Random geometric graphs 1,2,3 formalize the notion of “discretization” of a continuous geometric space or manifold. Nodes in these graphs are points, sprinkled randomly at constant sprinkling. Fun Graphing Games for Kids! - The Universe and More; Immerse Yourself in the World of Motion Graphs! Created by a teacher, Action Graphing employs.
- iPad
- Android (coming soon)
- $4.99
The Action Graphing app, from Universe & More, is a game that can help you understand motion graphing. (Motion graphs portray and record time, positioning, and velocity of objects in order to identify and predict their movements.) It does this by challenging you to analyze and interpret graphs in order to predict real events and model the motion of physical objects.
To use this app, you will follow the adventures of Ruggles and Non Chompsky, who are on a mission to find ice cream. They head off in their space ship to explore three different worlds, each of which represents progressively more challenging motion activities. Once Ruggles is on the surface of a planet, he moves along on a hoverboard that floats above a number line with each meter distance marked out on it. The bottom half of the screen displays a motion graph. It is up to you to move Ruggles across the screen and match the illustrated motion on the graph.
You'll need to know his initial position and his velocity in order to match coordinates on the graph below. A one-finger movement represents Ruggles’ initial placement on the graph, and a two-finger swipe sets his velocity. You are then able to see his actions once you’ve hit play.
This fully interactive app guides you along by providing information about how to play the game in snippets, when and if you need them. There are 60 or more challenges in the game and each one builds on the success of the previous ones. For each challenge, though, you will be given guidance, if needed, on how to control Ruggles on his hoverboard. In addition, coordinates can be displayed by tapping on the grid. Also, a triangle appears if you tap and hold on one coordinate and drag to the next coordinate. The dimensions of the triangle, the rise and the run, can then be used to calculate the slope.
Going Further
For EducatorsYou can use this app in your classroom to help supplement a unit on motion and forces or to help reinforce concepts you are teaching in class. The app can be particularly useful in helping students learn how to analyze and interpret graphs to model the motion of real objects. You can use the app to differentiate your instruction to meet the needs of individual students during graphing lessons. It also could be used by students who need extra practice at home.
The developer's website, Universe & More, also provides several worksheets that can be used in addition to the game to help reinforce learning.
Foucault's Pendulum
9-12 Website
Organisms in Motion: Practical Applications of Biological Research
9-12 Video
Gravity Launch App
3-12 Interactive
General Relativity
9-12 Interactive
To give you a taste of how to prove that the universe is expanding, and to give you some practice using SkyServer for astronomy research, this page will show you how to make a simple Hubble diagram, with only six galaxies.
The first step in creating a Hubble diagram is to find the distances to several galaxies. Unfortunately, measuring distances in astronomy is difficult; but fortunately, all you need for a Hubble diagram are relative distances to galaxies, not their actual distances measured in miles or light-years. Relative distances are measured with respect to some standard location, like the Andromeda galaxy or the Virgo cluster. For example, suppose you find that the Perseus cluster is five times further away from Earth than the Virgo cluster. If the Virgo cluster had a relative distance of 1, the Perseus cluster would have a relative distance of 5.
To measure relative distance, astronomers need some way to compare galaxies. Since galaxies are so similar, astronomers assume that they all have the same average properties - that each galaxy is just about as bright and just about as big as any other galaxy. When we assume that two galaxies' intrinsic brightness and size are the same, then any differences in brightness or size between them are due only to their distances from us. If galaxy A is brighter and larger than galaxy B, then it must be closer to us.
| Nearby galaxies appear large and bright, while distant galaxies appear small and faint. |
One of the easiest ways to compare galaxies is to compare their magnitudes. Magnitude is a measure of how bright a star or galaxy looks to us - how much light from that star or galaxy reaches Earth. In magnitude, higher numbers correspond to fainter objects, lower numbers to brighter objects; the very brightest objects have negative magnitudes. Universal shift register verilog.
The scale is set up so that if object A is 2.51 times fainter than object B, then object A's magnitude will be higher by one number. For example, a magnitude five galaxy is 2.51 times fainter than a magnitude four star. The sun has magnitude -26. The brightest star in the Northern sky, Sirius, has magnitude -1.5. The brightest galaxy is the Andromeda Galaxy, which has magnitude 3.5.
The faintest object you can see with your eyes has a magnitude of about 6. The faintest object the SDSS telescope can see has a magnitude of about 23. SDSS measures magnitudes in five wavelengths of light: ultraviolet (u), green (g), red (r), near infrared (i), and infrared (z).
Question 1: Why can magnitudes be used as a substitute for distances in the Hubble diagram? |
Exercise 2: In this exercise, you will find the magnitudes of six galaxies in SkyServer's database. The table below shows the object IDs and positions (right ascension and declination) of the six galaxies. To find a galaxy's information, click on its object ID. The Object Explorer tool will open in a new window, displaying the galaxy's data.
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Look at the close-up picture in the main frame of the Object Explorer. Just to the right of the picture, you will see a data table containing values for u, g, r, i, and z (the data are under the labels). These are the magnitudes of the galaxy. Save each galaxy in your online notebook by scrolling to the bottom of the left-hand frame and clicking 'Save in Notes.' Write down each galaxy's magnitude in the wavelength of green light, which is given by 'g.' |
When Slipher looked at the redness of light given off by a galaxy, he was measuring the galaxy's 'redshift' - a measure of how fast a star or galaxy moves relative to us. If you have ever stood by the side of the road as a car passed by, you have an idea of what redshift is. As the car moves toward you, its engine sounds higher-pitched than the engine of a stationary car. As the car moves away from you, its engine sounds lower than the engine of a stationary car. The reason for this change is the Doppler effect, named for its discoverer, Austrian physicist Christian Doppler. As the car moves toward you, the sound waves that carry the sound of its engine are pushed together. As the car moves away from you, these sound waves are stretched out.
The same effect happens with light waves. If an object moves toward us, the light waves it gives off will be pushed together - the light's wavelength will be shorter, so the light will become bluer. If an object moves away from us, its light waves will be stretched out, and will become redder. The degree of 'redshift' or 'blueshift' is directly related to the object's speed in the direction we are looking. The animation below schematically shows what a redshift and blueshift might look like, using a car as an example. The speeds of cars are much too small for us to notice any redshift or blueshift. But galaxies are moving fast enough with respect to us that we can see a noticeable shift.
Click on animation to play |
Astronomers can measure exactly how much redshift or blueshift a galaxy has by looking at its spectrum. A spectrum (the plural is 'spectra') measures how much light an object gives off at different wavelengths, from X-rays and ultraviolet light, through visible and infrared light, and into microwaves and radio waves.
The spectra of stars and galaxies almost always show a series of peaks and valleys called 'spectral lines.' These lines always appear at the same wavelengths, so they make a good marker for redshift or blueshift. If astronomers look at a galaxy and see one spectral line at a longer wavelength than it would be on Earth, they would know that the galaxy was redshifted and was moving away from us. If they see the same line at a shorter wavelength, they would know that the galaxy was blueshifted and was moving toward us.
By the end of its survey, the SDSS will have looked at the spectra of more than one million galaxies. Each spectrum is put into a computer program that automatically determines its redshift. The program outputs a picture like the one below, with spectral lines marked. The 'z' number at the bottom of the spectrum (before the +/-) shows the redshift. Positive z values mean the galaxy has a redshift; negative z values mean the galaxy has a blueshift.
Click on the image to see it full size |
Spectra of galaxies are stored in the SDSS's 'spectroscopic database.' The spectra are organized into plates and fibers, as they were when the SDSS telescope measured them.
Exercise 3: Find redshifts for the galaxies you looked up in Exercise 2. Click on the links below to return to the Object Explorer. Scroll down in the main frame until you see a miniature spectrum. This is the spectrum of the galaxy. Click the spectrum to see it full size. Click 'Summary' in the left-hand frame to return to the display. Just above the spectrum, you should see a data entry called 'z'. This z is NOT the z you saw in Exercise 2; this z represents the redshift. Write down the redshift (z) next to the g magnitude from Exercise 2.
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Now that you have magnitudes and redshifts for six galaxies, you are ready to make a Hubble diagram.
Exercise 4: Use a graphing program such as Microsoft Excel to make a Hubble diagram for the six galaxies you studied in Exercises 2 and 3. What does your diagram look like? For help with Excel, see SkyServer's Graphing and Analyzing Data tutorial. |
Can you really draw a straight line through your data? When scientists try to figure out what data mean, they often talking about making a 'model': in this case, the model you are using to relate magnitude to redshift is a straight line. Scientists often speak of the 'fit' - how well the model fits the data. The fit can be described with a percentage that shows how close the points lie to the place where they should lie if the model were true. Because every experiment has some error and every observation has some uncertainty, the fit is never 100% accurate. Generally, astronomers consider a fit above 90% to show that the model probably really does describe the data.
Exercise 5: Find the fit of a straight line model in your Hubble diagram. Excel (or another plotting program) can find the fit automatically using a 'trendline.' The program tries to find a straight line that passes as close to all the data points as possible, then measures how far each point falls from this straight line. See SkyServer's Graphing and Analyzing Data tutorial to learn how to make a trendline in Excel. Multiply the R-squared value by 100 to find the fit as a percentage. What is this number? Is a straight line a good fit to your data? |
You have now made a simple Hubble diagram, with six galaxies. The data in the diagram fit well with a straight line. Now, try making the same diagram with six different galaxies.
Exercise 6: Repeat Exercises 2 and 3 for the following galaxies: | |||||||||||||||
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Exercise 7: Repeat Exercise 4 for these six galaxies. Graph these data on the same scale you used in Exercise 4. What do your data look like now? Repeat Exercise 5. What is the percentage fit of the data? |
In Exercise 7, you used the same method you used in Exercises 4 and 5: you graphed magnitude as a function of redshift. So why does the graph you got look so different? When you made both Hubble diagrams, you assumed that magnitude could substitute for distance. When you assumed that magnitude could substitute for distance, you assumed that every galaxy had the same average brightness.
Galaxies do have average properties. But if every galaxy were exactly the same, astronomy would be quite a boring subject. Galaxies come in many different types, and many astronomers make their careers studying the different types of galaxies. Unfortunately, the different types make a Hubble diagram trickier than simply graphing magnitude and redshift.
In the next section, you will learn some other ways astronomers measure relative distance to other galaxies. In the section after that, you will learn more about how to measure redshift. Then, you will use this knowledge to construct a better Hubble diagram, one that does not fall into the trap that this simple diagram fell into.