Exam 2

Rules of the Epicycle theory

-All planets orbit the Earth in purely circular orbits called a deferent. -The Earth is slightly off center of the deferent -The planets do not sit on the deferent but on a purely circular orbit which is centered on the deferent called an epicycle. -The planet orbits the epicycle, while the epicycle orbits the deferent.

Inverse Square Law

A force gets weaker inversely as the square of its distance, or as 1/r2.

Ellipse

A geometric figure that is related to a circle. It is drawn around to points known as foci, such that if you drew a line from one focus to any point on the surface of the ellipse and then from that point to the other focus, the sum of the lengths of those two lines is constant.

Momentum

A property of constant, straight-line motion identified by the symbol p. Its a body's mass times its velocity.

Epicycle

A theory created by Ptolemy to help maintain the geocentric view and explain retrograde motion. This theory had the following basic rules: - All planets orbit the Earth in purely circular orbits called a deferent. - The Earth is slightly off center of the deferent. - The planets do not sit on the deferent but on a purely circular orbit which is centered on the deferent called an epicycle. - The planet orbits the epicycle, while the epicycle orbits the deferent.

Velocity

A vector that tells us not only the how quickly an object moves a particular distance, but also the direction of that motion.

Explain how Galileo's telescopic discoveries contradicted the Ptolemaic theory.

According to the Ptolemaic system, all of the heavenly bodies were perfect, texture-less spheres. Galileo showed that this was not the case when he observed mountains and valleys on the surface of the Moon. The Ptolemaic system held that there was a fixed number of stars in the sky, Galileo's discovery that, when viewed through a telescope, the Milky Way is comprised of a large numbers of stars that are too faint to see with the naked eye contradicts this. Galileo observed the four largest moons of Jupiter and determined not only that they orbited Jupiter, but also that the closer the moon, the shorter its orbital period. This showed that the Jupiter system behaved in the same manner as the planetary system around the sun. This contradicted the Ptolemaic system because it held that the Earth was the center to all motion in the universe, and that the Earth couldn't move because it would then loose all objects that orbited it. When pointing his telescope at Mercury and Venus, Galileo found that they go through phases (e.g., full, gibbous, quarter, crescent and new). This could not be possible in a geocentric model. In the diagram below, we see that in the Geocentric model, one can only have new and crescent phases of the planets.

What is the resolving power of a 25-cm (10-in) telescope at a wavelength of 550nm? What do two stars 1.5 arc seconds apart look like through this telescope?

As mentioned in the problem, D for our telescope is 25 cm = .25 m. Using the equation for resolving power on page 115 in Seeds, and as mentioned in the problem, D for our telescope is 25 cm, we have: a = 11.6/D = 11.6/25 = 0.46 seconds of arc Since this is less than the separation of the stars in the problem (1.5 seconds of arc), the telescope is able to resolve them as two stars.

If you build a telescope with a focal length of 1.3 m, what eyepiece focal length is needed for a magnification of 100 times?

Even though the theoretical resolving power of a 5 m telescope is better than twice as good as the 2.4 m Hubble Space Telescope (see the equation for a above), it is limited by atmospheric disturbances on the Earth. Thus Hubble is able to operate at the diffraction limit (a = 11.6/D = 11.6/240 = 0.023 seconds of arc), while the Earth-bound telescope's actual resolution is much worse (only about 0.5 seconds of arc).

The extra solar planet 51 Pegasi b has an orbital period of 4.23 days. What is its average distance from its parent star? How does this compare to other planets in our solar system?

First convert the days to years: 4.23 days = 0.012 years. Then using P2 = a3 we have P2 = 0.0122 = 0.000144 = a3 thus a = 0.05 AU. This planet orbits its parent star faster and closer than any other planet in our solar system. Mercury is the closest star to the Sun and it only has an average distance of 0.387 AU.

When Tycho observed the new star of 1572, he could detect no parallax. Why did that undermine belief in the Ptolemaic system? In the perfect heavens idea of Aristotle?

Keppler's laws state that the planets do not orbit the Earth, and do not move at constant speeds. This contradicted the beliefs from classical astronomy, where the idea of perfect, constant circular motion around the Earth was held as truth

Electromagnetic Radiation

Known as light, it is comprised of packets of energy called photons that mediate the electromagnetic force. Photons are not truly particles, nor are they waves, but rather a combination of both wave and particle. Under certain circumstances light will behave like a wave, and in other circumstances it will behave like a particle.

Kinetic Energy

Known as the energy of motion, this is a measure of how much energy is contained in a moving object. given by the equation T = (1/2)mv2.

Photon

Light packets that are a combination of both wave and particle, and are the means by which electromagnetic energy is transmitted.

How does the resolving power of the Mount Palomar 5-m telescope compare with that of the 2.4-m Hubble Space Telescope? Why does HST generally still outperform the ground-based telescope?

M = Fo/Fe where Fo = 1.3 m and using our desired M = 100, we have: 100 = 1.3/Fe Fe = 1.3/100 = 0.013 m.

If you lived on Mars, which planets would exhibit retrograde motion like that observed for Mars from Earth? Which would never be visible as crescent phases?

Planets exhibiting retrograde motion: Mars, Jupiter, Saturn, Uranus and Neptune. Planets never seen in crescent phase: all planets outside of the Earth's orbit: Mars, Jupiter, Saturn, Uranus and Neptune.

Parallax

The angular shift in the apparent position of an object (e.g., a planet or star) due to the observer's motion.

If a space probe is sent into an orbit around the Sun that brings it as close as 0.4 AU and as far away as 5.4 AU, is the orbit a circle or an ellipse? What will be its orbital period?

The average orbital distance is 4 AU. Plugging this into P2 = a3 we have P2 = a3 = 64, thus P = (64)^1/2 = 8 years.

Refraction

The bending of light as it passes from one medium into a another. The law that governs how this path changes is Snell's Law: n1sin<theta>1 = n2sin<theta>2

Black Body Radiation

The case where any light emitted by the body is due to the thermal energy contained within that body. In order for this to happen all of the light that happens to fall on that body (i.e., that star), is absorbed by the star (none is reflected or passes through it).

Emission Line

The case where instead of viewing a broad, continuous spectrum, you are instead are viewing light emitted by a specific element or group of elements. In this case, you would see the individual bright lines (where each line is a specific color), corresponding to the energy levels of the corresponding to the elements or molecules. Like the absorption lines, these emission lines are a finger print identifying the elements & molecules that are present.

Absorption Line

The case where you are viewing a broad, "white" light source and for very specific colors (corresponding to the energy levels of the atom) there are gaps in its light (because the photons for that corresponding color were absorbed before they had a chance to reach your telescope). The absorption lines you view are a finger print identifying the elements & molecules that are present.

Deferent

The circular path along which a planet's epicycle was thought to orbit the Earth.

Why is the binding energy of an electron related to the size of the electron's orbit?

The electrons in an atom can only orbit the atom in a specific set of levels (orbitals). As a result, an atom can only absorb or emit a photon whose energy (E = hc/l) matches the energy difference between the level in which the electron is starting, and the level in which it ends up. From this equation we see that the energy of this photon determines its wavelength l.

What evidence suggests that early human cultures observed astronomical phenomena?

The evidence that supports the observation of astronomical phenomena by early civilizations includes the alignment of buildings with astronomical events such as solstices and equinoxes, artwork depicting astronomical events (e.g., comets), and the writings in the Maya codices.

Semimajor Axis

The line that runs from the center of an ellipse to its edge, along its longest diameter.

Semiminor Axis

The line that runs from the center of an ellipse to its edge, along its shortest diameter.

Geocentric Universe

The model for the universe where the Earth is the center of all motion, and all celestial bodies orbit around the Earth.

Heliocentric Universe

The model for the universe where the sun is the center of the solar system. All planets and many other bodies orbit the sun, however, this model allows for bodies to orbit objects other than the sun (e.g., the moons of Jupiter orbiting Jupiter).

Retrograde Motion

The motion where as planets TRAVEL "forward" across the background stars, then (over a period of time lasting a few weeks) stop, move "backwards" a short distance, stop again and resume their forward motion.

Frequency

The number of wave crests that pass a fixed point in space per second. They symbols used for frequency are the Greek letter v or f.

Objective

The primary mirror or lens in a telescope.

Acceleration

The rate at which the velocity of a body changes.

Why do hot stars look bluer than cool stars?

The reason for this is that the light emitted by the star is due to the thermal energy contained within star and is governed by Wien�s Law: lmax = 3,000,000/T This says that the temperature is inversely proportional to the wavelength. That is, if the temperature is larger, then it's being in the denominator makes the wavelength shorter. Looking back at the spectrum chart, we see that shorter wavelengths are closer to blue in color.

Escape Velocity

The velocity necessary to escape the Earth's gravity completely (not just to achieve orbit). Escaping the Earth's gravity is required if you want to TRAVEL to the Moon, other planets or beyond.

Resolving Power

This is a measure of the finest detail that the telescope can help you see

Potential Energy

This is the energy that a body has available to it if it is in a field such as a gravitational field. Its identified by the symbol V, and may be calculated using the formula: V = mgh.

Conservation of Energy

This states that T + V = constant. For example, the kinetic energy that an object has when it strikes the floor is not created, but rather converted from the potential energy.

Eccentricity

This value measures how much the ellipse deviates from a perfect circle (basically, how long it is). Its calculated by dividing the distance of the focus from the center by the semimajor axis. The of a circle is zero.

What consequence would this have for someone trying to walk on this asteroid?

To get an intuitive understanding of this velocity, I would convert it to more familiar units 9.3 m/s = 33.5 km/hr = 21 miles/hr. This is the speed of a person at a dead-run. It would mean that a person wouldn't be at risk of escaping the asteroid's gravity by walking or jumping, however, they would likely bounce very high off of the ground with each step.

Uranus orbits the Sun with a period of 84.0 years. What is its average distance from the Sun?

Using P2 = a3 we have P2 = 164.82 = 27,159 = a3, thus to get a, take the cube root of 27,159: a = 30 A.U..

Sedna, a dwarf planet in our solar system, has a semimajor axis of 526 AU. What is its orbital period?

Using P2 = a3 we have P2 = 5363 = 145,531,576 thus P = 12,063 years.

If a star has a surface temperature of 20,000 K, at what wavelength will it radiate the most energy? Is it a cool or hot star?

Using T = 20,000 K and Wien's law: lmax = 3,000,000/T = 3,000,000/20000 = 150 nm

Human body temperature is about 310 K (3.0 x 10 ^2 K). At what wavelength do humans radiate the most energy? In which part of the electromagnetic spectrum do we emit?

Using T = 310 K and Wien's law: lmax = 3,000,000/T = 3,000,000/310 = 9,700 nm = 9.7 mm Looking back at the spectrum chart, we see that this is in the infrared.

What is the escape velocity of Asteroid433Eros?

Using the escape velocity formula from the lesson (click here for the formula): Plugging in G = 6.67 x 10-11 m3/s2kg and M = 7.2 x 1015 kg, and r = radius of moon = 1.1 x 104 m, we get 9.3 m/s.

What is the wavelength of radio waves transmitted by a radio station with a frequency of 100 million cycles per second?

We are given the frequency f = 100 million Hz = 100 x 10 6 Hz = 1 x 10 8 Hz (where 1 Hz / 1 cycle per second which is notated as 1 s-1) Using the wavelength equation, and using c = 3 x 108 m/s: l = (3 x 108 m/s)/( 1 x 10 8 s-1) = 3 m.

What is the orbital velocity of the Earth around the Sun?

We can use the equation above for orbital velocity for a circular orbit. The difference this time is that the M we will use is not the mass of the Earth, but rather the mass of the Sun. r in this case is the distance to the sun. So plugging in M = 2 x 1030 kg and r = 1AU = 1.50 x 1011 m, gives us v = 29,800 m/s (a very fast speed indeed!)

How does the light-gathering power of one of the 10-m Keck telescopes compare with that of the human eye? Assume that the pupil of your eye can open to a diameter of about 0.8 cm in dark conditions

da2/db2 and using the diameter of the Keck telescope as da = 10 m = 10,000 mm, and the diameter of your pupil to be db = 0.8 cm = 8 mm. We then have: da2/db2 = (10000)2/(8)2 = 1,562,500 times more light.

Wavelength

his is the distance between two crests of a wave (or two troughs). The symbol for wavelength is the Greek letter l (lambda).

Infrared observations of a star show that the star is most intense at a wavelength of 2000 nm. What is the temperature of the star's surface?

lmax = 3,000,000/T Except this time, we know lmax = 2000 nm, so putting this into the equation gives: 2000 nm = 3,000,000/T Dividing both side by 3,000,000 gives: 1/T = 2000/3,000,000 T = 3,000,000/2000 = 1500 K