Astronomy 121 - Some Important Mathematics

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

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Examples: The Earth orbits the Sun at a distance of 1AU with a period of 1 year. tex2html_wrap_inline87 = tex2html_wrap_inline89 .

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?

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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

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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 tex2html_wrap_inline97 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...

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?

tex2html_wrap_inline225 . 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 ( tex2html_wrap_inline99 ) and wavelength ( tex2html_wrap_inline101 ). 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.

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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 tex2html_wrap_inline105 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 ( tex2html_wrap_inline99 ) and wavelength ( tex2html_wrap_inline101 ).

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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.

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Remember red light has a wavelength of 0.7 tex2html_wrap_inline117 m = tex2html_wrap_inline119 cm and blue light has a wavelength of 0.4 tex2html_wrap_inline117 m = tex2html_wrap_inline123 cm).

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

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Star A is twice as hot as Star B. Star B appears red ( tex2html_wrap_inline127 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 tex2html_wrap_inline117 m/2=0.35 tex2html_wrap_inline117 m. Light with a wavelength of 0.35 tex2html_wrap_inline117 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 = tex2html_wrap_inline135 . 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= tex2html_wrap_inline135 the increase in emitted energy is tex2html_wrap_inline139 or tex2html_wrap_inline141 .

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 tex2html_wrap_inline147 where R is the radius of the star. Thus the luminosity of a star is

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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 tex2html_wrap_inline151 = tex2html_wrap_inline153 = 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 tex2html_wrap_inline139 = tex2html_wrap_inline141 .

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

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tex2html_wrap_inline161 is the original wavelength at which the light was emitted, also known as the rest wavelength. tex2html_wrap_inline163 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 tex2html_wrap_inline163 must be larger than tex2html_wrap_inline161 . 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, tex2html_wrap_inline163 , must be smaller than the original wavelength, tex2html_wrap_inline161 . 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

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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 tex2html_wrap_inline233 to cover. The surface area of a sphere is tex2html_wrap_inline235 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

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where tex2html_wrap_inline241 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

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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:

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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:

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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 tex2html_wrap_inline255 km and it is tex2html_wrap_inline257 km from the Earth. What is the angular size of the Sun on the sky?

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Mike Skrutskie
Wed Feb 21 15:07:04 EST 1996