ASTR001 HW4

Why are extrasolar planets hard to detect directly?

Extrasolar planets are difficult to detect directly because they are small when viewed from Earth, many light-years away. In addition to their tiny sizes, the light from the stars near the planets is much, much brighter than the planets themselves. As a result, the tiny, dim planets get lost in the glare of the stars so that we have great difficulties seeing them

Identical Planets? Imagine two planets orbiting a star with orbits edge-on to the Earth. The peak Doppler shift for each is 50 m/s, but one has a period of 3 days and the other has a period of 300 days. Calculated the two minimum masses and say which, if either, is larger.

If P is in years and a is in AU, we can use p^2=a^3. Planet 1 P=0.0821 yrs=0.041AU Velocity (p1)=[2(pi)r]/[(0.00821 years)(365 days)(24 hours)(60 minutes)(60 seconds)=1.49x10^5 m/s Mass(p1)=M(star)v(star)/v(p1)=M(star)x50/(1.49x10^5)=0.00033557 M(star) Planet 2 P=0.821 yrs=0.88AU Velocity (p2)=[2(pi)r]/[(0.88 years)(365 days)(24 hours)(60 minutes)(60 seconds)=3.2x10^4 m/s Mass(p2)=M(star)v(star)/v(p2)=M(star)x50/(3.2x10^4 m/s)=0.0015625M(star) Planet two is larger

Summarize the current state of knowledge about extrasolar planet masses and sizes. Based on their evidence, is it more likely that smaller planets or larger planets are more common?

We can mainly see large planets because of our methods of detection, however based models we know that small planets are more common. When the data is adjusted for the fact that our observations are skewed by the methods we use, we find that statistically small worlds are more common.

Briefly summarize the planetary properties we can in principle measure with current detection methods.

We can measure orbital eccentricity and inclination using the Doppler method data, in the future the astrometric data may also provide similar info. We use also use these two to get info on planetary mass, Doppler data is the main method once again. Planetary size is provided by transit observations. Planetary Density can be gotten from transit method-discovered planets by using the Doppler method.

Impact Energies. A relatively small impact crater 20 kilometers in diameter could be made by a comet 2 kilometers in diameter traveling at 30 kilometers per second (30,000 m/s). a) Assume that the comet has a total mass of 4.2x10^12 kilograms. What is its total kinetic energy? (Hint: The kinetic energy is equal to (1/2)(m)(v)^2, where m is the comet's mass and v is its speed. If you use mass in kilograms and velocity in m/s, the answer for kinetic energy will have units of joules.) b) Convert your answer from part a to an equivalent in megatons of TNT, the unit used for nuclear bombs. Comment on the degree of devastation the impact of such a comet could cause if it struck a populated region on Earth. (Hint: One megaton of TNT releases 4.2x10^15 joules of energy.)

a) (1/2)(4.2x10^12kg)(30,000m/s)=6.3x10^16 joules b) (6.3x10^16 joules)/(4.2x10^15 joules)=15 megatons of TNT

Relative Motion Practice I. In all the following, assume that you and your friends are in free-float reference frames. a) Bob is coming toward you at a speed of 75 km/hr. You throw a baseball in his direction at 75 km/hr. What does he see the baseball doing? b) Marie is traveling away from you at a speed of 120 km/hr. She throws a baseball at 100 km/hr (according to her) in your direction. What do you see the ball doing? c) José is traveling away from you at 99% of the speed of light when he turns on a flashlight and points it in your direction. How fast will the beam of light be going when it reaches you?

a) Bob sees the ball traveling towards him at 150 km/hr b) You see Marie traveling away from you at 120 km/hr and the ball traveling away from you at 20 km/hr c) It will be at 1% of the speed of light when it reaches you

Gravity and the EM Force. In this problem, we compare the strength of gravity to the strength of the electromagnetic (EM) force for two interacting electrons. Because both electrons are negatively charged, they repel each other through the EM force. Because electrons have mass, they attract each other through gravity. Let's see which effect will dominate. You will need the following information for this problem: The force law for gravitation is F(g)=GM(1)M(2)/(d^2) and G=6.67x10^(-11) *M(1) and M(2) are the masses of the two objects, d is the distance between them, and G is the gravitational constant The force law for electromagnetism is F(EM)=kq(1)q(2)/(d^2) and k=9x10^9 *q(1) and q(2) are the charges of the two objects in coulombs, d is distance between them, and k is constant. The mass of an electron is 9.10x10^(-31)kg The charge of an electron is -1.6x10^(-19) Coul. a) Calculate the gravitational force, in newtons, that attracts the two electrons if a distance of 10^(-10)m (about the diameter of an atom) separates them. b) Calculate the electromagnetic force, in newtons, that repels the two electrons at the same distance. c) How many times stronger is the electromagnetic repulsion than the gravitational attraction between the two electrons?

a) F(g)=GM(1)M(2)/(d^2) F(g)=[6.67x10^(-11)][9.10x10^(-31)kg][9.10x10^(-31)kg]/[10^(-10)m]^2 F(g)=5.523x10^(-52) N b) F(EM)=kq(1)q(2)/(d^2) F(EM)=[9x10^9][-1.6x10^(-19)][-1.6x10^(-19)]/[10^(-10)m]^2 F(EM)=2.304x(10)^(-9) c) 4.17x10^(-42) F(EM) is 6.5 times larger than F(g)

How do rules of geometry differ depending on whether the geometry is flat, spherical, or saddle shaped?

Triangle: Sum of angles is 180 deg (Flat), Sum of angles is greater than 180 deg (Spherical), Sum of angles is less than 180 deg (Saddle-Shaped) Parallel Lines: Remain parallel (Flat), Eventually converge (Spherical), Eventually diverge (Saddle-Shaped) Straightest Possible Path: A straight line (Flat), A piece of a great circle (Spherical), A piece of hyperbola (Saddle-Shaped) Circle: C=2(pi)r (Flat), C<2(pi)r (Spherical), C>2(pi)r (Saddle-Shaped)

Time Dilation with Subatomic Particles. A pi^(+) meson produced at rest has a lifetime of 18 nanoseconds (1.8x10^(-8)s. Suppose a pi^(+) meson is produced in a particle accelerator at a speed of 0.998c. How long will scientists see the particle before it decays? Briefly explain how an experiment like this helps verify the special theory of relativity.

t'=t*sqrt(1-(v/c)^2) t'=(1.8x10^(-8)s)*sqrt(1-(0.998c/c)^2) t'=(1.8x10^(-8)s)*sqrt(1-(0.002)^2) t'=(1.8x10^(-8)s)*sqrt(1-(0.00004)) t'=(1.8x10^(-8)s)*sqrt(0.99996) t'=(1.8x10^(-8)s)*(0.999979) t'=0.000000017999 s You see time passing only slightly slower than if there were no time dilation. In a spaceship with two different light colors on either end of the spaceship, depending on how fast it is going you will see both rays of light at the same time because the light takes the same amount of time to travel to you that it does to reach the same position on the spaceship.

Adding up Asteroids. It's estimated that there are a million asteroids in 1 kilometer across or larger. If a million asteroids 1 kilometer across were all combined into one object, how big would it be? How many 1-kilometer asteroids would it take to make an object as large as Earth? (Hint: You can assumed they're spherical. The expression for sphere of a radius is (4/3)(pi)(r)^3, where r is the radius.)

1,000,000 cubic km=(4/3)(pi)(r)^3 r^3=238,853.503 r=62.05 km (4/3)(pi)(6371km)^3=258,596,602,811 km^3

List five major ideas that come directly from the general theory of relativity.

1. Gravity arises from distortions of spacetime. Mass causes the distortions, and this distortion effects how other bodies move through spacetime. 2. Time runs slowly in gravitational fields. Stronger gravity = slowed time 3. Black holes can exist in spacetime. 4. It is possible for the universe to have a finite volume without having a center or boundaries. 5. Large masses that undergo rapid changes in motion or structure emit gravitational waves that travel at the speed of light.

List five major predictions of the spatial theory of relativity.

1. No information can travel faster than the speed of light (in a vacuum), and no material can even reach the speed of light. 2. If you observe anyone or anything moving by you at a speed close to the speed of light, you will conclude that time runs more slowly for that person or moving object. 3. If you observe two events to occur simultaneously, such as flashes of light in two different places at the same time, a person moving by you at a speed close to the speed of light may not agree that the two events were simultaneous. 4. If you carefully measure the size of something moving by you at a speed close to the speed of light, you will find that its length is shorter than it would be if the object were not moving. 5. If you could measure the mass of something moving by you at a speed close to the speed of light, you would find its mass to be greater than the mass it would have if it were stationary. As we will see, Einstein's famous equation, E = mc 2 , follow from this fact.

The "Near Miss" of Toutatis. The 5-kilometer asteroid Toutatis passed a mere 1.5 million kilometers from Earth in 2004. Suppose Toutatis were destined to pass somewhere within 1.5 million kilometers of Earth. Calculate the probability that this "somewhere" would have meant that it slammed into Earth. Based on your result, do you think it is fair to call the 2004 passage a "near-miss"? Explain. (Hint: You can calculated the probability by considering an imaginary dartboard of radius 1.5 million kilometers in which the bull's-eye has Earth's radius, 6378 kilometers.)

6378km/1500000km=0.004252 0.4252% I think it's hard to call this a "near-miss" due to such a low probability, however for the scope of the universe, considering there is infinite open space for the asteroid to hit, that is a relatively high probability.

How do we know the Kuiper belt and Oort cloud exist? Describe each in terms of location, the orbits and numbers of comets within them, and their likely origins.

A ring of comets that orbit the Sun beyond the orbit of Neptune is the Kuiper Belt. Oort cloud is in the outer solar system. They orbit the Sun. Oort cloud comets are from planetismals that were flung outward after forming between the Jovian planets. Kuiper Belt comets that formed from remnants of the creation of the universe and still remain in the outskirts of the planetary realm.

What is antimatter? What happens when a particle and its antiparticle meet? Why is antimatter always produced along with matter in pair production?

Antimatter: Any particle with the same mass as a particle of ordinary matter but whose other basic properties, such as electrical charge, are precisely opposite. -When a particle and its antiparticle meet, the result is mutual annihilation and the release of energy.

Briefly define asteroid, comet, dwarf planet, meteor, and meteorite. How did the discovery of Eris force astronomers to reconsider the definition of a planet?

Asteroid: A relatively small and rocky object that orbits a star; asteroids are officially considered part of a category known as "small solar system bodies." Comet: A relatively small, icy object that orbits a star. Like asteroids, comets are officially considered part of a category known as "small solar system bodies." Dwarf planet: An object that orbits the Sun and is massive enough for its gravity to have made it nearly round in shape, but that does not qualify as an official planet because it has not cleared its orbital neighborhood. The dwarf planets of our solar system include the asteroid Ceres and the Kuiper belt objects Pluto, Eris, Haumea, and Makemake. Meteor: A flash of light caused when a particle from space burns up in our atmosphere. Meteorite: A rock from space that lands on Earth. Since Eris was more massive than Pluto and additional planets were only slightly smaller than Pluto, so there needed to be some sort of distinction between what size constitutes a planet.

Under what circumstances do we see a comet with a nucleus, coma, and tails? In what direction do the tails point?

Comet approaching Sun, temperature increases, dirty ices of the nucleus begin to sublimate into gas. Gas drags away dust particles. Gas and dust create coma, Jets and gas shooting out at great speeds creating tails. UV light from the Sun ionizes the gas and the solar wind carries the gas outward away from the Sun. Tails point away from the Sun.

According to general relativity, what is gravity and why does Earth orbit the Sun? Describe both the use and the limitations of the rubber sheet analogy for picturing gravity.

Earth is following the straightest path possible through spacetime, but this path happens to go around and around the Sun. Massive objects cause a distortion in space-time, which is felt as gravity The rubber sheet analogy demonstrates how matter influences the "fabric" of four dimensional spacetime in a manner similar to how heavy weights distort a taut two dimensional rubber sheet. The greater the distortion of spacetime.

Briefly describe several observational tests that support general relativity, including Mercury's orbit, examples of gravitational lensing, and measurements of gravitational redshift.

Many possible answers, including: experiments in gravitational time dilation, observations of gravitational redshift, observations of gravitational lenses, evidence from binary pulsars for gravitational waves. Gravitational Lensing is the distortion of the appearance of distant objects due to gravity and occurs because their light passes through regions of space that are curved. Study of Mercury's precession reveals that it precesses more slowly than Newtonian gravity would predict. It takes a general relativity-based approach, accounting for the curvature of spacetime, to correctly predict the precession rate. General Relativistic corrections to time have also been tested by synchronizing clocks that then are flown on airplanes and satellites.

Distinguish between primitive meteorites and processed meteorites in terms of both composition and origin.

Primitive being first - Stony primitive meteorites - rocky mineral with small but noticeable fraction or pure metallic flakes - Remnants from birth of solar system Carbon-rich primitive meteorites - also contain substantial amounts of carbon compounds and sometimes small water Processed - Part of a larger object that "processed" original material of the solar nebula - younger than primitive meteorites - metal rich processed - high density iron and nickel mixed with small amounts of other metals Rocky processed - lower densities and are made of rock with compositions resemble mantles and crusts.

List the six quarks and six leptons in the standard model. Describe the quark composition of a proton and of a neutron.

Quarks: up, down, strange, charmed, top, bottom Leptons: electron, electron neutrino, muon, mu neutrino, tauon, tau neutrino Each have up two different quarks: up quark (electric charge of +1/3) & down quark (-1/3); two up quarks & one down quark make up protons. Two down quarks & one up quark make up a neutron.

Explain why observers in different reference frames will not necessarily agree about the order of two events that occur in different places.

Relativity of Simultaneity - the light flashes (e.g.) that are simultaneous in your reference frame occur at different times in a different observer's reference frame. It depends on your frame of reference.

What is the theory of relatively? How does special relativity differ from general relativity?

Special theory of relativity is Einstein's theory that describes the effects of the fact that all motion is relative and that everyone always measure the same speed of light. It deals only with the special case in which we ignore the role of gravity. General theory of relativity is Einstein's generalization of his special theory of relativity so that the theory also applies when we consider effects of gravity or acceleration. It applies with or without gravity.

List the four fundamental forces in nature, and name the exchange particles for each.

Strong force - gluon Electromagnetic force - photon Weak force - W & Z Gravity - graviton

What is the equivalence principle? Give an example that clarifies its meaning.

The equivalence principle is what says acceleration and gravity are the equivalent. If you were to be in a glass box and hurled into space, you wouldn't notice any difference.

How are meteor showers linked to comets, and why do they recur at about the same time each year?

The gas escaping from comets also carries sand to pebble size pieces of rocky material. Orbiting Earth passes through a particular comet's orbit at the same time each year.

What is spin? What are the two basic categories of particles based on spin?

The inherent angular momentum of a fundamental particle. The two categories are fermions and bosons.

How do the orbits of known extrasolar planets differ from those of planets in our solar system? Why are these orbits surprising?

The orbits of many known extrasolar planets are much more eccentric and much nearer their stars than the jovian planets of our solar system. This is surprising since our planet formation model suggests that planets should have nearly circular orbits and that jovian planets, which require ice to form, should form only farther out in the solar system.

What do we mean by the straightest possible path on Earth's surface?

The straightest possible path on the surface between the points (aka geodesic curves).

Mass Increase. a) A spaceship has a rest mass of 500,000 tons. If you could measure its mass when it was traveling at half the speed of light, what would it be? b) A fly has a mass of 1 gram at rest. How fast would it have to be traveling to have the mass of a large SUV, which is about 3000 kilograms?

a) momentum=mass x velocity p=500,000 tons x 1 m/s p=500,000 tons x m/s [500,000 tons x m/s]/(0.5c)=250,000tons/c b) momentum=mass x velocity p=1g x 1 m/s 3000000 m/s