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#E = mc^2#


This is calculated using the famous equation of Einstein,

#E = m c^2#

In Fusion reaction like the ones taking place in the core of a Star, there is enough pressure to fuse hydrogen nuclei to form one helium nucleus.

So, 4 hydrogen nuclei are fused together to form one Helium nucleus. But, where does the energy come from that keeps the Sun from collapsing?.

When 4 Hydrogen nuclei are merged together they show a certain discrepancy in the mass when a Helium atom is formed, i.e the mass of 4 Hydrogen atoms before Fusion is less than the mass of the Helium atom after the reaction this mass defect is converted into energy by #E = mc^2#.

#"mass of a hydrogen atom " = " 1.00794 u"#

#"mass of one helium atom " = " 4.002602 u"#

Where #u = 1.6605 xx 10^(-27)"kg"#.

The mass defect, #Deltam#, will be

#Delta m = 4 xx m_H - m_(He)#

#= 4 xx "1.00794 u" - "4.002602 u"#

# = "0.029158 u "# or #" "4.8416859 * 10^(-29)"kg"#

In the equation #E = mc^2#, #c# is the speed of light in a vacuum, approximately equal to #3 * 10^8"m s"^(-1)#.

This means that you have

#E = 4.8416859 * 10^(-29)"kg" * (3 * 10^8)^2"m"^2"s"^(-2)#

#E = 4.35751731 * 10^(-12)" J per reaction"#


It takes thousands of years for a photo to get from the sun's core to escape from the surface.


When a photon is created in the core of the sun it will collide with many protons and electrons in its path to the surface.

The photon effectively has to find a path to the surface by randomly being absorbed and re-emitted by particles on the way. This is an example of the drunkard's walk problem.

The number of steps, between particles on the way #n# can be calculated by the formula #n=R^2/d^2#. Where #R=6.96*10^8#m is the distance to travel - radius of the Sun and #d# is the free mean path - the distance between particles. The value of #d# is estimated to be about 1 centimetre.

Putting #d=0.01# into the formula gives #n=4.8*10^(21)# steps.

The time taken for each step is #t=d/c# where #c=2.99*10^8#m/s is the speed of light. This gives #t=3.3*10^(-11)# seconds.
This gives a total time #T=nt=1.58*10^(11)# seconds. Given that a year is #3.1*10^7# seconds. This gives #T=5109# years.

This calculation assumes that the free mean is actually about a centimetre, it could be much smaller, which would make the time longer. In any case it takes thousands of years to make the journey.


#M_(mars)=6.479xx10^23# kg (I used Google calculator so you may get a slightly different number if you approximate #pi#).


The equation that relates the mass of Mars with the orbital information provided about Phobos is the following:


T is the period. Let's express that in seconds:

#7*3600+39*60=25200+2340=27540# seconds

Let's plug in and solve for M:


#M=6.479xx10^23# kg (I used Google calculator so you may get a slightly different number if you approximate #pi#).

A quick check of "the internet" tells me that the actual number for the mass of Mars is #6.39xx10^23# kg, so we're pretty close.

Thanks to the following link for the equation: http://www.physicsclassroom.com/class/circles/Lesson-4/Mathematics-of-Satellite-Motion

It all started with Beta decay. An electron is emitted in the decay of a nucleus. There are no electrons in the nucleus, lepton number is not conserved unless another lepton is formed. Since electric charge is conserved, the particle needed to be neutral, it was called a neutrino. Associated with each lepton, we have one neutrino. One for the electron, one for the muon, and one for the Tau.

Neutrinos interact via gravity, weak interaction, but not electromagnetic interaction. Because they are neutral, their mean free path is larger than that of charged particles. Neutrinos are produced during Proton-Proton reactions. These reaction are fusion reactions, by which nuclei of increasingly large atomic number are produced. Starting with Hydrogen and moving up the Mendeleev ladder.

Fusion is the process that heats up the sun. It also prevents it from collapsing under its own weight. In the sun, conversion Hydrogen into Deuterium takes place through the chain PP I. P P III generates the most energetic neutrinos. It contains the reaction

#Be_5# gives #Be_4 + e + nu_e#.

These reactions were the ones for which John Bacall designed his experiment. When he observed about 1/3 of the expected number of neutrinos, the standard model had to be revised and a tiny mass was shown to give rise to neutrino oscillations. Electron neutrinos can turn into muon neutrinos. In general, all species of neutrinos can turn into each other. With a detector targeting only one species, the neutrino count was off by 2/3.
With this problem now solved, we can and we need to use neutrinos to probe the sun's interior.

Why we need neutrinos to know what is going on inside the sun is because the sun's core has 150 times the density of water. Assuming the path of a photon to be a random walk, the average size of each step was estimated to be 9/100 centimeters. Considering that the photon has almost 700000 kms to travel to reach the surface and that it does so through a diffusion process, one finds that it takes 1.7 10^5 years for photons to emerge out of the sun,

Photons are a poor way of probing the sun's interior. Neutrinos travel close to the speed of light and reach detectors much faster.


Spectroscopy is measuring light to find out what something is made of on an atomic scale.


Spectroscopy is measuring the light emitted by stars to tell what sort of atoms the star is making, or what a substance is made up of.

Each atom is a nucleus surrounded by a load of electrons. The electrons have specific energy levels, like stairs, and can move between the levels by absorbing or emitting photons of certain frequencies.

When an electron absorbs a photon, it gains energy and moves up the stairs. When it emits a photon, it loses energy and moves down the stairs again. Luckily for us, photons are also light, so we can see it when electrons lose energy in atoms.

Also, each atom has very specific energy levels that we can measure precisely here on earth, so we know from the frequency of light emitted what energy is lost and so what atom it is in the star, or any other substance.


[The above image is an example of spectroscopy.]

Energy of a photon is given by:

#E=hf# or #E=hnu#

where #h# is called Planck's constant and is approximately #6.626*10^-34#, and #f# is the frequency of the light emitted in Hertz (oscillations per second).

We may not be able to directly measure the frequency, but we know that the speed of light is constant and that it is given by:

#c = flambda#

where #c# is the speed of light, #lambda# is wavelength, and #f# is, again, frequency. The speed of light is about #3*10^8m/s# approx. The real value is taken as #2.99792458*10^8m/s#. These values are speed of light in a vacuum.


#f=c/lambda#, so


This is great because the colour of light is how humans see wavelength, so just from the colour of something we can determine the energy of light emitted, and the energy of light emitted is specific to each atom.

We can also go the other way, rather than measuring the light that something emits, measure the light that it absorbs, which, although it is exactly the opposite, gives us exactly the same result. We can do this by shining white light on a gas in the lab and seeing what light comes through the other side. The light that doesn't come through is absorbed, and from this we can tell precisely what the gas is.


Yes, black holes are formed when large stars collapse.


When a large star's core runs of of fusible material, the core is mainly iron. Fusion reactions involving iron require more energy than is produced so fusion reactions in the core stop.

Once fusion reactions stop the core starts to collapse under gravity. If the core is more than a few solar masses, nothing can stop the collapse due to gravity.

The Schwarzschild radius #r_s# defines the size of a black hole of a given mass.

#r_s = (2GM)/c^2#

Where #G# is the gravitational constant, #M# is the mass of the object and #c# is the speed of light.

Once the stellar core collapses to its Schwarzschild radius it become a black hole.

There are considered to be three types of black hole. Primordial black holes were created soon after the big bang. Stellar black holes are formed from the collapse of large stars. Supermassive black holes reside at the centres of most galaxies.

So, the relationship between stars and black holes is that stellar black holes are what remains when a large star's core collapses.

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