Escape Velocity and Black Holes

Gravitational Potential Well

The changing Gravitational Potential around a Massive object can be modelled as a three dimensional Potential Well. The diagram below shows the Potential Well for a Massive object (such as a planet or star) :-

Note - The above diagram shows a vertical slice through a three dimensional funnel shape.

The above diagram allows an understanding of why satellites remain in orbit around a Massive object. The Potential Well can be thought of as a three dimensional funnel, with the satellite moving in a circular path at a set height. As the Gravitational Field is a conservative field, the satellite experiences no loss of energy ( due to friction ) . This means that the satellite will continue to move in a stable circular orbit indefinitely. 

The video below shows an example of a simulated non-conservative Field :-

In the above video, the coin follows a decaying orbit around the centre, as Friction causes the coin to lose Kinetic Energy. This continues until the coin falls into the central hole. This is a good approximation of what occurs when satellites re-enter the Earth's atmosphere, due to atmospheric drag. 

The diagram below shows the relative depth of the Gravitational Potential Wells of objects within our Solar System :-

Examinable Derivation - Escape Velocity

In order for an object to escape from a Gravitational Potential Well, it must be given sufficient Energy to move from its starting radius to infinity. This minimum Energy required can be calculated by the following derivation.

The Potential Energy of an object in a Gravitational Field is given by :-

As the Gravitational Potential Energy is defined as equal to 0 J at infinity, therefore to escape the Gravitational Potential Well, the object must be given Energy equal to :-

The lower limit to escape completely would be to provide the object with an initial Kinetic Energy such that at infinity, all Kinetic Energy has been converted to Gravitational Potential Energy :-

Ek  +  Ep  =  0 

Note - The above formula for escape velocity gives the required velocity in the absence of atmospheric Friction. This would cause energy to be lost due to air resistance, and as such is beyond the scope of the Advanced Higher course. 

Example 1 -

If the Mass of the Earth is equal to 6x1024 kg and has a mean radius of 6371 km, calculate the escape Velocity at the Earth's surface.

Ve  =  ( ( 2 x 6.67x10-11 x 6x1024 ) / 6371x103 )0.5 

Ve  =  11209 ms-1

Ve  =  11.2 kms-1

Composition of Planetary Atmospheres

Each planet within our Solar System has its own unique atmospheric composition (Or no Atmosphere). The differences in composition are due to the different escape Velocities for each planet. 

Atmospheric gases are bound to a planet due to Gravitational attraction. In order for a gas to be lost to space, the molecules of gas must achieve escape velocity. Due to collisions with other gas molecules, each element within the atmosphere travels with a range of Velocities.  The spread of these Velocities follows a Maxwell-Boltzmann Distribution, as can be seen in the diagram below  :-

As can be seen in the diagram above, the lighter the element, the higher the average Velocity of the molecules within the gas. Due to this, lighter gases ( such as Hydrogen or Helium ) are more likely to reach escape Velocity and escape the atmosphere than heavier gases. 

The larger the planet, the higher the escape Velocity, therefore small rocky planets such as the Earth can only retain relatively heavy molecules within their atmosphere, whilst gas giants can retain much lighter molecules. 

The diagram below shows how the differing atmospheres of the planets in the Solar System are caused by the escape Velocity of atmospheric gases :-

Note - As can be seen from the above diagram, it is not just planetary Mass which determines the composition of their atmospheres, Temperature also plays a role. This is due to an increased Temperature giving a corresponding increase to average Kinetic Energy within a gas. A higher average Kinetic Energy will again give a higher probability that molecules will reach escape velocity. This is why mercury has no atmosphere, whereas the smaller dwarf planet Pluto does. 

The video below shows how atmospheric gases can escape the Earth :-

Note - Only the first method ( thermal escape ) Is examinable as part of the Advanced Higher course.

Non Examinable Derivation - Escape Velocity and Black Holes

As can be seen from the above formula, the more Massive an object, the higher the escape velocity. This implies that there is a value of Mass and radius at which the escape Velocity would be equal to the speed of light. If the escape Velocity is equal to the speed of light, then light Itself would not be able to escape the object, and the object would be defined as a Black Hole. 

In 1916, the concept of a Black Hole was first postulated by the German scientist Karl Schwarzschild, in his off-duty time whilst serving as a lieutenant of artillery in the German Imperial Army. By utilising the formula for escape Velocity above, and setting the escape Velocity equal to the speed of light, Schwarzschild calculated the radius of a spherical Mass from which light could not escape :-

Where :-

r  =  Radius of Object ( m ) 

G  = Universal Gravitational Constant ( 6.67x10-11 m3kg-1s-2 ) 

M  = Mass of Object ( kg ) 

c  =  Speed of Light ( ms-1 ) 

Note - The Schwarzschild radius is also sometimes called the Event Horizon, as no information can leave the Black Hole.

Example 2 - 

Calculate the Schwarzschild radius of a stellar core of Mass 6.36x1030 kg  ( 3 times more Massive than the Sun ) collapsing under its own Gravity to form a Black Hole. 

G  =  6.67x10-11 m3kg-1s-2

M  =  6.36x1030 kg

c  =  3x108 ms-1 

r  =  ( 2 x 6.67x10-11 x  6.36x1030 ) / ( 3x108 x 3x108 ) 

r  =  9427 m 

r  =  9.43 km 

Note - As can be seen from the above calculation, a huge Mass must be compressed to a very small volume to create a Black Hole. Most stars will never collapse enough to form a Black Hole, most will end their lives as cooling White Dwarfs. It requires the remains of the dying star to have a Mass of at least ~ 3 Solar Masses. 

This does not mean a star of 3 Solar Masses, but the remains of the star after the outer layers have been lost having a mass of ~ 3 Solar Masses. To have a remaining core of this size, the original star must have had a Mass of ~ 15 Solar Masses or greater ( See The Life Cycle of a Star ).