Astrophysics

Image shows an approximation of the shape (Equipotentials) 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

=

G
M

r

{\displaystyle v\_o = \sqrt{\frac{GM}{r}}}

Orbital period:

# T
2
π

r
3

G
M

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

Orbital angular momentum:

m
v
r

=

m

(

G
M

r

)

r

=

m

G
M
r

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

Rotational angular momentum of solid sphere:

# ω

2
5

m

r
2

v
r

=

2
5

m
v
r

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

where:

* *r* is the orbit's semi-major axisw
* *G* is the gravitational constantw,
* *M* is the mass of the more massive body.
* *m* is the mass of the less massive body.

Moons orbital angular momentum is 28.73 \* 10^33 Js

Earths rotational angular momentum is 7.079 \* 10^33 Js

Correcting for Earths uneven mass distribution: 3.85 + 0.44 + 0.25 + 0.073 = 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.

Roche limit 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:

G
⋅
1
 Earth mass
⋅

(

4
 hours

2
π

)

2

3

{\displaystyle \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
π

(
18
,
381
 km

)
3

G
∗
1
 Earth mass

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

 = 7.554 hours
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)
1.42\*10^24 Js/yr (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 = +2.667 ms/century
620 million yrs ago the Moon had 1.42\*10^24 \* 620\*10^6 = 0.88\*10^33 Js 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

G
⋅
1
 Solar mass

(

336.134453
 days

2
π

)

2

3

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

 = 0.9461 au
Therefore Earth must be receding from the sun at 13 m/yr

Thats 0.395 au / 4.543 billion years
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### Planets

From Wikipedia: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 (Electron degeneracy pressure) is inversely proportional to the sixth power of its radius:

# P

1

r
6

{\displaystyle 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:

plot (-(1.83\*r3/x -1.83\*r2)+(x2/2-r2/2)=0.5) for r=0 to 1 and x=0 to 1
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 3% smaller then it would be otherwise.

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### Dark matter

From Wikipedia: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 Wikipedia: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:

θ
E

=

4
G
M

c
2

d
S

−

d
L

d
S

d
L

{\displaystyle \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 0.00024 arcseconds (angle subtended by 1 au at 4000 parsecs). 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: Stellar evolution, Helium flash, Schönberg–Chandrasekhar limit, Coronal\_heating\_problemw

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
π

R
2

σ

T
4

{\displaystyle L = 4 \pi R^2 \sigma T^4}

where σ is the Stefan–Boltzmann constant:

# σ

2

π
5

k

B

4

15

h
3

c
2

=

π
2

k

B

4

60

ℏ
3

c
2

=
5.670373
(
21
)

×

10

−
8

J

m

−
2

s

−
1

K

−
4

{\displaystyle \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 3.828×10^26 Watts

Thats 6,297,000 watts/m2
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 distribution function has temperatures greater than around 260 keV, or 3.0 \* 109 K. 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 plasma frequency. 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.

# ω

n

e

q
e

2

m
e

ε
0

{\displaystyle \omega = \sqrt{\frac{n\_\mathrm{e} q\_e^{2}}{m\_e \varepsilon\_0}}}

where
ne = number of electrons / volume.
See also: Bremsstrahlung#Thermal\_bremsstrahlung

From Wikipedia: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 Wikipedia: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 Wikipedia: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 Wikipedia: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 Kappa mechanism.

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 Wikipedia: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 first phase change of hydrogen in the universe was recombination due to the cooling of the universe to the point where electrons and protons form neutral hydrogen. The universe was opaque before the recombination, due to the scattering of photons (of all wavelengths) off free electrons, but it became increasingly transparent as more electrons and protons combined to form neutral hydrogen atoms. The Dark Ages of the universe start at that point, because there were no light sources.
The second phase change occurred once objects started to condense in the early universe that were energetic enough to re-ionize neutral hydrogen. As these objects formed and radiated energy, the universe reverted to once again being an ionized plasma. (See Warm–hot intergalactic medium). At this time, however, matter had been diffused by the expansion of the universe, and the scattering interactions of photons and electrons were much less frequent than before electron-proton recombination. Thus, a universe full of low density ionized hydrogen will remain transparent, as is the case today.

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:

plot (-(15\*r^3/x -15\*r^2)+(x^2/2-r^2/2)=0.5) for r=0 to 0.5 and x=0 to 1
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

G
⋅
0.6
 solar mass

(
8900
 km

)
2

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

 = 103,000 g's
Its density is 404,000 g/cm3 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

G
⋅
1.13
 solar mass

(
4500
 km

)
2

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

 = 755,000 g's
Its density is 5.887 \* 106 g/cm3 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: 3.24 \* 106 g's
Density: (sqrt(2)9)3 \* 3844.75 g/cm3 = 4.454 \* 107 g/cm3
Core pressure: 1.8 \* 1019 bars
The core of a white dwarf with a mass greater than the Chandrasekhar limit (1.44 solar masses) will undergo gravitational collapse and become a neutron star.

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### Neutron stars

See also: Gravitoelectromagnetism
Neutron star radiation peaks in the x-rays which corresponds to a temperature of about 3 million kelvins. For a 10 km radius neutron star that gives a total luminosity of 15 solar luminousities

Assuming a solid honeycomb array of neutron pairs with radius 1 fm, a sheet of neutronium (if such a thing existed) would have a density of 1.2893598 g/mm2.

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 12.5544 km

Its gravitational binding energy would be 0.282 solar masses of energy

The Tolman–Oppenheimer–Volkoff limit (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 equation of statew 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:

G
⋅
2
 solar mass

(
12.5544
 km

)
2

{\displaystyle \frac{G \cdot 2 \text{ solar mass}}{(12.5544 \text{ km})^2}}

 = 1.717 1011 g's
The pressure in its core would be

3

8
π

G
⋅
M
a
s

s

a
c
t
i
v
e

⋅
M
a
s

s

p
a
s
s
i
v
e

r
4

=

{\displaystyle \frac{3}{8 \pi} \frac{G \cdot Mass\_{active} \cdot Mass\_{passive}}{r^4} =}

5.072 \* 1028 bar =
5.071 \* 1028 bar

Its moment of inertia is: 0.4\*2 solar masses\*(12.5544 km)^2 = 2.507×10^38 kg m^2

From Wikipedia:Glitch (astronomy)

A glitch (See Global\_resurfacing\_event) 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

r
a
d
i
u
# s
1

/

d
e
n
s
i
t
y

3

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

If angular momentum is constant then

f
r
e
q
u
e
n
c
# y
v

/

# r
1

/

r
2

=
1

/

(
1

/

d
e
n
s
i
t
y

3

)
2

=
d
e
n
s
i
t

y

2
3

{\displaystyle 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 = 1.197×10^33 kg m^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 Wikipedia:Supermassive black hole

A **supermassive black hole** (**SMBH** or **SBH**) is the largest type of black hole, on the order of hundreds of thousands to billions of solar masses (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 Wikipedia: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 6.66 billion years to reach full size. See TON 618

Growing at a rate of 17.6/2.8^2 solar mass per year a 240 billion solar mass Black hole would take 107 billion years to reach full size.

Masses of supermassive black holes in billions of solar masses:

1. 240? (Hypothetical based on Zipf's law)
2. 120? (Hypothetical)
3. 80? (Hypothetical)
4. 66
6. 40
7. 33
8. 30
10. 23
11. 21
12. 20
13. 19.5
14. 18
15. 17
16. 15
17. 14
18. 14
19. 13.5
20. 13
21. 12
22. 12.4
23. 11
24. 11
25. 10
26. 10
27. 9.8
28. 9.7
29. 9.1
30. 7.8
31. 7.2
32. 7.2
33. 6.9

From Wikipedia: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

L

E
d
d

=

4
π
G
M

m

p

c

σ

T

≅
1.26
×

10

31

(

M

M
⊙

)

W

=
1.26
×

10

38

(

M

M
⊙

)

e
r
g

/

s

=
3.2
×

10
4

(

M

M
⊙

)

L
⊙

{\displaystyle \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

⊙

{\displaystyle M\_{\odot} }

 is the mass of the Sun and

L
⊙

{\displaystyle L\_\odot}

 is the luminosity of the Sun.

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### Gamma-ray bursts

From Wikipedia: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 GRB 080319B 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 Wikipedia: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 (1,635,000 light years)3

There are 12 galaxies per cubic megaparsec. Thats 1 galaxy per (1,425,000 light years)3

One short grb per 3 days at average redshift of 0.5 (4.6 \* 109 light years) gives 1 grb per (1,300,000 light years)3

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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 Wikipedia:Cosmic ray

Cosmic rays 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 Wikipedia:Ultra-high-energy cosmic ray

In astroparticle physics, an **ultra-high-energy cosmic ray** (**UHECR**) is a cosmic ray particle with a kinetic energy greater than than 1×1018 eV, far beyond both the rest mass 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 Greisen–Zatsepin–Kuzmin limit (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 cosmic microwave background (CMB). However, if an EECR is not a proton, but a nucleus with

A

{\displaystyle A}

 nucleons, then the GZK limit applies to its nucleons, each of which carry only a fraction

1

/

A

{\displaystyle 1/A}

 of the total energy.

These particles are extremely rare; between 2004 and 2007, the initial runs of the Pierre Auger Observatory (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 5.46 \* 1018 particles will fall onto a star with radius 1 million kilometers every hundred million years.

From Wikipedia: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 ultra-high-energy cosmic ray particles.

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### Expansion of the universe

From Wikipedia: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 Wikipedia: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

{\displaystyle t}

 to their distance at some reference time

t
0

{\displaystyle t\_0}

. The formula for this is:

d
(
t
# )
a
(
t
)

d
0

,

{\displaystyle d(t) = a(t)d\_0,\,}

where

d
(
t
)

{\displaystyle d(t)}

 is the proper distance at epoch

t

{\displaystyle t}

,

d
0

{\displaystyle d\_0}

 is the distance at the reference time

t
0

{\displaystyle t\_0}

 and

a
(
t
)

{\displaystyle a(t)}

 is the scale factor. Thus, by definition,

a
(

t
0

# )
1

{\displaystyle a(t\_0) = 1}

.

The scale factor is dimensionless, with

t

{\displaystyle t}

 counted from the birth of the universe and

t
0

{\displaystyle t\_0}

 set to the present age of the universe:

13.799
±
0.021

G
y
r

{\displaystyle 13.799\pm0.021\,\mathrm{Gyr}}

 giving the current value of

a

{\displaystyle a}

 as

a
(

t
0

)

{\displaystyle a(t\_0)}

 or

1

{\displaystyle 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 Friedmann equations.

The Hubble parameter is defined:

H
≡

a
˙

(
t
)

a
(
t
)

{\displaystyle 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
)

{\displaystyle d(t) = d\_0 a(t)}

 one can see that

d
˙

(
t
# )

d
0

a
˙

(
t
)

{\displaystyle \dot{d}(t) = d\_0 \dot{a}(t)}

, and also that

d
0

=

d
(
t
)

a
(
t
)

{\displaystyle d\_0 = \frac{d(t)}{a(t)}}

, so combining these gives

d
˙

(
t
# )

d
(
t
)

a
˙

(
t
)

a
(
t
)

{\displaystyle \dot{d}(t) = \frac{d(t) \dot{a}(t)}{a(t)}}

, and substituting the above definition of the Hubble parameter gives

d
˙

(
t
# )
H
d
(
t
)

{\displaystyle \dot{d}(t) = H d(t)}

 which is just Hubble's law.

From Wikipedia: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

{\displaystyle v = H\_0 \, D}

where

* v

{\displaystyle v}

 is the recessional velocity, typically expressed in km/s.
* *H*0 is Hubble's constant and corresponds to the value of

H

{\displaystyle H}

 (often termed the **Hubble parameter** which is a value that is time dependent and which can be expressed in terms of the scale factor) 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

{\displaystyle 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 *r*HS , objects recede at a rate faster than the speed of light:

r

H
S

=

c

H
0

.

{\displaystyle r\_{HS} = \frac{c}{H\_0} \ . }

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

The Hubble constant

H
0

{\displaystyle 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

≡

1

H
0

=

1

67.8

(km/s)/Mpc

=
4.55
⋅

10

17

s

{\displaystyle 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

{\displaystyle q}

, which is defined by

# q
−

(

1

H
˙

H
2

)

.

{\displaystyle 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 = 70.9 (km/s)/Mpc

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### Scale height

From Wikipedia: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

k
T

M
g

{\displaystyle H = \frac{kT}{Mg}}

where:
* *k* = Boltzmann constant = 1.38 x 10−23 J·K−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.

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