- 2995 - CRAB NEBULA - Neutron Star math? There are over 1000 “Pulsars” that astronomers have identified. Probably the most studied of these is the “Crab Nebula” which was a supernova that exploded in the year 1054 and today is a rotating pulsar.
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------------------------------ 2995 - CRAB NEBULA - Neutron Star math?
- The Crab Nebula is 6,000 lightyears away in the Constellation Taurus the Bull. It is 4.4 lightyears across having expanded for 967 years. The July 4, 1054 supernova was recorded in Chinese history.
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- The total luminosity of the nebula is 5*10^31 watts, compared to the luminosity of the Sun at 3.8*10^26 watts, it is 100,000 times brighter. It contains a remnant star at its core that is a Pulsar with a period of 33.085 milliseconds. That means the star is rotating 30 revolutions per second.
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- The period of rotation is so accurate that it represents one of the most accurate clocks in the Universe In fact, our system of atomic clocks on Earth use several of these pulsars to synchronize the clocks around the globe to the correct time.
- It is amazing to me how the astronomers concluded that these Pulsars were rotating Neutron Stars. This review will take you through their calculations. The last calculation is how the 500,000,000,000,000,000,000,000,000,000,000 watts of power comes from a spin down rate of this star by 1 millisecond in 75 years. That is a very accurate clock.
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- It is one today’s the most accurate clocks. Let’s start with why the pulses every 33 milliseconds can not be coming from two binary stars in close orbit around each other.
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- We can calculate the orbits of stars using Kepler’s formula for the period of orbit squared = the radius of orbit cubed:
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------------------ p^2 = a ^3
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----------------- Where the constants needed to calculate this in metric terms are:
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----------------- p^2 = 4*pi^2* a^3 / G * (m1 + m2)
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----------------- where: G is the constant force of gravity = 6.7*10^-11 m^3/kg*sec^2
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---------------- where: (m1+m2) = mass of 2 Solar Mass stars = 4*10^30 kg
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---------------- where: p = 33.085*10^-3 seconds
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---------------- where: a = the radius of orbit separating the 2 stars.
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-------------- a^3 = p^2 * G * (m1+m2) / 4*pi^2
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-------------- a^3 7.48 * 10^15 m^3
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-------------- a = 196 kilometers.
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- Therefore the distance separating these two stars with this period of rotation is under 400 kilometers. The diameter of our Sun is 7,700 kilometers. This means that the theory that binary stars could be creating these pulses is not possible. Pulsars are not binary stars.
- Pulsars could simply be pulsating stars. There are many examples of variable stars, many are pulsating. However, normal stars pulsate with periods from hours to months. The denser the star, the faster it pulsates.
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- White Dwarf stars are much denser and have been found with pulsation periods from 100 to 1,000 seconds. Neutron stars are the densest stars known and they can have periods from 100 to 800 milliseconds.
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- The Crab Nebula’s Neutron Star pulses are 33 milliseconds. There is no known class of stars with the density required to have pulsations with this short a period of pulsation. So, a Pulsar that is a pulsating star is unlikely.
- The next theory is that the Pulsar is a rotating Neutron Star. To have a period of pulses 33 milliseconds per cycle, the angular velocity would have to be 190 radians per second, which would mean the star would have to be rotating 30 revolutions per second. That is spinning extraordinarily fast.
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- So, we have to ask how fast can a star spin before its centrifugal force tears it apart. To learn this we need to calculate the centrifugal force at the star’s surface and compare that with the force of gravity at the star’s surface. If we set these two calculations equal to each other we will have the boundary where the star can not spin any faster.
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-------------- The force of gravity = G* M*m / r^2
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-------------- The centrifugal force = m*w^2 *r
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------------- where: M is the mass of the star
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------------- where: “w” is the angular velocity at the surface
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------------- where: “r” is the radius of the star
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------------- where: “p” is the period, 2*pi/w = 3.3^10^-4 seconds
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-------------- where: “G” is the force of gravity = 6.7*10^-11 meters^3/(kg*sec^2)
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--------------- M / r^3 = 4*pi^2 / G*p^2
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--------------- note that M / r^3 is very close to the formula for mass divided by volume which is density. Let’s divide both sides of the equation by 4/3 * pi to get volume:
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----------------M /4/3*pi* r^3 = 4*pi^2 / 4/3*pi*G*p^2
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--------------- Therefore density at the boundary = 3*pi / G*p^2
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----------------- Density at the boundary = 1.28*10^14 kg / m^3
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- Therefore the density needed to keep the Neutron Star from flying apart is 10^14 kg/m^3. Neutron Stars have densities of 10^18 kg/m^3 which is 10,000 times greater density than what is needed. Therefore, Pulsars can be rotating Neutron Stars.
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- Neutron stars have radius’ from 10 to 15 kilometers. They have masses from 1.4 to 3.0 Solar Mass. Calculating mass/volume gives us densities from 0.67*10^18 to 0.43*10^18 kg/m^3.
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- The last calculation has to do with what powers the intense radiation of the Crab Nebula. A Neutron Star does not have a thermal nuclear core like ordinary stars. It is very dense and rapidly spinning.
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- Could the spin down rate of the star be the power needed? But, the spin down rate is so small, only 1 millisecond in 75 years. Can the loss of rotational energy equal the 5^31 watts of the Crab Nebula luminosity?
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------------- The slow down rate is the rate of change of the period = 10^-3 sec / 75 years.
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------------- dp /dt = 4.2 *10^-13 sec / sec.
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------------ The rate of change of angular velocity = dw/dt = - 2*pi / p^2 * dw/dt
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----------- dw/dt = - 2.42*10^-9 radians / sec^2
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---------- w = 2*pi / p = 190 radians / sec.
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- The angular velocity is slowing down 0.000,000,00242 radians per second per second.
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- The Energy of Rotation is the same as the Energy of Angular Momentum
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- The Energy of Rotation is ½ * Inertia * ( angular velocity)^2
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------------------------- E = ½ I*w^2
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- The moment of Inertia of a solid sphere = I = 2/5 M*r^2
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------------------------ E = ½ * 2/5 M*r^2 * w^2
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- We are trying to find if the total energy (luminosity) is equal to the rate of change of rotational energy.
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----------- The rate of change of rotational energy = dE /dt
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----------- The rate of change of rotational energy = - I *w* dw/dt
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------------ dE / dt = - I* dw / dt
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------------ dE / dt = - 2/5 M*r^2*w* dw / dt
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------------- Set the total luminosity, “L” equal to this change in rotational energy,”dE/dt“:
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------------- L = - 2/5 M*r^2*w* dw / dt
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------------ M*r^2 = - L* 5 / 2* w*dw/dt
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-------------M*r^2 = - 5*10^31 kg*m^2/sec^3* 5 / 2 * 190 (2.4*10^-9)/sec^3
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----------- M*r^2 = 2.7 * 10^38 kg* m^2
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- This result of mass times radius squared is exactly equal to a 1.4 Solar Mass Neutron Star with a radius of 10 kilometers.
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------------ M*r^2 = 1.4*2*10^30 kg * ( 10^4))^2 = 2.8 * 10^38 kg* m^2
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- It is amazing that a giant gyroscope, spinning Neutron Star can contain that much energy. Pulsars are spinning Neutron Stars. It is in the math. But, how did it get spinning that fast in first place?
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- The same way an ice skater spins faster when she pulls her extended arms in close to her body. The star was spinning like our Sun at say a period of 25.4 days, there are 8.7 *10^4 seconds in a day.
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- The radius was 7*10^8 meters but gravity compressed the star down to a Neutron Star that is 10^4 meters. The Conservation of Angular Momentum means, the Inertia * angular velocity^2 is constant which means, Mass*radius^2*angular velocity^2 is constant:
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---------------- I*w^2 = I * w^2 = 2/5*M*r^2 *w^2= 2/5* M*r^2*w^2
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---------------- r^2* (2*pi/p)^2 = r^2* (2*pi/p)^2
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--------------- (10^4 meters)^2 / p^2 = (7*10^8 meters)^2 / ( 2.2^10^6 seconds)^2
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--------------- period = 33 seconds
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- The period of the Crab Nebula Neutron Star is 33 seconds
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- Note: The rate of change is the same as the derivative in calculus. dE/dt is the derivative of the formula for Energy versus time. It is the rate of change of Energy with time. It is the slope of the curve of Energy plotted versus time.
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- Where did the gravitational energy come from to collapse the star? It came from the Big Bang. All matter and energy came from the Big Bang. And, it came out of Nothing.
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- My theory is that there is an anti-Universe out there somewhere that if we ever ran into we would annihilate each other and return to Nothing. Don’t tell me that studying this stuff isn’t interesting!!!
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-------------------------------------- see these Reviews for more information:
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- 2993 - PULSARS - are used to find Dark Matter? - Pulsars can be used to measure tiny changes of acceleration within the Milky Way, scanning internally for Dark Matter and Dark Energy.
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- This Review 2293 also lists 15 more reviews about Pulsars, available upon request.
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- 2960 - NEUTRON STARS - to measure expansion of the universe. How fast is the Universe expanding? Ever since the expanding Universe was first discovered nearly 100 years ago, it’s been one of the biggest questions plaguing cosmology. If you can measure how fast the Universe is expanding right now, as well as how the expansion rate is changing over time, you can figure out everything you’d want to know about the Universe as a whole.
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- 2950 - NEUTRON STARS - it dosn’t get any stranger? Astronomy has many, many strange things to try to figure out. Let’s start with “neutron stars“. The crushing gravity, intense magnetic fields, and lightning-fast rotations place “neutron stars” among the most exotic beasts in the universe. Next come the most powerful magnetic fields in the universe wrack the searing surfaces of neutron stars called “magnetars“. These magnetic monsters form one of the most eccentric branches on the neutron star-pulsar family tree.
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- This Review 2950 lists 8 more reviews abut neutron stars.
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- 720 - Crab Nebula. In 1744 a 14 year old boy in France was looking through his telescope searching for comets. His name was Charles Messier (1730 - 1817). His frustration was that he kept finding fuzzy patches of light that looked like comets but watching them night after night he would discover that they did not move with respect to background stars. They were not comets.
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January 22, 2021 CRAB NEBULA - Neutron Star math? 862 2995
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--- to: ------ jamesdetrick@comcast.net ------ “Jim Detrick” -----------
--------------------- --- Saturday, January 23, 2021 ----------------------------------------
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