**Kepler's Third Law** - Kepler discovered that the size of a
planet's orbit (the semi-major axis of the ellipse) is
simply related to sidereal period of the orbit. If the size of the
orbit (a) is expressed in astronomical units (1AU equals the average
distance between the Earth and Sun) and the period (P) is measured in
years then Kepler's Third Law says

Examples: The Earth orbits the Sun at a distance of 1AU with a period of 1 year. = .

Suppose a new asteroid is discovered which orbits the Sun at a distance of 9AU. How long does it take this object to orbit the Sun?

**Newton's Law of Gravitation** - Any two objects, no matter how
small, attract one another gravitationally. The attractive force
depends linearly on the mass of each gravitating object (doubling the
mass doubles the force) and inversely on the square of the distance
between the two objects

G is the gravitational constant which is just a number which makes the results of the equation match-up with our system of measurement.

Example: Halley's Comet travels on an eccentric elliptical orbit which carries it as close as about 1AU to the Sun and as far as about 20AU. Compare the gravitational force between Halley's Comet and the Sun at these extremes in its orbit.

*The comet changes its distance from the Sun by a factor of 20.
Since the gravitational force depends on the inverse square of the
distance, then the force when the comet is far from the Sun is
or 400 times weaker than when the comet is near the Sun.*

**Magnitudes** - a measure of stellar brightness. In the Greek
system the brightest stars were of the *first* magnitude. Stars that
were not bright enough to be *first* magnitude but close were
*second* magnitude stars. Stars fainter still were *thrid*
magnitude and so on down to the faintest stars visible to the
naked eye which were *sixth* magnitude stars. Perverse astronomers
of modern times mathematized this relationship between numbers and
stellar brightness. You don't have to know the exact equation
because it will only make you miserable. What *is* important is...

- Fainter stars have numerically large magnitudes. The magnitudes of bright stars are small numbers or possibly even negative numbers. For example, an 11th magnitude star is fainter than a 9th magnitude star, and a -4th magnitude star is brigher than a 0th magnitude star.
- Each magnitude corresponds to a difference in brightness
of a
*factor*of 2.512. More importantly, stars that differ in magnitude by 5 differ in brightness by a factor of 100 (= ).

Examples: Star A is 100 times brighter than Star B. Star B has a magnitude of 9. What is the magnitude of Star A?

*Since the difference in brightness is a factor of 100, the difference
in magnitude must be 5. Since Star A is brighter than Star B its
magnitude must be 5 numbers smaller that B's magnitude. 9-5=4.
Star A has a magnitude of 4.*

What if Star A was 10,000 times brigher than Star B?

* . Remember each factor of 100
corresponds to a difference in magnitude of 5. The total magnitude
difference here then is 5+5=10, one 5 for each factor of 100.
Star B has a magnitude of 9, so Star A has a magnitude of 9-10=-1.*

**The Propagation of Waves** Waves of light, sound, or even water
each have a well defined frequency ( ) and wavelength
( ). If you think carefully about the definition of these
two numbers it should be obvious that their product
is the velocity of propagation of the waves.

where c is the symbol for the velocity of the wave. For light c=300,000 km/s.

Example: On a quiet pond you observe water waves with a wavelength of 10 cm which pass a floating cork at the rate of 2 wave crests per second. How fast are the waves moving along the water.

*The frequency of the waves is 2 per second and their wavelength
is 10 cm. 10 cm 2/second = 20 cm/second. Note how
important proper units are to the final answer. *

**The Energy Carried by a Photon** - Each packet of
electromagnetic energy, or photon, carries a specific amount of
energy which is simply related to the photon's frequency ( )
and wavelength ( ).

High frequency (short wavelength) photons carry the most energy. Low
frequency (long wavelength) photons are the least energetic. The
constant, *h*, is called Plank's constant and once again its purpose is
to scale the equation so that it agrees with our system of
measurement.

Example: Which type of photon carries the most energy, ultraviolet photons or gamma rays?

*Since gamma rays have a much higher frequency than ultraviolet
light they carry the most energy per photon.*

**Blackbody Radiation** - A blackbody is an object that appears
perfectly black (reflects no radiation at all) when cold. There are
virtually no perfect blackbodies in the real world. On the other
hand,
almost any solid object has properties very similar to perfect
blackbodies. Common everyday experience tells you that when you heat
something up it begins to glow, first a dull red, then yellow, then
white hot, and eventually even blue. Two laws describe the color
and quantity of radiation emitted by a blackbody at a given
temperature:

**Wien's Law** - This law permits you to
calculate the peak wavelength (*i.e.* color)
of the continuous spectrum emitted by a hot
object if you know the temperature.

Remember red light has a wavelength of 0.7 m = cm and blue light has a wavelength of 0.4 m = cm).

Examples: The temperature of the Sun is about 5800K. What is the peak wavelength of its blackbody spectrum?

*
*

* *

*
*

Star A is twice as hot as Star B. Star B appears red ( m). What color is Star A?

*Since the peak wavelength depends inversely on the temperature,
if the temperature doubles the peak wavelength is halved.
0.7 m/2=0.35 m. Light with a wavelength of 0.35 m is blue.*

**The Stefan-Boltzmann Law** This law quantifies the amount of
energy emitted by one unit of surface area of a hot blackbody.
Energy = . Note that the amount of energy emitted
depends very strongly on the temperature.

Example: How much more energy does a blackbody emit if its temperature triples?

*Since E= the increase in emitted energy is or
.*

**The Luminosity of a Star** - The *luminosity* of a star is a
measure of the total amount of energy it emits. Since stars behave
like blackbodies we can use the Stefan-Boltzmann law for the amount
of energy emitted by one square centimeter of the surface of the
star with the equation for the surface area of a sphere. The surface
area of a sphere, *i.e.* the number of square centimeters emitting
radiation, is where R is the radius of the star. Thus the
luminosity of a star is

Examples: Suppose the size of a star doubles while it maintains the same temperature. How does the luminosity of the star change?

*Since the luminosity depends on the radius squared the new
luminosity will be = = 4 times greater. If the
radius tripled instead, the luminosity would increase by a factor of
9.*

Two stars have the same size. One star is 3 times hotter than the other. How much more luminous is the second star?

*The luminosity of a star depends on the temperature raised to
the fourth power. If the temperature increases 3 times the
luminosity increases by = .*

**The Doppler Equation**
Just as the perceived wavelength (pitch) of sound waves vary depending on
whether their source is approaching you or receding from you, the
perceived wavelength (color) of light depends on the relative motion
of the source with respect to the observer. The shift in wavelength
is given by

is the original wavelength at which the light was emitted, also
known as the rest wavelength. is the measured
wavelength,
which will be shifted from the original wavelength due to the motion
of the source. *v* is the velocity with which the source is moving
with respect to the observer and *c* is the velocity of light. *v*
can be either positive or negative. A
positive value of *v* corresponds to a source which is moving away
from the observer. Negative values of *v* imply approach.
If the source is moving away (*v* is positive) then must
be larger than . The observed wavelengths are longer than
the emitted wavelengths. Such a shift to *longer* wavelengths is
called a *redshift* regardless of the original wavelength of the
light. When we talk about the spectrum we will often use the terms
*red* and *blue* to indicate directions toward longer and shorter
wavelengths respectively.
Conversely, if a source of light is approaching, *v* is negative and
the shifted wavelength, , must be smaller than the
original wavelength, . This shift towards smaller
wavelengths seen in approaching objects is called a *blueshift*, a
shift in the direction of shorter wavelengths.

Example: The most prominent emission line of hydrogen gas occurs at a wavelength of 6563Å. Suppose you observe this line in a distant galaxy at a wavelength of 6600Å. Is the galaxy approaching or receding and how fast is it moving?

*The original (emitted) wavelength is 6563Å. The shifted
(observed) wavelength is 6600Å. Using the Doppler Equation
*

* *

*
*

* *

*
*

* *

*
Since v is positive the galaxy is moving away from the Earth and
Sun.*

**The Inverse Square Law** - Distant lights appear fainter than
nearby lights of the same intrinsic brightness.
Imagine that a light bulb emits a
brief burst of light in all directions. That light moves away from
the bulb like the expanding *surface* of a balloon. As the surface
expands the light must become more dilute because it has more
to cover. The surface area of a sphere is
where *R* is the radius of the sphere or, in this
case, the distance from the light bulb. As the sphere grows, as the
distance from the light bulb increses, the light becomes diluted
(fainter) in proportion to the area over which it spreads. Thus

where means ``is proportional to".

Examples: Star A and Star B are equally luminous.
Star A is 5 times further away than Star B. How much fainter does
Star A *appear* to be than Star B?

*Using the above relation *

* *

*
Star A appears 25 times fainter than Star B. *

**Angular measurement:** We measure distances on the sky
in terms of angles. The apparent (angular) size of an
object depends on both its true size and its distance from us.
A large object can *appear* quite tiny if it is far away. Conversely
a small object can *appear* quite large if it is close to us.
The apparent angular size of an object is:

A radian is a unit of angular measure equal to 57.29 degrees. Most often in astronomy we use angular units of arcseconds. There are 3600 arcseconds in a degree. Thus the angular size of an object, when measured in units of arcseconds is:

since there are 206265 arcseconds in a radian. These equations will give you the right answer only if the diameter and distance are expressed in the same units of measure.

Example: The diameter of the Sun is km and it is km from the Earth. What is the angular size of the Sun on the sky?

Wed Feb 21 15:07:04 EST 1996