$ =

Its density is which is 12,625 times denser than oxygen in its ground state. Thats 23.2853 times denser. Sqrt(2)9 = 22.63

A 1.13 solar mass White dwarf is 4500 km in radius which Is 15.9 times smaller than Jupiter which suggests a composition of sulfur. It has a surface gravity of


 * $$\frac{G \cdot 1.13 \text{ solar mass}}{(4500 \text{ km})^2}$$ =

Its density is which is 11,498 times denser than sulfur in its ground state. Thats 22.573 times denser.

For a white dwarf made of iron:


 * Radius: 2,553 km
 * Surface area: 8.2*107 km2
 * Mass per surface area: 3.8 * 1013 g/mm2
 * Mass: 4.454 * 107 g/cm3 * (4/3)*pi*(2553 km)3 in solar masses = 1.56 solar masses.
 * Surface gravity:
 * Density: (sqrt(2)9)3 * 3844.75 g/cm3 = 4.454 * 107 g/cm3
 * Core pressure:

The core of a white dwarf with a mass greater than the (1.44 solar masses) will undergo gravitational collapse and become a neutron star. Back to top



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Neutron stars

 * See also:

Assuming a solid honeycomb array of neutron pairs with radius 1 fm, a sheet of (if such a thing existed) would have a density of.

Density of a liquid neutron star made of neutron pairs with radius 1 fm would be 479.8×1012 g/cm3

The maximum observed mass of neutron stars is about 2.01 M☉.

At that density a 2 solar mass neutron star would have a radius of

Its gravitational binding energy would be

The (or TOV limit) is an upper bound to the mass of cold, nonrotating neutron stars, analogous to the Chandrasekhar limit for white dwarf stars. Observations of GW170817 suggest that the limit is close to 2.17 solar masses.

The for a neutron star is not yet known.

A 2 solar mass neutron star with radius of 12.5544 km would have a surface gravity of:


 * $$\frac{G \cdot 2 \text{ solar mass}}{(12.5544 \text{ km})^2}$$ =

The pressure in its core would be $$\frac{3}{8 \pi} \frac{G \cdot Mass_{active} \cdot Mass_{passive}}{r^4} =$$ =

Its moment of inertia is: 0.4*2 solar masses*(12.5544 km)^2 =

From Glitch (astronomy)

A glitch (See ) is a sudden increase of up to 1 part in 106 in the rotational frequency of a rotation-powered pulsar. Following a glitch is a period of gradual recovery, lasting 10-100 days, where the observed periodicity slows to a period close to that observed before the glitch.

If mass is constant then


 * $$radius = 1/\sqrt[3]{density}$$

If angular momentum is constant then


 * $$frequency = v/r = 1/r^2 = 1/(1/\sqrt[3]{density})^2 = density^{\frac{2}{3}}$$

The moment of inertia of a solid crust 1 cm thick is: 0.666*((1.2*479.8×10^12 g/cm3)*1 cm*4*pi*(12.5544 km)^2)*(12.5544 km)^2 =. Thats 1/209,440 of the total moment of inertia. 1 cm doesn't seem like much but if each Neutron were the size of an atom then that one centimeter would be one or two km.

External link: Pulsar glitches and their impact on neutron-star astrophysics

From Supermassive black hole

A supermassive black hole (SMBH or SBH) is the largest type of, on the order of hundreds of thousands to billions of ( M ☉), and is found in the centre of almost all currently known massive galaxies.

The mass of the SMBH in a galaxy is often close to the combined mass of the galaxy's globular clusters.

The mean ratio of black hole mass to bulge mass is now believed to be approximately 1:1000.


 * The most massive galaxy known is 30 trillion solar masses.

Some supermassive black holes appear to be over 10 billion solar masses.

From Quasar:

A quasar is an active galactic nucleus of very high luminosity. A quasar consists of a supermassive black hole surrounded by an orbiting accretion disk of gas. The most powerful quasars have luminosities exceeding 2.6×1014 ℒ☉ (1041 W or 17.64631 M☉/year), thousands of times greater than the luminosity of a large galaxy such as the Milky Way.

Growing at a rate of 17.6/1.4^2 solar mass per year a 60 billion solar mass Black hole would take to reach full size. See

Growing at a rate of 17.6/2.8^2 solar mass per year a 240 billion solar mass Black hole would take to reach full size.

Masses of supermassive black holes in billions of solar masses:
 * 1) 240? (Hypothetical )
 * 2) 120? (Hypothetical)
 * 3) 80? (Hypothetical)
 * 66
 * 40
 * 33
 * 30
 * 23
 * 21
 * 20
 * 1) 19.5
 * 18
 * 17
 * 15
 * 14
 * 14
 * 1) 13.5
 * 13
 * 12
 * 1) 12.4
 * 11
 * 11
 * 10
 * 10
 * 9.8
 * 9.7
 * 9.1
 * 7.8
 * 7.2
 * 7.2
 * 6.9
 * 7.2
 * 6.9

From Eddington luminosity

The Eddington luminosity, also referred to as the Eddington limit, is the maximum luminosity a body (such as a star) can achieve when there is balance between the force of radiation acting outward and the gravitational force acting inward. The state of balance is called hydrostatic equilibrium. When a star exceeds the Eddington luminosity, it will initiate a very intense radiation-driven stellar wind from its outer layers.

For pure ionized hydrogen
 * $$\begin{align}L_{\rm Edd}&=\frac{4\pi G M m_{\rm p} c} {\sigma_{\rm T}}\\

&\cong 1.26\times10^{31}\left(\frac{M}{M_\odot}\right){\rm W} = 1.26\times10^{38}\left(\frac{M}{M_\odot}\right){\rm erg/s} = 3.2\times10^4\left(\frac{M}{M_\odot}\right) L_\odot \end{align} $$

where $$M_\odot$$ is the mass of the Sun and $$L_\odot$$ is the luminosity of the Sun.

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Gamma-ray bursts


From Gamma-ray burst

Gamma-ray bursts (GRBs) are extremely energetic explosions that have been observed in distant galaxies. They are the brightest electromagnetic events known to occur in the universe. Bursts can last from ten milliseconds to several hours. After an initial flash of gamma rays, a longer-lived "afterglow" is usually emitted at longer wavelengths (X-ray, ultraviolet, optical, infrared, microwave and radio).

Assuming the gamma-ray explosion to be spherical, the energy output of would be within a factor of two of the rest-mass energy of the Sun (the energy which would be released were the Sun to be converted entirely into radiation).

No known process in the universe can produce this much energy in such a short time.

From GRB 111209A

GRB 111209A is the longest lasting gamma-ray burst (GRB) detected by the Swift Gamma-Ray Burst Mission on December 9, 2011. Its duration is longer than 7 hours.

On average two long gamma ray burst occurs every 3 days and have average redshift of 2. Making the simplifying assumption that all long gamma ray bursts occur at exactly redshift 2 (9.2 * 109 light years) we get one gamma ray burst per

There are 12 galaxies per cubic megaparsec. Thats

One short grb per 3 days at average redshift of 0.5 (4.6 * 109 light years) gives 1 grb per

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Ultra-high-energy Cosmic rays
External links: http://hires.physics.utah.edu/reading/uhecr.html, The cosmic ray energy spectrum as measured using the Pierre Auger Observatory

The highest energy gamma rays ever detected from space were around 16 TeV which corresponds to a wavelength of 1/13,000 fm.

From Cosmic ray

are high-energy radiation, mainly originating outside the Solar System and even from distant galaxies. Upon impact with the Earth's atmosphere, cosmic rays can produce showers of secondary particles that sometimes reach the surface. Composed primarily of high-energy protons and atomic nuclei, they are of uncertain origin. Data from the Fermi Space Telescope (2013) have been interpreted as evidence that a significant fraction of primary cosmic rays originate from the supernova explosions of stars. Active galactic nuclei are also theorized to produce cosmic rays.

From Ultra-high-energy cosmic ray

In, an ultra-high-energy cosmic ray (UHECR) is a cosmic ray particle with a kinetic energy greater than than 1 × 1018 , far beyond both the and energies typical of other cosmic ray particles.

An extreme-energy cosmic ray (EECR) is an UHECR with energy exceeding 5 × 1019 eV (about 8 joule), the so-called   (GZK limit). This limit should be the maximum energy of cosmic ray protons that have traveled long distances (about 160 million light years), since higher-energy protons would have lost energy over that distance due to scattering from photons in the (CMB). However, if an EECR is not a proton, but a nucleus with $$A$$ nucleons, then the GZK limit applies to its nucleons, each of which carry only a fraction $$1/A$$ of the total energy.

These particles are extremely rare; between 2004 and 2007, the initial runs of the (PAO) detected 27 events with estimated arrival energies above  5.7 × 1019 eV, i.e., about one such event every four weeks in the 3000 km2 area surveyed by the observatory.

At that rate particles will fall onto a star with radius 1 million kilometers every hundred million years.

From Oh-My-God particle:

The Oh-My-God particle was an ultra-high-energy cosmic ray detected on the evening of 15 October 1991 by the Fly's Eye Cosmic Ray Detector. Its observation was a shock to astrophysicists, who estimated its energy to be approximately 3×1020 eV. It was probably a cluster of 6 ultra-high-energy cosmic ray particles.

mv2 = (205,887*128^2*2 neutron mass * (2.807*c)^2) =

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Expansion of the universe
From Expansion of the universe

The expansion of the universe is the increase of the distance between two distant parts of the universe with time.

The expansion of space is often illustrated with conceptual models. In the "balloon model" a spherical balloon is inflated from an initial size of zero (representing the big bang).

From Scale factor (cosmology)

Some insight into the expansion can be obtained from a Newtonian expansion model which leads to a simplified version of the Friedman equation. It relates the proper distance (which can change over time, unlike the comoving distance which is constant) between a pair of objects, e.g. two galaxy clusters, moving with the Hubble flow in an expanding or contracting FLRW universe at any arbitrary time $$t$$ to their distance at some reference time $$t_0$$. The formula for this is:


 * $$d(t) = a(t)d_0,\,$$

where $$d(t)$$ is the proper distance at epoch $$t$$, $$d_0$$ is the distance at the reference time $$t_0$$ and $$a(t)$$ is the scale factor. Thus, by definition, $$a(t_0) = 1$$.

The scale factor is dimensionless, with $$t$$ counted from the birth of the universe and $$t_0$$ set to the present age of the universe: $$13.799\pm0.021\,\mathrm{Gyr}$$ giving the current value of $$a$$ as $$a(t_0)$$ or $$1$$.

The evolution of the scale factor is a dynamical question, determined by the equations of general relativity, which are presented in the case of a locally isotropic, locally homogeneous universe by the.

The Hubble parameter is defined:


 * $$H \equiv {\dot{a}(t) \over a(t)}$$

where the dot represents a time derivative. From the previous equation $$d(t) = d_0 a(t)$$ one can see that $$\dot{d}(t) = d_0 \dot{a}(t)$$, and also that $$d_0 = \frac{d(t)}{a(t)}$$, so combining these gives $$\dot{d}(t) = \frac{d(t) \dot{a}(t)}{a(t)}$$, and substituting the above definition of the Hubble parameter gives $$\dot{d}(t) = H d(t)$$ which is just Hubble's law.

From Hubble's law



The discovery of the linear relationship between redshift and distance, coupled with a supposed linear relation between recessional velocity and redshift, yields a straightforward mathematical expression for Hubble's Law as follows:


 * $$v = H_0 \, D$$

where
 * $$v$$ is the recessional velocity, typically expressed in km/s.
 * H0 is Hubble's constant and corresponds to the value of $$H$$ (often termed the Hubble parameter which is a value that is and which can be expressed in terms of the ) in the Friedmann equations taken at the time of observation denoted by the subscript 0. This value is the same throughout the Universe for a given comoving time.
 * $$D$$ is the proper distance (which can change over time, unlike the comoving distance, which is constant) from the galaxy to the observer, measured in mega parsecs (Mpc), in the 3-space defined by given cosmological time. (Recession velocity is just v = dD/dt).

Hubble's law is considered a fundamental relation between recessional velocity and distance. However, the relation between recessional velocity and redshift depends on the cosmological model adopted, and is not established except for small redshifts.

For distances D larger than the radius of the Hubble sphere rHS, objects recede at a rate faster than the speed of light:


 * $$r_{HS} = \frac{c}{H_0} \ . $$

Its radius is the Hubble radius and its volume is the Hubble volume.

The Hubble constant $$H_0$$ has units of inverse time; the Hubble time tH is simply defined as the inverse of the Hubble constant, i.e. $$t_H \equiv {1 \over H_0} = {1 \over 67.8\textrm{(km/s)/Mpc}} = 4.55\cdot 10^{17}\textrm{s}$$ = 14.4 billion years. The Hubble time is the age it would have had if the expansion had been linear.

The value of the Hubble parameter changes over time, either increasing or decreasing depending on the value of the so-called deceleration parameter $$q$$, which is defined by


 * $$q = -\left(1+\frac{\dot H}{H^2}\right).$$

In a universe with a deceleration parameter equal to zero, it follows that H = 1/t, where t is the time since the Big Bang.

The age of the universe is thought to be 13.8 billion years.

1/13.8 billion years =

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Scale height
From Scale height

Scale height is the increase in altitude for which the atmospheric pressure decreases by a factor of e. The scale height remains constant for a particular temperature. It can be calculated by


 * $$H = \frac{kT}{Mg}$$


 * where:


 * k = = 1.38 x 10&minus;23 J·K&minus;1
 * T = mean atmospheric temperature in kelvins = 250 K for Earth
 * M = mean mass of a molecule (units kg)
 * g = acceleration due to gravity on planetary surface (m/s²)

Approximate atmospheric scale heights for selected Solar System bodies follow.
 * Venus: 15.9 km
 * Earth: 8.5 km
 * Mars: 11.1 km
 * Jupiter: 27 km
 * Saturn: 59.5 km
 * Titan: 21 km
 * Uranus: 27.7 km
 * Neptune: 19.1–20.3 km
 * Pluto: ~60 km

If all of Earths atmosphere were at 1 bar then the atmosphere would be 8.5 km thick.

See also:
 * precipitable water
 * Madden-Julian Oscillation monitoring
 * National Weather Service

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