Topics in Astronomy: Lesson 3

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   :   Black Holes
Assignment
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Galactic black hole
consuming smaller galaxy --
photo by
NASA

Learning Objectives:

Black Holes

This week we will look at the ultimate in high energy physics, black holes. Imagine if you will, a point in space. This point is a star that was so massive when it died that the pull of its own gravity caused it to become more and more dense, first past the point of electron degeneracy, past the point of being a neutron star, and finally collapsing down to a mere point in space called a singularity.

The gravity close to the singularity (the infinitely dense point) is so strong that not even light can escape it. Instead, anything that comes close to our singularity gets sucked down never to return. We even have a special name for the point of no return around a black hole, the event horizon, or the Schwartzschild Radius. Imagine a bullseye with an infinitely small center surrounded by a larger ring. The ring is the point at which even light cannot escape if it crosses.

Perhaps the analogy of a camp stew is appropriate here. I was once a Boy Scout leader and one of the boys in my Troop was a fairly accomplished cook. Ben’s specialty was stew, and his method of cooking was somewhat unique. ANYTHING that fell into the pot was instantly stirred under with the exclamation “Oops, sucked it under!” With a black hole, the “Oops, sucked it under!” point is the Schwartzschild Radius.

It gets its name from a brilliant German mathmetician, Karl Schwartzschild. Schwartzschild was probably one of the first persons in the world to truly understand Einstein’s equations dealing with mass, space, and time. By using the equations of Einstein, Schwrtzchild was able to determine mathematically the radius of the event horizon (Oops, sucked it under!). In his honor, it was named the Schwartzschild Radius. Schwartzschild, who only lived to his late 20s, was just one of the many casualties in World War I. It’s sobering to wonder what he would have been able to accomplish if he had grown to his full intellectual stature.

Schwartzschild’s formula was quite simple: Rsch=2GM/C^2. Rsch is the radius of the event horizon. G is Newton’s universal constant of gravitation. This is commonly accepted as being .0000000000667 M^3/kg.sec^2. As you can see, this is a fairly tiny little number, even when we multiply it by two as we do in this equation. M is the mass of the star in kilograms, and lastly C is the speed of light 300,000,000 meters per second. Looks like this one will really give the old calculator a workout. Now the mass of stars is commonly given in multiples of the mass of our sun, so we must also know this number. The mass of the sun is commonly given as 1,989,000,000,000,000,000,000,000,000,000 We will work though a problem using this formula.

However, before we begin this, let’s learn something about a math called the scientific notation. It works because our math is based on the number 10. Very small numbers like the gravitational constant can be written as 6.67 x 10 to the -11th power, or more commonly 6.67 x 10^-11. The speed of light, that huge number of 300,000,000 m/sec is written as 3x10^8, and the mass of the sun is 1.989 x 10^30 Kg. Note the ^ sign is the way that a power of a number can be expressed without actually using supersets.

Now the really neat part is that addition, subtraction, multiplication, etc. are the same, only the power of the number is treated differently. For instance, C^2 can be written as 300,000,000x300,000,000, and your hand calculator will have a stroke, or you can simply write it as 3x10^8 x 3x10^8 where you multiply the threes to get nine, but then add the powers to give a total of 9x10^16 instead of 90,000,000,000,000,000 M/Sec.

Division involves subtraction of the exponents, while addition and subtraction does not affect the exponents. The only trick to adding and subtracting is to get the exponents to the same value before you simply add and subtract the base number. For example 3 x10^3 +2x10^4=? You must convert the 3 to .3 in order to get the exponent of the 3 to a 4 like the exponent of 2. Then you simply add to get 2.3x10^4.

OK, I trust that is clear as mud, let’s go with an example. We have a 5 solar mass star that one fine night decides it is tired of its existence and proceeds to implode down to a black hole. How close can we get to the black hole without being sucked under? Assuming a very, very powerful spacecraft then, G=6.67 x 10-11, so 2xG would be 13.34 x 10^-11. M would be 5 times the mass of the sun, or 5x 1.989 x 10^30 or 9.945 x 10^30. Thus, we have our main terms defined.

The formula is Rsch=(13.34x10^-11)x(9.945x10^30)/9x10^16. Doing all the multiplying on the top gives us: Rsch= 1.3267 x 10 ^21/9x 10 ^16. Doing the division gives us 14740.7 meters, or 14.7407 kilometers. In other words, our example sun who was 5 times as massive as our own star would have a Schwartzchild Radius of only a little over 14.5 kilometers. The point is that the Schwartzchild Radius is really small even for a truly massive star.

All right then, that is good for a stellar black hole, what about the real monsters that are thought to lurk in the center of galaxies? Galactic black holes are thought to exist with between one and two million solar masses. Let’s work out the math for a one million stellar mass hole.

The formula is the same, indeed we only change one term, that of the mass of the star Rsch=2x(6.67x10^-11)x(1 million x 1.989x10^30)/9x10^16. Working the top of the fraction first yields 2.6533 x 10^29/9x10 ^16 or 2.9481 x 10^9km. Thus a gigantic, galactic black hole would only have a Schwartzchild Radius of 2,948,100,000 kilometers. The orbit of the planet Uranus is 2,875,000,000 Km, so from this we can see that the radius of even a huge, galactic black hole is really relatively small compared to the vastness of space.

Why do astronomers bother with the concept of black holes, what is their fascination? Perhaps because it is the ultimate unknowable in our universe. We can by definition never know what the inside of a back hole is like. As nothing can leave once it is sucked under, then any probe we might manufacture and launch would fail, just at the most interesting time.

Now there are all kinds of theories about what black holes are like; it is mathematically possible that black holes can provide wormholes through space that will allow us to move from point A to point B at near-instantaneous speed.

Theoretically, it is possible at least according to some astrophysicists, that such a worm hole could provide us with time travel. All of these possibilities are interesting, but by definition at this time impossible to prove or disprove.

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Assignment

Explore the Falling into a Black Hole Website.

Then, speculate on the likelihood of a Capt. Kirk and his good ship Enterprise being sucked into a black hole as they ramble around the Galaxy. What would happen to them if they were sucked into a black hole?

Write your predictions (no more than two or three paragraphs, please) and email them to Janet Hallmark.

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Reminder of Optional Activity: You may, if you choose to do so, use the starchart you printed from the downloaded software or information from other sources to locate and observe objects in the night sky.

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webspinner: Janet Hallmark
© 1998 -- 2006 jhallmark@hughes.net
URL: http://webspace.webring.com/people/bu/um_9502
Netword: Janet Hallmark

K.B. & Janet Hallmark's photos taken at Loblolly Observatory. The Loblolly Observatory logo was created by Pogebait. NASA images used with permission for educational purposes. Some clipart images by Absolute Web Graphics Archive. If I have forgotten to credit any clipartist for his/her work, I apologize and will gladly make corrections as soon as notified. Special thanks to K.B. Hallmark III, Subject Matter Expert for this course. K.B. holds a Masters' Degree in Astronomy at the University of Western Sydney in Nepean, Australia. Janet has received a Masters' Degree in Education at the University of Houston -- Clear Lake, Texas and a Educational Specialist Degree at Nova Southeastern University, Fort Lauderdale, Florida.

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