Using the relative distance table students should be able to answer what types or variables of information are given in the table. The answer is two, distance and time. This can be expressed as a rate if we place distance over time, for example 60mi/hr. In my classes, I seldom use anything more complicated than a rate formula such as velocity = distance/time. Once a student realizes the proper units for each variable complicated word and formula problems can be simplified by use of what I call the pie method. Draw a circle and cut it into three parts, on the top and two on the bottom like so
Plug the equation into the pie exactly as it is given to you. If a variable is on top, division, put it on top. If it is next to each other, multiplication, put it in likewise. Ask students to devise a way to never plug it in wrong! The trick is to always plug the side with two variables in the first and then place the third variable in the remaining spot. Now that all the slots are filled, cover what you are looking for and solve the equation. For example, using the pie method to find the speed of a jet to Mercury, the first thing we would do is put the speed or velocity equation (velocity = distance )
In to the pie diagram. Using the hint given earlier, I would then plug in distance first, since they
are on the side with two variables. Your pie should now look like this.
In the remaining space, plug in velocity.
Earlier, you were asked to cover what you are looking for. If we were to cover velocity, the resulting equation would be velocity = distance.
Using our distance/time table for Mercury in a jet form Earth, the result would be
velocity = distance or 5,700,000 miles or 5,277,777.8 miles
Time 10.8 yrs year
I will ask students to use the pie method to find the speed or velocity for the moon, Sun, and Mercury for a jet, rocket, and different forms of light from the previous table of distance and time travel from earth.
It should be demonstrated, the faster the velocity, the less time it would take to travel! This statement seems intuitive, but most students don't take the time to process this fact. Have students ponder what would happen to time if an object traveled at the speed of light.
Partial List Formulas that can be Applied to the Pie Method
Area = Length x Width
Pressure = Force
Density = Mass
% Rate = Base x Portion
Weight = Gravity x Mass
Force = Mass x Acceleration
Mechanical Advantage = Resistance Force
Voltage = Amperage x Resistance
Work = Force x Distance
Power = Work
Potential Energy = Weight x Height
E = mc^2
Acceleration = Change in Speed
Wave Speed = Frequency x Wave Length
Storytelling Using the Powers of Ten
As we discuss distances and properties of the planets the need for a concise and simplified way of handling very large and very small numbers becomes evident. Scientific notation and operation of numbers to a power are one method to achieve this.
One of my favorite ways to teach numbers to a power is to arrange the students in a group of three or four and "make believe" or create a story and present it to the class using the picture as a focal piece. Some restriction on vulgarity and language must be emphasized as imaginations do tend to run wild. If students buy into the process, it can be used as a tool to help them write. After each group has presented their fanciful tale, inform the students that all of these pictures are of the same subject at different perspectives (Heights!) When Archimedes could not solve the dilemma of the composition of the gold crown, he changed his perspective to achieve a rather revealing result. But that is another story for another day.
I have enclosed some sample pictures from the book
The Powers of Ten
. It shows the range dimensions from the micron of cells to the vastness of outer space. This difference is accomplished by the movement of a decimal point. For every space the decimal point is moved to the left, the exponent is increased by +1. For every space the decimal point is moved to the right, it is decreased by - 1. We use this information when we put numbers into scientific notation. Scientific notation is a form in which the decimal of a number is moved until there in one number to the left of the decimal point and the exponent is established by the rules above. For example, .00004 microns - 4x10^-5 microns and 800,000km = 8x10^5km. By placing numbers to a power into scientific notation a comparison or order can be established.
Rules to Operate a Number to the Power of Ten
Where does the author of
The Powers of Ten
get this title? A powerful way of using our ten fingers or toes to do Math. Here are the rules for addition, subtraction, multiplication, and division. Before we do our four math functions it is important to realize 8x10^5 = 8 8 x 10 10 x 10 x 10 x 10 and 4 x 10^-4 = 4.
Addition & Subtraction - If exponents are the same then add or subtract and keep the exponent. If the exponents are not the same, change them by moving the decimal point so that they are now the same, then add or subtract and keep the exponent.
5x10^3km + 65x10^3km = 70x10^3km
7x10^20km - 30x10^19km = 7x10^20km - 3x10^20km = 4x10^20km
Multiplication - Multiply the non-exponent portion of the number and add the exponent.
Example: (40x10^17km)(2x10^4km) = 80x10^21km
Division - Divide the non- exponent portion of the number and subtract the exponent
Example: 40x10^16km = 20x10^12
I have enclosed a practice sheet to reinforce the concept of numbers to a power.
Contributions of Einstein
Albert Einstein once stated, "Do not worry about your difficulties with mathematics; I can assure you mine are greater".
Albert Einstein's was a patent clerk for Switzerland. He had wanted to be a physics professor but could not find a job. He alienated his professors and they would not give him a recommendation. As a patent clerk, he applied all of his knowledge of science to new inventions. He would sit down with a stack of documents and determine if the object violated central ideas of physics. Is the basic principle new or was it known? Einstein threw his self into the problem and stripped away all of its complications.
This same tenacity led him to question everyday events and fundamental laws that we take for granted. For example, we assume a mile is a mile everywhere. If there was a race, we assume the person who finished in four minutes was faster than the one who finished in five. Einstein questioned everything. He stated if we were able to catch up with light, it would not be light. In a series of thought experiments, which took place in his mind, he stated that if you were in a train that moves 186,000 mi/sec and it caught up to the light on the train, the person outside the train would measure distance short and time outside the train, slow. In the television series "Lost In Space", as the Robinsons traveled around the galaxy for a year when they returned, everyone they knew would have been dead for a thousand years. Time and distance are relative, E=mc^2. This is only applicable when the speed of an object nears the speed of light.
In another thought experiment, he imagined an elevator in a deep well with no window to connect you to the outside world. If the elevator accelerated uniformly, you could not tell if you were moving or not. He concludes acceleration and gravity are the same in curved space. An application of this is the Sun curves the space of planets orbiting it such as the Earth. Space and time are the same. This must be taken into consideration as you calculate the velocity from the table of Distance and Time.
Nothing is as it seems on a small scale either. Rutherford & Bohr postulates an atom is made up of mostly empty space. Therefore, the principle behind Honey I Shrunk The Kids, is true (in theory anyway). When we sit in a chair we don't actually sit on it. Your matter actually hovers angstroms above the chair. Why don't our hands pass through another when we clap? A field of charges (similar to the Robinson force field) prevents it. Nothing is as it seems!! Scientists discovered uncertainty is a principle in which the very nature of matter is written.
Applications of the Theory of Relativity
Everything in the universe emits radiation. A telescope can pick us its signal. In 1967, Jocelyn Bell operated a radio telescope outside Cambridge, England. She noticed a strange blip on the chart paper used to record what the telescope observed. An advisor told her the signal was interesting but probably man made. She returned to her chart and calculated when she should see it again. She discovered it was a string of pulse 1.3 seconds apart. This had never been seen before. Heavenly bodies don't pulse on and off that fast. She thought for a second it might have been little green men sending a signal. She found a second signal on the opposite side of the sky. This eliminated the little green men theory and heated up debate in the astro-physics community. It turns out that she was the first to discover the previously untested theory of what happens when a star dies. When a large enough star runs out of fuel, it will collapse and crush the atoms within it. What is left is a neutron star, every available space is squeezed out. If matter that makes up the sun, would squeeze down to a section of New Haven, a spoonful would weigh 2x10^11 lbs. Neutron stars can rotate 700 times per second while they blast powerful radio waves outward. It is this signal that Jocelyn Bell picked up with her telescope. Theory had predicted the existence of neutrons stars but most scientists thought they were too weird to be real. The universe is strange and more fantastic than we can begin to realize.
The reality of a neutron star also begs the question is a black hole possible? Carl Swartzchild was fighting on the Russian Front in WWI when Einstein's theory of relativity was published. He noticed that Einstein equations allowed for the possibility of a star so dense even light could not escape it. He made some calculation of his own and mailed them to Einstein. Unfortunately, a week after receiving Einstein's reply, Swartzchild died in battle. Though Einstein marveled at his work, he did not accept it, stating, "...It just doesn't smell right". Yet the same theories that allow for neutron stars allow for black holes. It is just a matter of taking it a step further to over shooting. Sometimes the collapse of neutron stars is so powerful that it over shoots and becomes a black hole. Professor Walter Lewin of MIT, states, "if you take the earth and squeeze it in a large vice to 3cm, it would become a black hole!"
Finding something that does not give off radiation is difficult. In the 1970s, there is indirect evidence found by Paul Merten and Louise Webster of a black hole. In the constellation Cygnus, they saw a super giant star circling something they could not see. They supposed that it must be circling a small, denser star because its gravity was sucking huge amounts of matter from its large neighbor. The only explanation is a black hole.
Big Bang or Steady State
How did the universe begin? Most scientists believe that at one point, all matter was together if we could go back far enough in time. It had infinite density. At some point, it exploded, sending matter throughout the universe as we see it today. This belief is called the Big Bang Theory.
Why should this theory be accepted? Were there others? Not everyone believes in the Big Bang Theory. Some believe in a universe that has not changed since the creation of time-the Steady State Theory.
In 1960 Robert Dicke proposed a way to settle this argument. He stated that if the universe was created with a big bang, it should be filled with radiation. He believed this because it had to be very dense and hot and there was no way to get rid of the radiation. This heat is converted to radio static.
Robert Wilson and Arnold Penzias worked on a seemingly unrelated problem 30 miles away. They wanted to study our galaxy and make a contribution to astronomy. Although he believed in the Steady State Theory, he neither set out to prove or disprove either theory. The radio telescope they used was specifically set up to exclude all non-specific signals. Yet every time and in every direction he pointed his radio telescope, all he picked up was radiation, in agreement with Robert Dicke's theory. It should be noted that Einstein equations show that the universe is expanding. Einstein did not believe his own equations, as he did in the previously mentioned Swartzchild application of the theory of relativity. He inserted a constant in the equation to reflect a steady state universe. He latter called this the biggest blunder he ever made!
In the beginning, some 15 billion years ago, the universe exploded from a single point. Less than 1 minute later, it was a million billion miles across. Though it was cooling rapidly, it had an average temperature of a billion degrees. Over time, the universe continues to cool. Gravity formed clumps of atoms to form stars, (time lapsed: a billion yrs). Eventually planets formed and on the third planet from our Sun, life began.
Science. Non-science and the Scientific Method
As fantastic as the reality of science is, some things can not be tested by the scientific method. I learned in this astronomy course that the scientific method is used to disprove a result based on laws and experiments, rather than proving it. Albert Einstein states, "What hopes and fears does the scientific method imply for mankind? I do not think that this is the right way to put that question. Whatever this tool in the hand of man will produce depend entirely on the nature of the goals lived in this mankind. Once these goals exist, the scientific method furnishes means to realize them. Yet it cannot furnish the very goals. The scientific method itself would not have led anywhere, it would not even have been born without a passionate striving for clear understanding".
Astrology is a practice followed by many on a daily basis. Based on constellations, many would take risks where, usually, they would not. This would no more withstand the scrutiny of science than a fortune cookie reading. Yet certain predicted events may come true.
In 1910, Haley's comet returned to Earth. In this return of the comet, the Earth was said to pass perilously close to the tail. French astronomer Erin Delant warned of disastrous weather. Comets were always linked with catastrophes such as the fall of kings and kingdoms. Sure enough, just before the arrival of the comet, Paris experienced the worst flooding in thirty years. Delant's colleagues predicted an even direr situation. He postulated the comet's tail contains enough nitrous oxide, (laughing gas) to produce extreme joy, then widespread madness and death.
In the end its up to science to determine fiction from fact. Scientists analyzed the photographic and spectrographic data of the tail of the comet and determined it was to diffuse to have any effect on the Earth.
One of the questions asked in this astronomy course was "What type of experiment do astronomers perform?" None, was the answer. An astronomer's lab is his observatory. Astronomy is an observing science. Sight is the primary sense used in this science. The instrument that enhances this endeavor is the telescope.
Dutch optician Hans Lippershay invented the first telescope. Before this time, even as far back as the 1200's scientists experimented with magnifying lenses. After all his efforts, unfortunately, Lippershay was not granted a patent on his invention. (Obviously the patent clerk was not Einstein.)
In 1906, Italian scientist, Galileo, after hearing of Lippershey's invention, made a crude telescope. His telescope could only magnify objects thirty times and only had a small field of view. Although the principle of his telescope is only applied to opera glasses today, he is credited as the first human to see the rings of Saturn, four of Jupiter's moons, and the mountains and craters of Earth's moon.
Types of Telescopes
Although a radio telescope is used less
Frequently than an optical telescope, it can
be used in any type of weather. It can also
tune in on stars that give off no light at all.
Types of Telescopes
Exercise: How to Make a Simple Telescope
The number of times an object is magnified can be calculated by knowing the focal length of the objective, light rays are bent until they come to the focal length. The distance between the center of the lens and the focal point is the focal length. The magnification (cm) of a telescope can be found by dividing the focal length of the focal objective (fl) by the focal length (f2) of the eye piece or m = fl
Using the above diagram, I will have the students calculate the magnification of the telescopes they have made.
Some Great Accomplishments of Astronomers
For centuries since antiquity, the night sky has been the greatest show on Earth. Part of its allure is its mystery.
Many men and women have worked to reveal the mysteries of the heavens. Their pursuits did not dim the beauty of the midnight sky, but only deepen our appreciation for it. This is not unlike removing the veil of and incredibly beautiful woman. My words do not do the process justice.
George Hale set out in 1919 to build the world's biggest telescope. It was to be twice as big as any that had existed at that time.
He was ahead of his time in vision and in terms of technology available. His enthusiasm overshadowed the fact that no one was able to pour that much glass at one time. There were no roads on the proposed mountain site. Some pieces of equipment were so big that they had to be shipped by boat. The glass they poured for the mirror cracked twice. Rather than replacing the glass, he had it ground and polished. This process of removing the cracks from the glass took four years.
He was beset by so many problems, he stated little green men visited him and gave him advice. On November 2, 1917, his trials, tribulations (and the advice of the little green men) paid off and the observatory opened to prove the worth of his efforts. It's 9,000 ton mirror could detect a candle 5,000 miles away. It would be the dream of astronomers (of that time) to spend a night on Mt. Wilson. Astronomers stated that the dome opened like thunder and it would be you and God alone, to enjoy the night sky. The air was described as crisp and sometimes so cold it would freeze a teardrop. Though the temperatures were brisk, he would not allow coffee nor women in the observatory. He must have though both were poisonous to the system. He stated that wives were a distraction to their monk-like scholarly pursuits.
Ironically, without the discovery of a woman Henrieta Leavitt, we would not be able to calculate the distance of the most far off stars. If a star is close, then parallax can be used to ascertain the distance. However, if a star is more than 10 parsecs, the Leavitt method must be employed. This was the first great discovery using a 100-inch telescope. Women were hired to do the menial tasks at the Harvard College Observatory. She noticed a pattern in a class of stars called Cepheids and realized the time it took to reach their maximum brightness can be used to judge their distance. It is a measure by which out stars distances are judge and without it astronomers would be clueless. She discovered four novae and more than 2,400 variable stars.
Henrietta Leavitt's discovery was of particular use to Edwin Hubble. Hubble was a star athlete and won a Rhode Scholarship to study law at Oxford. Upon his return to the U.S., he decided to study astronomy rather than law. Hubble attempted to put away his Missouri roots and become the quintessential English Scholar. Because, he was an excellent astronomer, and had the propensity to ask the right questions, he became a respected member of the Mt. Wilson Team. Hubble wanted to unlock the secret of the nebulae, faint smears of light that have puzzled astronomers for a thousand years. Their true nature eluded him for four years, even with Hales' 100 in telescope. In October of 1923, he took a forty-minute plate and developed it. He thought he saw a "nova" - stars that brighten unexpectedly. The next night, he took a deeper photographic plate. This plate had what he thought were three novae. When he began to compare the plates, he discovered that one of the three novae was not a nova at all. He discovered that it was a Cepheid and it was a Eureka moment. Thanks to Leavitt's discovery about Cepheids, he realized that this star and the system that it is part of must be larger than any he had dreamed. We now know that there are billions of galaxies, each containing billions of stars. He discovered that Andromeda was not a part of our galaxy. It is about 2 million light years away. Andromeda galaxy and others are big systems equaled to or surpassing the Milky Way.
Like Newton before him, this discovery alone would have assured his place in halls of science. Not satisfied to rest with his prior accomplishments, he makes an even greater discovery, the ratio between the distance of a galaxy and its speed. For five years he gathered information on galaxies, speed and direction. If a galaxy is moving away the wavelengths of its light are stretched and its frequency is decreased. Its light appears to redden. The faster it moves, the redder the light. If a galaxy is moving towards the earth, the frequency of its light is increased and it appears bluer. He then plotted the nebula's motion again the distance and found a straight line! This means a galaxy's distance is proportional to its velocity. If a galaxy is twice the distance from another, it is moving twice as fast.
This also means that the universe is expanding. In the entire history of man, the universe had been seen as a fixed quantity. Hubble's discovery drastically shifted the way the universe was viewed. If you have an expanding universe it means that it may have had a beginning and that it also may have an end.
Stars to the human eye appear to crowd the midnight sky, but in terms of population density, they are less dense than the population of the most desolate regions of the earth, for example, the Antarctic or the Great Sahara plains. Neil McAleer makes the comparison, if our Sun were represented by a basketball in New York City, then on the same scale the solar system, would have a diameter of about 2 miles and the next star would be a basketball 5,000 miles away, in Hawaii.
The next star to our Sun is Alpha Centauri, 4.3 light years away. A light year is defined as the distance it takes light to travel in one year at a speed of 186,000 miles/hr or 3x10 m/s. It would take an Apollo spacecraft 850,00 years to reach Alpha Centauri.
It is because of these great distances between stars that light from all the stars equals about 1/15 the light of the full moon or 1/6,000,000 the light of the sun. It would be equal to shining a 100-watt bulb about 614 ft away, which are approximately two football fields away.
A star's lifetime is determined by its mass, which is determined by the size and dynamics of the parent gas cloud from which it was made. In the animal kingdom sometimes the larger the animal, the longer the lifetime. McAleer states the opposite is true in the celestial kingdom. Massive stars burn out quickly and have short lifetimes, while less massive stars live the longest.
Stars lifetime's range from one million or 1x10 years for the most massive stars to 100 billion years or 100x10 years for the least massive stars. Comparing these stars lifetimes is like comparing a single afternoon to the lifetime of seventy two-year-old humans. (average life span)
If I state (as I have previously) that the number of stars far outnumber the grains of sand on all the beaches in the world, or one of the other billions of references when it comes to stars, who would be mad enough to try to categorize them? Astronomers Meg Nad Saha and Russell.
Probably the largest amount of information that is attainable for a single star is obtained from it's spectrum. Spectra secured with a slit spectrograph can reveal whether the star is a member of a close binary system, if it is a highly luminous supergiant, a moderately bright giant or a dwarf star likes the sun. The main purpose of spectrograms is to determine the chemical composition of a star. They will also tell if the star is in a rapid rotation and if it has a strong magnetic field.
The multitude of stars falls in a small number of spectral classes ranging from hottest to coolest. In 1920 Saha arranged the categories O, B, A, F, G, K, M. R and N stars (sometimes referred to as the carbon or C stars) and the S stars supplement this group.
I learned a rather humorous pneumonic to remember the order as stated above. Oh Be A Fine Girl Kiss Me Right Now, Smack!
The O-M spectral sequence represents a group of stars of the same chemical composition but different temperatures and pressure. The R, N and S stars are different from others in chemical composition and are giant or super giants.
O and B are the bluest and the hottest. The M, R, N and S stars are the reddest and coolest.
In 1913 Henry N. Russell was the first to plot the absolute magnitude vs. the spectral types. He latter collaborated with Eignar Hertzberg to develop Russell - Hertzberg diagram which relates the brightness of stars to it's temperature. Russell found that the stars do not randomly fall on the graph, but tend to congregate in restricted domains.
Most of the congregational stars are dwarfs belonging to what is called the main sequence. They range from hot blue objects 10,000 or more times brighter than the Sun, down through the A stars such as Procyon, through the sun down to K stars and finally to the faint red dwarfs thousands of times fainter than the Sun.
Extensive astronomical research has been based on this diagram.