Name:_______________________________ Date:______________________
ASTR 1303/1304 – Basics of Astronomy (Formulas & Math)
ASTRONOMICAL DISTANCES & LAWS OF MOTION (10 points)
Student Learning Outcomes
Upon successful completion of this assignment, students will be able to:
• Understand the vast scale of the Universe in terms of cosmic distances, objects and events.
• Apply special units used in astronomy to measure distances between cosmic objects.
• Understand and apply the Laws of Motion in terms of elliptical orbits of planets, asteroids, or comets; the Universal Law of Gravitational Mutual Attraction; Circular Orbits; and Escape Velocity.
Image Credit: ThePlanets.org
Part 1 – Units of Measurement in Astronomy (2 points)
The distances in astronomy are so great that using miles or kilometers is insufficient, because the numbers become too large to handle. When dealing with the great distances within our solar system, we use Astronomical Units (AU), which are multiples of the distance from the Earth to the Sun.
However, that measurement – the Astronomical Unit – is not large enough when considering the distance to other stars or galaxies. In that case, distance is stated in light years (ly), which is how far light travels in a year based on the speed of light.
A third unit that is preferred by astronomers is the parsec (pc), which is even greater than a light year, and is based on the angle of parallax.
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A. The unit of measurement that is convenient for stating the large distances within our solar system is the Astronomical Unit (AU), defined as the average distance of the Earth to the Sun. That distance is approximately 150 million kilometers (150,000,000 km) or 93 million miles (93,000,000 miles).
Some distances within the Solar System in AU are:
• By definition, the Earth is 1 AU from the Sun.
• Mars is 1.52 AU from the Sun.
• The distance between planets depends on their orientation in their respective orbits. Mars can be between 2.52 AU and 0.53 AU from Earth, depending on their relative positions.
• As reported by NASA on Monday, February 27, at 8:30 a.m., the spacecraft Voyager 1 had traveled over 159.2 AU from the Sun, the furthest a manmade object has ever traveled. The distance of 156.29 AU equates to 23,815,412,000 km or 14,798,206,000 miles.
B. Although the astronomical unit is fine for our solar system, it is not sufficient to designate the greater distances to other stars and galaxies. Instead, the light year is used as a unit of measurement. A light year is the distance light travels in one year, based on the speed of light (which is a constant, represented by “c”). The speed of light is approximately 300,000 kilometers per second or 186,000 miles per second. This makes one light year equal to: • 9,500,000,000,000 km = 9.5 x 1012 km (about 10 trillion km) • 5,900,000,000,000 miles = 5.9 x 1012 miles (about 6 trillion miles)
• A light year also equals 63,241 AU.
Keep in mind that since it takes a specific time for light to travel a particular distance, we are seeing the light of celestial objects when it left those objects – as they were in the past. So, looking at the stars is, essentially, time travel!
Common large distances in space, measured in light years, include:
• Proxima Centauri, the nearest star in our Milky Way galaxy (after the Sun) =
4.22 light years away. (It took 4.22 years for the light to reach our eyes.)
• The Milky Way galaxy = about 100,000 light years in diameter. (It would take 100,000 years to travel across the Milky Way at the speed of light, roughly 67,000 miles per hour.).
• The Canis Major Dwarf Galaxy, the closest galaxy to us = approximately 25,000 light years away. (It took 25,000 years to reach our eyes!)
• The size of the Universe = estimated to be between 93 billion and 156 billion light years across.
• The time it takes light to travel from the Sun to the Earth (1 AU) = approximately 499 seconds or 8.32 minutes (about 8 minutes). You could say that 1 AU equals 8.32 light minutes. This means that when we see sunlight, we are seeing it in the past, as it was 8.32 minutes ago.
C. Astronomers prefer to use the parsec (pc) to designate extremely great distances in space, because it relates to the geometric method they commonly use to establish distance. Parsec stands for parallax of one arcsecond, which is 1 AU divided by the tangent of an arcsecond. One parsec is equal to approximately 3.262 light years.
Although astronomers prefer to use the parsec, other scientists and the general public commonly use the light year to designate large distances. The reason is that it is difficult for most people to visualize distance in terms of such a small angle, and since one parsec equals 3.26 ly, there isn’t much advantage in using the term as a unit of measurement unless doing astronomical research. However, those working in Astronomy need to be familiar with this unit of measurement.
Image Credit: EarthSky.org
Questions: – REMEMBER TO SHOW ALL CALCULATIONS!
1. Calculate the distance of a light year, based on the speed of light in kilometers. (Show calculations.) (2 points)
Part 2 Scientific Notation or Power of Ten (2 points)
Scientific Notation is used for extremely large or small numbers in science, whether Astronomy, Biology, Chemistry, or any of the other sciences. It helps to simplify large or small numbers and make computations easier.
The procedure is to take a large (or small) number and put it into the form of the ones unit, including any significant digits (i.e. any number other than zero), then multiplying times ten to an exponent (the exponent shows how many places to move the decimal point to the left).
For example: 5,104,000 in Scientific Notation = 5.104 x 106
(Note: some calculators may present this as 5.104E6.)
NOTE: Any questions involving large numbers must have the final answers converted into Scientific Notation for the full point. And include units!
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Questions:
2. The average radius of the Earth is approximately 6,371 km. In scientific notation, the average radius of the Earth is… (1 point)
3. The average distance from Jupiter to its closest moon, Io, is approximately 421,700 km. In scientific notation, the average distance from Jupiter to Io is… (1 point)
Part 3 Distance Unit Conversions (2 points)
• To convert light years to AU, multiply number of light years by 63,241.
• To convert AU to kilometers, multiply number of AU by 150,000,000.
• To convert light years to kilometers, multiply number of light years by 9,500,000,000,000.
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Questions:
4. Distance from the Sun to the nearest star, Proxima Centauri, is 4.2 light years. What is the distance from the Sun to the nearest star in AU? (Show calculations.) (1 point)
5. The Canis Major Dwarf Galaxy is the next galaxy closest to us at 25,000 light years away. How far is that in km? (Show calculations.) (1 point)
Part 4 Kepler’s Three Laws of Planetary Motion (2 points)
Description: The figure below shows several positions of a comet traveling in an elliptical orbit around the Sun. Four different segments of its orbit (A – D), and the corresponding triangular shaped area swept out by the comet, have been shaded in gray. Assume that each of the shaded triangular segments have the same area. (Note: Refer to Lecture PowerPoint: Chapter 2 – The Copernican Revolution.)
6. Ranking Instructions: Rank the speed of the comet during each segment (A – D) of the orbit, from slowest to fastest. (1 point)
Ranking Order: Slowest 1_______ 2_______ 3_______ 4_______ Fastest
-OR- The speed of the comet during each of the segments would be the same___ (indicate with check mark).
7. Explain which of Kepler’s Laws of Planetary Motion applies, and state why
(Kepler’s Law 1, 2, or 3?). (1 point)
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____________________________________________________________________ Part 5 Newton’s Laws of Motion (2 points)
|
F = GMm r2 |
Newton’s Law of Universal Mutual Gravitation:
This formula shows that any two bodies (M = object with greater Mass, m = object with less mass) attract each other through gravity (G = Gravitational constant = 6.67 x 10-20 km3/kg-s2), with a Force (F) proportional to the product of their masses (M * m), and inversely proportional to the square of their distance (r or R = radius).
This formula means that everything has gravity, from a speck of dust to a massive planet. More massive objects exert gravity over less massive objects; a high-mass planet’s gravity would overtake the mutual gravity of two lesser objects on its surface. The farther two objects are from each other, the less the force of gravity.
Your weight is your mass (how much space you take up) that is gravitationally pulled by the larger Mass of a planet. The more Mass the planet has, the more the gravitational pull on you would be, and the more you would weigh; the less Mass a planet has, the less the gravitational pull, and you would weigh less.
Circular Orbit:
This formula means that one must accelerate laterally around a planet at the same velocity as the planet’s rotation to stay in orbit (e.g. the International Space Station).
(Note: It is incorrect to use the terms “zero gravity” or “weightless”; there is simply less of a gravitational pull the farther away from an object one is. A better term to use is “microgravity”; while an astronaut in a space station is in “freefall” due to the space station accelerating laterally around the Earth at the same speed that the Earth is rotating.)
Escape Velocity:
This formula means that one must accelerate at a higher velocity than a planet’s rotation to escape the planet’s gravity and travel to space (e.g. rocket).
Questions:
8. Which formula would be used for a satellite orbiting the Earth? (1 point)
a. 
|
F = GMm r2 |
b.
c. F = m a
d. 
9. If the Earth’s mass is 6.0 x 1024 kg, its radius is 6,371 km, and G = the
Gravitational constant = 6.67 x 10-20 km3/kg-s2, determine the velocity of a rocket to travel to space. (Choose a formula from #9 above and show calculations; continue on back page, if needed). (1 point)
(Blank page for calculations, if needed.)
