Dr. Aaron Titus | Department of Physics, High Point University
PHY1050      Astronomy of Stars, Galaxies, and the Cosmos
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Schwartzchild radius

When you throw a ball, such as in the animation below, the ball rises stops and comes back again. You can click on the different links to give the ball different initial speeds to see how the initial speed affects how high the ball will travel before it stops and comes back to Earth.

Initial speed = 4 m/s | 6 m/s | 12 m/s

How fast would the ball have to travel so that it would never come back to Earth? That is, what initial speed is needed so that the ball can get infinitely far away from Earth and still be moving? That is called the escape speed and it only depends on the mass and radius of Earth. It does not depend on the mass or size of the ball at all. The equation for escape speed is

where G is just a constant (i.e. a number) called the Universal Gravitation Constant, M is the mass of the body (such as Earth in the example), and R is the radis of the body (such as Earth in this example).

Note that for more massive bodies and from smaller bodies (i.e. smaller radius), the escape speed is larger. For example, suppose that you keep the mass of a star the same, but decrease its radius. Then, this increases the escape speed needed to escape the star.

The simulation below demonstrates escape speed using an illustration that Isaac Newton himself came up with. Suppose there is no air resistance. A hypothetical cannon sits atop a high mountain and shoots cannonballs. The path of a cannonball is such that it would orbit Earth (if Earth did not get in the way). If its initial speed is great enough, then the cannonball will travel all the way around Earth and come back again. If the cannonball is even greater, then it will escape and will never come back.

The cannon in this case shoots lots of cannonballs at the same time, each at a different initial speed. Click the Start Animation link to view the simulation.

After viewing the motion of the various cannonballs, click the Show Escape Speed link. In this case, the initial speed of the cannonball is equal to the escape speed; therefore, the cannonball will never return.

Image from Isaac Newton's Principia (1687).

Simulation modified from Mario Belloni's simulation in Physlet Physics.

Start Animation | Show Escape Speed

For those of you who remember your high school algebra II class, you will be interested in the following observations. The paths of the cannonballs in the case where they orbit Earth are ellipses (or in one case a circle). For the cannonball that is fired at the minimum speed required to escape Earth, its path is a parabola. If the initial speed of the cannonball is greater than the escape speed, then its path will be a hyperbola. Note that hyperbolas, parabolas, ellipses, and circles are all cases of conic sections (i.e. slices of a cone).

Note that nothing travels faster than the speed of light. The speed of light is a fundamental, universal speed limit.

What if the ratio of mass to radius of an object was such that the escape speed from the object is greater than the speed of light? Then, nothing--even light--can escape the object. That's a black hole!

The radius inside of which nothing can escape is called the Schwartzchild radius. For a given mass M, we can use the equation above to calculate the Schwatzchild radius of a black hole by setting the escape speed equal to the speed of light, c, and solving for R.

The Schwatzchild radius of a black hole of 1 solar mass (which does NOT exist), is 3 km. Since Schwatzchild radius and mass are proportional, then we can easily calculate the Schwatzchild radius of any mass black hole by just multiplyiing its mass in solar masses times 3 km.

For example, the minimum mass of a black hole is approximately 3 solar masses. Thus, its Schwatzchild radius is about 3 x 3 = 9 km.

Note, that black holes have no "surface" in the traditional sense of the word. There is nothing to counterbalance the gravitational collapse of a black hole. Thus, it collapses to zero volume. If we speak of a "surface" of a black hole, we are probably speaking of its Schwatzchild radius, inside of which nothing--even light--can escape.

 

 

 

 

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