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Stars - Active and Dead incl. Black-Holes

Note: apart from the theories proposed by Isaac Newton and Henri Poincaré, all the formulas and proposals provided below are exclusively from CalQlata.

A star is a giant ball of >99% hydrogen gas with sufficient core pressure to generate atomic-nuclear fusion. Whilst its gravitational energy, which is governed by its mass, remains greater than its fusion energy a star will not burst and when all of its fusion energy is spent, gravitational energy will take over completely and the star will have collapsed into a ball of iron (and lighter elements).

A star's mass, and therefore its gravitational energy, does not [significantly] alter throughout its life, even after collapse. Gravitational attraction at its surface, however, decreases as it grows in size and increases as it shrinks.
i.e. the greater a star's density, the greater its gravitational attraction at its surface
and the greater a star's mass, the greater its gravitational energy

An Active Star

Throughout its life there are two opposing energies keeping an active star from; a) collapsing (Fusion) and b) bursting (gravitational).

At the instant of its birth, a star comprises sufficient gas to generate the pressure at its core required to initiate and maintain the nuclear fusion energy that compresses/combines the hydrogen atoms into larger elements, the largest of which that can be created in this way is iron. Towards the end of its life, the reduced pressure generated by the remaining hydrogen gas at the surface of its iron core can only create the lighter elements (lithium, beryllium, boron, carbon, etc.). The heavier elements such as cobalt, nickel, gold, etc. are created in high-energy explosions such as supernovae and the 'Big-Bang'.

During its life a star continually emits (radiates) the energy generated by its nuclear fusion in the form of light, heat and sub-atomic particles (electrons, positrons, photons, alpha, beta, gamma and X-ray particles, etc.).

Whilst a little mass is lost through the emission of sub-atomic particles as it ages, a star will retain the same mass throughout its life, even after collapse, and as long as its gravitational energy remains greater than its fusion energy, the star will not burst.

Apart from a little core heat, the only energy a star will possess at the end of its life will be gravitational.

A Dead Star

A dead star is one that has 'collapsed', i.e. virtually all the hydrogen atoms will have been converted into iron and/or lighter elements (at its outer surface). After collapse, a dead star may gradually increase its gravitational energy as it collects (attracts) matter from its surroundings. If it collects enough additional mass, its gravitational energy may eventually be sufficient to prevent light particles (photons) from escaping its surface, at which time it will have become a 'black-hole'.

Black Holes

A black-hole is simply a dead (collapsed) star with sufficient gravitational energy to prevent photons from escaping its surface, thereby making it impossible to observe visually/directly.

The very largest stars contain sufficient mass at their birth to ensure that at the instance of their collapse they will already possess enough gravitational energy to trap photons.

For example; VY Canis Majoris has a radius (9.879082E+11 m) almost 20 times larger than that of the minimum iron star that could be expected to collapse to an instant black-hole (5.3218365E+10 m) according to the Calculations below⁽¹⁾

In most cases, however, black-holes are constructed, i.e. they begin as large dead stars but without sufficient gravitational energy to hold onto photons. As they collect additional mass from their surroundings, however, they eventually generate the gravitational energy required to do so.

Formulas

Whilst CalQlata has adopted 'U' to symbolise energy, with due respect to Henri Poincaré we have adopted his symbol 'E' for this page

Henri Poincaré showed us that the limiting velocity (i.e. that of light in a vacuum {c}) for any mass is defined by his equation;
c = √(E/m)
which is more commonly known today as E = m.c²

It was probably Kristian Huygens who told us that velocity and acceleration are related thus;
v² = 2.a.R

If we assume that there is also a limiting gravitational acceleration for any given mass (star or planet), it is probably equivalent to the limiting energy level defined by Henri Poincaré for velocity;
i.e. for the same mass; m.c² = m.2.a.R
where 'a' is the gravitational acceleration (g) at its outer surface (at radius 'R')⁽²⁾

Therefore, if we know the limiting energy for any specified mass according to the relationship:
E = m.c²
we should be able to apply this limiting energy to the gravitational energy equation to find the limiting mass from:
c² = 2.g/R → g = c² / 2.R
E = m.g.R → m.R.c² / 2.R → ½.m.c² (kinetic energy at light speed)

i.e. if 2.g.R ≥ c² for a given star, a photon will have insufficient energy to escape its surface

Isaac Newton defined the relationship between gravitational acceleration and mass with his gravitational constant 'G' (see Gravitational Constant below)
where: F = G.m₁.m₂ / R² → m.g = G.m₁.m₂ / R² → g = G.m / R²
The units for G are N.kg²/m² only because of the way the formula is constructed, but it breaks down thus: ᶜ/s².m³/kg

The following Table contains the principle formulas used in the Calculations (below) to determine the properties of collapsed and active stars and black-holes.

FormulaVariables & ConstantsComments
V = 4/3 π.R³ V = volume of a sphere
R = radius of a sphere
ρ = m/V V = volume of a sphere
ρ = density
m = mass
R = ³√[3.m / 4π.ρ] R = radius of a sphere
ρ = density
m = mass
Calculating a star's radius based upon its known density and mass
F = G.m₁.m₂ / R² F = centripetal force between a force centre and an orbiting body
G = Newton's gravitational constant
m₁ = mass of the force centre
m₂ = mass of the orbiting body
R = the straight-line distance between the centres of the force centre and the orbiting body
The attracting (gravitational) force between a force centre (a sun) and an orbiting body (a planet)
F = m.a = m.g F = force
m = mass
a = acceleration
g = gravitational acceleration
As described by Isaac Newton
g = G.m / R² g = gravitational acceleration at 'R'
G = Newton's gravitational constant
m = mass of force centre (star)
R = radius of the force centre (star)
Used to determine the gravitational acceleration of a mass at radial distance 'R' from its centre
p = ρ.R p = core mass-pressure of the hydrogen gas
ρ = average planet density
R = planet radius
Mass-pressure at planet core
p = g/G p = mass-pressure
g = gravitational acceleration
G = Newton's gravitational constant
special case: unknown
m = p.R² m = mass of body
p = mass-pressure at interface of iron core and hydrogen gas
R = radius of star
Used to determine the mass of a star of known radius from its core pressure⁽³⁾
p₁/R₁ = p₂/R₂ R₁ = radius of star₁
R₂ = radius of star₂
p₁ = mass-pressure at core of R₁
p₂ = mass-pressure at core of R₂
A relationship between any two masses of the same average density⁽³⁾
R₁/R₂ = g₁/g₂ R₁ = radius of star₁
R₂ = radius of star₂
g₁ = gravitational acceleration₁
g₂ = gravitational acceleration₂
A relationship between any two masses of the same average density⁽³⁾
R₁/ρ₁ = R₂/ρ₂ R₁ = radius of star₁
R₂ = radius of star₂
ρ₁ = density of star₁
ρ₂ = density of star₂
A relationship between any two masses of the same material (e.g. hydrogen)⁽³⁾
R₂ =
(3.m₂.R₁ / 4.π.ρ₁)⁰˙²⁵
R₁ = radius of star₁
R₂ = radius of star₂
ρ₁ = density of star₁
m₂ = mass of star₂
Used to find the radius of a star of known mass of the same material (e.g. hydrogen)⁽³⁾
E = m.c² E = relativistic momentum (energy) in 'm' at 'c'
m = mass of star
c = speed of light in a vacuum
Limiting momentum (energy) in a body of mass (m) when moving at the speed of light (c)
E = m.g.R E = gravitational energy in 'm'
m = mass of body
R = radius of 'm'
g = gravitational acceleration at 'R'
Gravitational energy in a body of mass (m) at radius 'R'
E = G.m²/R G = Newton's gravitational constant
E = gravitational energy of star
m = mass of star
R = radius of star
Gravitational energy
v² = 2.a.R c = speed of light
g = gravitational acceleration at 'R'
R = radius of body (star)
Relationship between velocity and acceleration
c² = 2.g.R c = speed of light
g = gravitational acceleration at 'R'
R = radius of body (star)
Used to determine the equivalent limiting radius 'R' below which light cannot escape a given mass
v² = 2.G.m/R G = Newton's gravitational constant
m = mass of star
R = radius of star
v = velocity
Newton's escape velocity for a planetary body
m = ¾(E/G)⁰˙⁶ / ρ⁰˙² m = mass of body (star)
E = gravitational energy
G = Newton's gravitational constant
ρ = density of body
Alternative formula for the mass of a body
m =
(3.c⁶ / 32.π.ρ.G³)⁰˙⁵
m = mass of body (star)
c = speed of light
G = Newton's gravitational constant
ρ = density of body
Minimum (limiting) mass of a star that can be expected to collapse directly to form a black hole

Calculations {© 15/02/17}

The following comprises a series of calculations that predict the expected ultimate destiny of three stars according to the above formulas and data developed in CalQlata's 'Laws of Motion' web page.

The Earth's Sun

Known Properties (active):
mass (m) = 1.98850E+30 kg
radius (R) = 6.95710E+08 m
Calculated Properties (active):
Volume (V) = 4/3 . π.R³ = 1.4105E+27 m³
average density (ρ) = m/V = 1409.7829 kg/m³
gravitational acceleration (g) = G.m/R² = 274.08066 m/s²
gravitational energy (E) = m.g.R = 3.7917E+41 J
Known Properties (dead) assuming the density of iron:
average density (ρ) = 7870 kg/m³ ⁽⁴⁾
mass (m) ≈1.98850E+30 kg
Calculated Properties (dead):
volume (V) ≈ 4/3 . π.R³ ≈ 2.52668E+26 m³
radius (R) ≈ ³√[3.m / 4π.ρ] ≈ 3.921818E+08 m
gravitational acceleration (g) ≈ G.m/R² ≈ 862.502 m/s²
2.g.R ⁽⁵⁾ ≈ 6.7652E+11 m²/s² (c² = 8.987552E+16)

Given that 2.g.R is so much less than c², it is unlikely that the earth’s sun will become a black-hole immediately after collapse

VY Canis Majoris

Known Properties (active):
Radius (R) = 9.87908E+11 m
mass (m) = 5.967E+31 kg
Calculated Properties (active):
volume (V) = 4/3 . π.R³ = 4.03867E+36 m³
average density (ρ) = m/V = 1.477467E-05 kg/m³
gravitational acceleration (g) = G.m/R² = 4.07880E-03 m/s²
gravitational energy (E) = m.g.R = 2.40439E+41 J
Known Properties (dead): (assuming the density of iron)
mass (m) = 5.967E+31 kg
average density (ρ) = 7870 kg/m³ ⁽⁴⁾
Calculated Properties (dead):
volume (V) ≈ 4/3 . π.R³ ≈ 7.58196E+27 m³
radius (R) ≈ ³√[3.m / 4π.ρ] ≈ 1.21870E+09 m
gravitational acceleration (g) ≈ G.m/R² ≈ 2680 m/s²
2.g.R ⁽⁵⁾ ≈ 6.53278E+12 m²/s² (c² = 8.9875518E+16)

Whilst VY Canis Majoris is very large, it has a mass little larger than our sun and therefore is unlikely to collapse into an instant black hole

Minimum Potential Black-Hole Star

Known Properties (dead):
c² = 2.g.R = 8.9875518E+16 m²/s²
average density (ρ) = 7870 kg/m³ (assuming the density of iron)
Calculated Properties (dead):
mass (m) = √(3.c⁶ / 32π.ρ.G³) = 9.6287854E+37 kg
volume (V) = m/ρ = 1.22347971E+34 m³
radius (R) = ³√(3.V / 4π) = 1.42945138E+11 m ⁽⁵⁾
gravitational acceleration (g) = G.m/R² = 314370.67118 m/s²
Known Properties (active):
mass (m) = 9.6287854E+37 kg
Calculated Properties (active):
radius (R) = ⁴√(R₁⁴ . m₁/m₂) = 5.80350E+10 m
volume (V) = 4/3π.R³ = 8.18762E+32 m³
average density (ρ) = m/V = 117601 kg/m³ ⁽⁴⁾
gravitational acceleration (g) = G.m/R² = 1907222 m/s²
gravitational energy (E) = m.g.R = 1.065768E+55 J
2.g.R = 8.9875518E+16 m²/s² (c² = 8.9875518E+16)

Th black-hole on collapse (see Neutron Star below)

The above results are summarised in the following Table (input data):

UnitsThe Earth’s SunVY Canis MajorisMinimum Potential Black-Hole Star
Active
masskg1.98850E+305.967E+319.6287854E+37
radiusm6.95710E+089.87908E+115.80350E+10
volume1.41050E+274.03867E+368.18762E+32
average densitykg/m³1.477467E-052001892117601
gravitational accelerationm/s²274.08066310.00407881907222
gravitational energyJ3.79168E+41 2.40439E+411.065768E+55
Dead
masskg1.98850E+305.967E+319.6287854E+37
radiusm3.921817841E+081.21870E+091.42945138E+11
volume2.52668E+267.58196E+271.22347971E+34
average density ⁽⁴⁾kg/m³787078707870
gravitational accelerationm/s²862.50188382680314370.67118
2.g.R ⁽⁵⁾m²/s²6.7652E+116.53278E+128.9875518E+16
m²/s²8.9875518E+168.9875518E+168.9875518E+16

Properties of a Black-Hole {© 15/02/17}

From our formulas above:
v² = 2.a.R
Special [limiting] case to trap photons: c² = 2.g.R → g = c² / 2.R
Gravitational (potential) energy: E = m.g.R → E = m.R.c² / 2.R
Limiting gravitational energy that will trap photons: E = m.c² / 2
i.e.: Kinetic energy of the mass at light speed: E = ½.m.c²
If we apply these calculations to the above (iron) limiting black-hole:
E = m.g.R = 9.6287854E+37 x 314370.67118 x 142945138000 = 4.32696038E+54 J
KE = ½.m.c² = 9.6287854E+37 x 2997924582² ÷ 2 = 4.32696038E+54 J
CalQlata has therefore shown that the gravitational energy in a planet that will trap photons is equal to its kinetic energy at light speed

Assuming that the limiting mass for a star that can be expected to collapse to form an instant black hole (trap high-energy photons) is governed by the relationship 2.g.R = c², its mass can be calculated thus:
m = √(3.c⁶ / 32.π.ρ.G³) = 9.6287854E+37
and its radius; R = 5.3218365E+10 m⁽¹⁾

However, it is claimed that all black holes are neutron stars, i.e. they collapse to the density of a neutron. If this is so its properties may be calculated as follows:

Neutron Star

The properties of a neutron:
m = 1.6749272928E-27 kg
R = 1.11328405737E-15 m
V = 5.77971706488E-45 m³
ρ = 2.8979399407E+17 kg/m³ ⁽⁶⁾

The radius of this star is related to its mass thus:
R³ = 3.m / 4.π.ρ
R = ³√(3.m / [4 x π x 2.8979399407E+17])
R = ³√m . ³√(3 / [4 x π x 2.8979399407E+17])
R = 9.37433933214E-07 x ³√m

According to Newton: v² = 2.G.m/R
Limiting condition: c² = 2.G.m/R
R = 2.G.m / c²
2.G.m / c² = 9.37433933214E-07 x ³√m
m/³√m = ¹˙⁵√m = 9.37433933214E-07 x c²/2.G
m = (9.37433933214E-07 x c² ÷ 2.G)¹˙⁵ = 1.58677073182E+31 kg
{check: m = √(3.c⁶ / 32.π.ρ.G³) = 1.58677073182E+31 kg}
which represents the minimum mass that could create a neutron star on collapse
Its radius: R = 2.G.m / c² = 23556.57050854 m
The gravitational acceleration at its surface is:
g = c² / 2.R = 1.90765285297E+12 m/s²
{check: g = G.m/R² = 1.90765285297E+12}
KE = ½.m.c² = 7.13059206345E+47 J
Eg = m.g.R = 7.13059206345E+47 J
confirming CalQlata's claim that the gravitational energy of a black-hole is equal to its kinetic energy at light speed

The problem is, however, whether or not iron can collapse to the density of a neutron with the above mass(es), because in order to do so, there can be no protons or electrons (or photons) within the body.
Moreover, its ultimate density is governed by the compressibility or iron.

Constants {© 02/02/17}

For all stars (and planets): 2.π.R.G.ρ / g = 1.5 {radians}
Proof: 2.π.R.G.ρ / 1.5 g = 1
4/3.π.R.G.(m/(4/3.π.R²)) / g
G.m / g.R² = 1
g = G.m / R²

We can check the above calculations as follows:

The Earth’s Sun
active: 2.π.G.ρ.R / g = 2.π.G x 1409.782932 x 6.95710E+08 ÷ 274.0806631 = 1.5 {radians}
dead: 2.π.G.ρ.R / g = 2.π.G x 7870 x 3.921817841E+08 ÷ 862.5018838 = 1.5 {radians}

VY Canis Majoris
active: 2.π.G.ρ.R / g = 2.π.G x 1.477467E-05 x 9.87908E+11 ÷ 4.07880E-03 = 1.5 {radians}
dead: 2.π.G.ρ.R / g = 2.π.G x 7870 x 1.21870E+09 ÷ 2680 = 1.5 {radians}

Minimum Potential Black-Hole Star
active: 2.π.G.ρ.R / g = 2.π.G x 117601 x 5.80350E+10 ÷ 1907222 = 1.5 {radians}
dead: 2.π.G.ρ.R / g = 2.π.G x 7870 x 1.42945138E+11 ÷ 314370.67118 = 1.5 {radians}

This constant also applies to planets (e.g. the earth):
2.π.G.ρ.R / g = 2.π.G x 5508.25830105342 x 6371000.685 ÷ 9.80663139027614 = 1.5 {radians}

A useful constant (K), based upon Newton's theories, for all matter is:
K = g ÷ ρ.R = 2.79446002926891E-10 m³/s²/kg
and
G = K ÷ 4/3π = 6.67135365544089E-11 m³/s²/kg

Dark Matter

Given that the outer part of the Milkyway galaxy is where most of this [dark] matter appears to reside, and that this is where most of the younger short-lived stars appear to proliferate, CalQlata suspects that most stars that have ever existed (in this region) are already dead/collapsed, making them difficult to see. Add this to the huge amounts of intergalactic elemental dust in this region and this is perhaps what we should be looking for our dark matter, as opposed to neutrinos.

Gravitational Constant {© 15/02/17}

Is 'G' equal to the reciprocal of [exactly] 50 times the speed of light ?
Under Investigation

Notes

  1. Compared with a radius of 3.26E+12 m for what is claimed to be a super-massive black hole at the centre of Messier 87 (spiral galaxy))
  2. At radii below (inside) the surface of a star (or planet), gravitational attraction is less than at its surface because there is less mass generating it. Gravitational attraction above (outside) the surface of a star is less than at its surface because its mass does not increase with radius.
  3. These relationships are representative at the birth of each star and/or stars of similar age, but are approximate for stars of different ages due to the variability of core size and material
  4. The density of a collapsed star will define its ultimate size, not its gravitational energy. Therefore, whilst the average density of the collapsed star may not be as defined above, any such variation will not affect its ability to trap photons
    The average density of iron (7870 kg/m³) for a dead star is estimated based upon this value appearing approximately half-way through its radial distance with lighter elements above this point and denser iron below it.
  5. Schwartzschild formula R = 2.G.m/c² predicts a radius of:
    R = 2 x 6.671282E-11 x 9.6287854E+37 / 299792458² = 1.42945138E+11m
  6. Whilst this calculation is based upon a neutron star comrising compacted neutrons throughout its radial distance, this is highly unlikely. It is most probable that whilst the core would generate this density, the average density should be slightly above half this value. Resulting in a Neutron star having the same mass but a greater radius.

Further Reading

You will find further reading on this subject in Earth's Properties & Laws of Motion

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