The first thing to remember is that the lengths of the sides of a right triangle are related by the

This says that the sum of the squares of the "legs" of a triangle (the two sides that meet in the 90° angle) are equal to the square of the hypotenuse (the side that is opposite the 90° angle).

The next thing to remember is the trigonometric functions that relate the ratios of the lengths of the sides of the right triangle to its angles. For angle A in the figure the relations are:

SIN(A) = opposite/hypotenuse

TAN(A) = Cos(A)/SIN(A) = opposite/adjacent

The length around a circle is called the *circumference*. The ratio of circumference, C, to the diameter, D (D = 2r), is the transcendental number represented by the Greek letter . Hence,

The ratio of the area, A, of a circle to the square of the radius is again . Hence,

For a sphere the 3-D equivalent of Circumference is *Surface Area*, SA. It is given by

Finally, the volume, V, of a sphere is given by

An ellipse is the set of points whose sum of distances to two fixed points (called foci, focus singular) is the same. One way to draw an ellipse is to attach the ends of a string to two fixed points and pull the string tight with a pencil and circle around the points. The long axis of the ellipse is called the major axis, the short axis is called the minor axis. The distance from the center out to the ellipse along the major axis is called the semimajor axis, usually denoted with the letter a. Circles are just special ellipses. They are ellipses with both foci at the same point. |

Understanding of these simple geometric figures and relations allowed the ancient Greeks to make measurements of the size of the Earth, Moon, Sun and their distances from one another. Review pages 39 - 40 of your text for how these ideas are used with the concept of angular size and distance of an object.

Acceleration is the amount of change in velocity per unit time. It has units of distance/time/time (i.e., m/s^{2}). The distance that an object moves in a span of time, t, with a constant acceleration, a, and an initial velocity, v_{0}, is given by

- (A): Planets move in elliptical orbits with the Sun at one focus of the ellipse (not the center!).
- (B): The orbital speed of a planet varies so that a line joining the Sun and the planet will sweep over equal areas in equal time intervals.
- (C): The amount of time a planet takes to orbit once around the Sun is called the period, P, and it's related to the orbit's size, a. The period, P, squared is proportional to the semi-major axis, a, cubed:
P ^{2}a^{3}This says that the farther away a planet is from the Sun it will take a longer time to go around once.

Newton discovered that the constant of proportionality is dependent on the masses of the objects in orbit, and he wrote Kepler's 3rd law as

P Where G is the universal gravitation constant. If m^{2}= 4^{2}/G(m_{1}+ m_{2}) * a^{3}_{2}<< m_{1}then we may ignore m_{2}and then the relation is justP ^{2}= 4^{2}/Gm_{1}* a^{3}To see an example of the use of this relation see problem #1 of HW #2 and the lecture notes for lecture 5.

Since F = ma also, then object 1 experiences an acceleration, a_{1} found by setting the two expressions equal to one another.

a_{1} = - Gm_{2}/r^{2}

The uses of these formulae are many (e.g., surface gravity). See the lecture notes and HW #2 for examples.

**Kinetic Energy**: the energy of motion

for any object with mass, m, and velocity, v, its kinetic energy, KE, is given byKE = (1/2)mv ^{2}**Potential Energy**: the amount of work that can potentially be done

This has no simple definition, it is defined in a mathematical way. For an object, with mass, m, under the gravitational attraction of another with mass, M, and a distance, r, away the potential energy, PE, isPE = - GMm/r

E is a constant and so KE and PE can only change in such a way that as one becomes smaller the other becomes larger by the same amount. In other words we can convert potential energy into kinetic energy and vice versa.

(Recall that raising something to the 1/2 power is the same as taking the square root.)

So, light with shorter wavelengths has more energy than light with long wavelengths (blue light is more energetic than red light, X-rays are more energetic than microwaves)

Where c is the speed of light, _{0} is the wavelength of the light as seen at rest, and is the measured change in wavelength ( - _{0}).

An object with a temperature, T, will radiate energy away with an energy spectrum like that of a theoretical construct called a blackbody (shown in the figure above). The hotter the object the more light energy (photons) it radiates at all wavelengths. But, depending on its temperature it will radiate more photons at certain wavelengths. Thus the color of emitted light from an object can reveal its temperature.

The relationship between temperature, T, and the wavelength at which most of the energy (photons), _{peak}, is emitted is called Wien's Law:

The relationship between temperature, T, and the total amount of radiation being emitted per unit surface area per unit time, , is called the Stefan-Boltzmann Law:

So this tells us that for any given wavelength we want bigger telescopes to resolve smaller details. It also tells us what size telescope we need to resolve an arbitrary angle at a specific wavelength. (From the surface of the Earth we are limited to how small an angle can be resolved because of the blurring of light as it passes through the atmosphere.) At larger wavelengths we will need larger telescopes to resolve details at the same level as at smaller wavelengths.

In summary. Big Telescopes = Good

Big Telescopes in Space = Even Better

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