Student Participation (Spring 2023)
Contents
Student Participation (Spring 2023)#
Students will work in pairs to produce summaries of each class meeting. These summaries should list the main topics discussed and emphasized ideas/examples during the respective lecture. Students are encouraged to be as detailed as possible without being overly verbose. The summaries should be submitted as a pull request from your forked repository before the next class meeting.
Jan 10#
Class Overview#
Introductions
Dr. Quarles told us about his research.
He has contributed to several exoplanet discoveries with Kepler and even an exomoon candidate.
He studies planets with 2 stars in particular, and his latest research has investigated how many moons that could potentially orbit the Earth. The answer is 3!
He has studied how obliquity can affect a planet’s climate.
Syllabus
The class syllabus provides us with a roadmap for the course.
The course is structured for the students to learn through class participation/activities and homework.
Student learning will be assessed through exams and in-class presentations.
Course website
Class notes are provided on the course website, which includes web urls (for jargon) and video explanations of some topics.
GitHub accounts
GitHub is a convenient way to store and share information.
Create GitHub account and fork repository.
Jan 12 (Freeman)#
Intro to Cosmology I#
1.1 What is Cosmology?
Cosmology is the science of the origin and development of our universe and other universes.
Need to know knowledge of the universe because there is chaos in the world.
1.1.1 How does the Universe work?
Einstein-one of the most famous physicist in modern physics
To answer why the Sun is responsible, many cultures deify the Sun and create rituals (Whenever you see the Sun in the sky, you get warm)
There were some other stars that appeared to wander through the night sky, which they called the planets.
The heliocentric description of how the Universe works would eventually overtake the geocentric view because of technological advancements that improved the observations.
1.1.2 Where did everything come from?
Another purpose of cosmology is to explain the origin of everything.
Ancient cultures developed their stories using observations in their daily lives.
Galaxies form through clusters of stars interacting with a special substance called dark matter, but the details of the dark matter are still being determined.
Larger structures form clusters of galaxies that all originated from an immense expansion of the Universe from a big bang event.
1.1.3 How does the origin story explain the current structure?
Around 100 years ago, the Universe consisted of only the Milky Way, and it was static (neither expanding nor contracting).
The Universe was much larger than assumed and it was expanding. This meant that the Universe could have a beginning and thus, an origin story was not completely a human invention.
A changing Universe means that we can map effects to potential causes using observations of the faintest galaxies.
Dark matter affects the formation and rotation of galaxies.
Dark energy makes up around 70% of everything
1.2 Cosmology among many cultures
1.2.1 Ancient Egypt
Early Egyptian culture developed a cosmology during the 2nd and 3rd millennia B.C.
The key to the rediscovery of ancient Egyptian culture was with the Rosetta Stone, which was a monument made of stone or wooden slab
The pharaohs (i.e., Egyptian kings) used some of the Egyptian gods (e.g., Osiris, Isis, Ptah, horus and Anubis) to establish their right to rule. From there, the monuments of Egypt were built, which included the pyramids and temples.
Osiris controlled fertility, agriculture, and the afterlife representing the forces of order, while Isis represented motherhood and is the sister-wife of Osiris. The brother of Osiris was Set, who controlled the deserts, violence, and foreigners representing the forces of chaos, with his sister-wife Nephthys who is associated with mourning, night, childbirth, and service to the temple.
Jan 17 (Scarlett & Norlund)#
Intro to Cosmology II#
1.2.3. Aristotelian Cosmology
Aristotle’s writings influenced progress of science for 1000 years because they were systematic and had brought forth a range of knowledge.
Most important source of cosmology from Aristotle was his work De Caelo, in which the longest chapter was over the celestial sphere and how the Earth rotates at the center of the universe.
Aristotle didn’t believe that the Earth rotated or orbited anything, but was persuaded that stars would be subject to parallax shifts if the Earth did rotate and move in an orbit.
He thought the universe to be built layer upon layer over a spherical Earth.
He believed the Sun was a bright big planet.
Possibly thought that planets were self-luminous.
In Aristotle’s Metaphysics he accepted the theory of Callippus’ 55 concentric spheres.
Callippus viewed the universe being made of spherical shells in which some of the shells carry planets.
The first sphere was “heaven” which had a perpetual circular movement that it shared with the other spheres.
These spheres were moved by “gods”.
1.2.4. Ancient China
Most of the creation story is over Pangu and the cosmic egg.
Astronomy was related to the government/state with the positions of the imperial astronomer and the imperial astrologer.
Astronomer observed planetary motions (for good and bad omens)
Astrologer observed the weather (clouds)
These positions lasted for over 2000 years and they headed large staffs as they were the keepers of time.
Water clocks were developed to improve measurements relative to the motion of the heavens.
4th century BCE onwards saw an increase of astronomical works
Kai Thien cosmology had the heavens as a hemispherical bowl over the hemispherical earth.
Oldest surviving description of the heavens as completely spherical is by Zhang Heng.
The sphere’s circumference is divided into 365.25 units corresponding to the distance the Sun moves in one day.
Zhang was the first to turn an armillary sphere by a water wheel.
By 960-1126 CE there were two observatories in the capital that would compare observations and results.
When a new imperial astronomer took their position they discovered that the previous two sets of astronomers had copied each others’ reports.
1.2.5. Pre-Columbian America
Limited information on pre-Colombian America, and is subject to change when new information is discovered.
Olmec, Zapotec, Aztec, Maya had a similar cosmology of a layered universe.
Layers went from Earth as the bottom to the Moon, clouds, stars, the Sun, Venus, comets, and on until layer 13 which contained the domain of “god”.
The Mayans developed ways to analyze astronomical events using math, and they produced a series of books and only 5 survived Spanish conquests/neglect.
Each book was a piece of folded bark about 6.7 m long
One book contained a lunar, solar, and a Venus calendar.
Another book the Dresden Codex from the 13th to 14th century contained drawings of the Mayan gods
The Mayans had any gods including gods of rain, the Moon, death, reactions, maize, and the Sun (in rough order of frequency).
Their almanacs contained information on various things net-making, fire drilling, and marriage.
1.2.6 Medieval and Early Renaissance Europe
The beliefs and practices of th the early Christian Church had an impact on the development of cosmology in Medieval and Early Renaissance Europe centuries later.
Cosmology went from having a flat Earth to a spherical Earth, but only after a lot of discourse as in the book of Genesis “waters were suspended above the firmament.”
Boethius adopted Aristotle’s cosmology and proposed that the universe was governed by cause and effect.
In the 13th century, Richard of Wallingford constructed a mechanical clock model of the universe which purportedly showed planetary movements and changing tides.
Clock struck on a 24 hour astronomer system rather than common seasonal hours.
It had spiral gears, and an oval wheel to give a carefully calculated variable velocity for the Moon’s motion around an astrolabe dial.
The Astrarium was later constructed by Giovanni de Dondi which had a seven-sided frame, with a dial for each planet, the Sun, and the Moon.
The Astrarium used a Ptolemaic diagram for the planetary mechanism which failed to represent the universe as a single system.
1.2.7 Galileo, the Telescope, and Cosmology
By the time of Copernicus, Aristotelian cosmology was begin seriously questioned, where the cometary observations worked against Aristotle.
Geocentrism vs. Heliocentrism
Kepler introduced new phenomena which weren’t explainable by Aristotelian cosmology which were solar coronas and the appearance of new stars.
Galileo used his telescope to discover mountains on the moon, that the Milky Way was separate stars, absence of parallax in comets, and that Jupiter had moons.
Galileo saw the phases of Venus which caused problems because the various collinear arrangements of the Sun, Venus and the Earth are limited.
Only Ptolemaic, Copernican, and Tychonic systems allowed Venus to lie in between the Earth and the Sun, but only the Copernican model could show the full set of phases for Venus.
1.2.8 Cosmology in Bentley and Newton
1685 Newton had written De mundi systemate to show that stars lie at greater distances from the Sun than what was thought.
This photometric method compared the brightness of the Sun, with that of the star, and on the inverse-square law of photometry.
Newton used this method to discover that the distance of Sirius was a million times that of the mean distance of the Sun from the Earth.
He observed that the gravitational attraction of stars on one another was minimal or else everything would collapse in on itself.
1692 Richard Bentley asked Newton for advice of Boyle lectures over what happened if matter were spread uniformly throughout space and allowed to move under gravity
If space were limited, then it would fall into one large spherical mass.
If space were infinite, then all the masses could stay in place.
This problem would later be known as Bentley’s paradox.
1.2.9 The Renewal of Cosmology
Almost all modern theories and the overall structure of the universe can be traced back in part to the ideas of Albert Einstein
Einstein’s special and general relativity developed the idea that a system of bodies should be independent of the way an observer studying the bodies is moving.
In Special Relativity he introduced that the speed of light is a constant in a vacuum and doesn’t respond to relative motion.
The curvature f space-time in General Relativity developed into a feature such that a particle moving freely under gravity follows a geodesic (the shortest line between two points in curved space-time).
Introduced cosmological constant saying that the universe was static even though it’s not
1.3 Differences in Historical, Scientific, and Mathematical Cosmology
1.3.1 Connections between Astronomy and the State
Astronomy typically began as a way of understanding the world so that agricultural production could increase.
Individuals were trained in Astronomy to advise rulers to act when necessary as they knew the best times to plant and harvest crops.
Astronomy was indistinguishable from magic
Religion would seek to explain why things exist in the universe and how the universe works
Historical cosmology is associated with myth making
Stories would have to be changed based on public opinion.
European renaissance period
Astronomy is correlative which means they fall within the natural pattern recognition of human observers.
If A only happens when B is present, we might infer that B causes A to occur.
People thought astrology predicted people’s lives and lived by horoscopes
Telescope was developed for nautical purposes
Galileo was hired to make a telescope for a noble so he could be a creep and look at people across a lake.
Change from empiricism to the fore which allowed cosmology to be corrected
1.3.2 Models of the Universe
In 1918 Sir Arthur Eddington wrote a report later called Mathematical Theory of Relativity.
No solution for field equations that described the universe a being without matter
Cosmological constant was a Trojan horse bc it had the answer to an undiscovered phenomenon
Universe has three separate fates
Closed, gravity dominated leading to Big Crunch
Open, whatever is responsible for the cosmological constant wins and the universe continues to expand
Static, gravity and comic repulsion balance out
Jan 19 (Carey & Kennedy)#
Special Theory of Relativity I#
Jan 24 (Freeman & Leonard)#
Special Theory of Relativity II#
2.3.2 Proper Length and Length Contraction
Both time dilation and the downfall of simultaneity contradict the Newtonian notion of absolute time.
Length contraction is the relativistic phenomenon where the length of a moving object is measured to be shorter than in its rest frame. It occurs only in the direction of motion, and its effect is significant only when the object is moving at speeds close to the speed of light.
Only lengths or distances parallel to the direction of the relative motion are affect by length contraction; distances perpendicular to the direction of the relative motion are unchanged.
2.3.3 Time Dilation and Length Contraction are Complementary
Time dilation and length contraction are not independent effects because they come together from the Lorentz transformation equations. They are complementary.
2.3.4 The Relativistic Doppler Shift
In 1842 Christian Doppler showed that as a source of sound moves through a medium (e.g., air), the wavelength is compressed in the forward direction and expanded in the backward direction. (Doppler Effect)
Doppler deduced that the difference between the observed wavelength \(\lambda_{\rm obs}\) for a moving source of sound and the rest wavelength \(\lambda_{\rm rest}\) measured in the laboratory for a reference source is related to the radial velocity \(v_r\)
If the source of light is moving away from the observer \((v_r>0)\), then \(\lambda_{\rm obs} > \lambda_{\rm rest}\). This shift to a longer wavelength is called a redshift.
Similarly, if the source is moving toward the observer \((v_r<0)\), then there is a shift to a shorter wavelength, or a blueshift.
2.3.5 The Relativistic Velocity Transformation
Because space and time intervals are measured differently by different observers in relative motion, velocities must be transformed as well.
2.4 Relativistic Momentum and Energy
According to the Principle of Relativity, if momentum is conserved in one inertial frame of reference, then it must be conserved in all inertial frames. (relativistic momentum vector)
2.4.1 The Derivation of \(E=mc^2\)
Mass and energy are two sides of the same coin; one can be transformed into the other.
2.4.2 The Derivation of Relativistic Momentum
The relativistic momentum vector has the form: p=fmv , where is a relativistic factor that depends on the magnitude of the particle’s velocity, but not its direction.
The y and y’ components of each particle’s velocity are chose to be arbitrarily small compared to the speed of light c.
Jan 26 (Norlund & Carey)#
General Relativity I#
3. General Relativity and Black Holes
3.1. The General Theory of Relativity
Newton’s law of universal gravitation between two bodies of mass \(m_1\) and \(m_2\) is \(F=\frac{Gm_1m_2}{r_{12}^3}r_{12}\), which depends on the displacement vector \(r_{12}=r_2-r_1\), and its magnitude \(r_{12}\).
This predicted the existense and position of the planet Neptune in 1846, but it couldn’t account for the large shift in the orientation of Mercury’s orbit.
The major axis of Mercury’s orbit swings around the Sun in a couter-clockwise diretion relative to stars due to the gravitational influences of the other planets.
The angular position of perihelion shifts at a rate of 574” per century.
3.1.1. The Curvature of Spacetime
Relativity deals with a unified spacetime, where both space and time must be described in a new way near an object.
Distances between points in the space surrounding a massive object are altered so that we can describe the spacetime as becoming curved through a fourth spatial dimension that is perpendicular to all of the usual three spatial dimensions. Consider this analogy:
Imagine four people holding the corners of a rubber sheet, stretching it tight and flat. This represents the flatness of empty space and exists in the absence of mass.
Also imagine that a polar coordinate system is painted on the sheet with evenly spaced concentric circles spreading out from its center.
Lay a heavy bowling ball (representing the Sun) at the center of the sheet, and watch the indentation of the sheet as it curves down and stretches in response to the ball’s weight.
Closer to the ball, the sheet’s curvature increases and the distance between points on the circles is stretched more.
The fourth spatial dimension has nothing at all to do with the role played by time as a fourth nonspatial coordinate in the theory of relativity.
Mass has an effect on the surrounding space.
Mass acts on spacetime, telling it how to curve.
Curved spacetime acts on mass, telling it how to move.
In general relativity, gravity is the result of objects moving through curved spacetime, and everything that passes through is affected (even massless particles such as photons).
Nothing can move between two points in space faster than light.
Time runs more slowly in curved spacetime.
3.1.2. The Principle of Equivalence
Laws of physics are the same in all inertial reference frames.
Accelerating reference frames are not inertial, because the introduce fictitious forces that depend on the acceleration.
For example, an apple at rest on the seat of a car will not remain at rest if the car suddenly brakes to a halt.
Consider two massive, charged objects separated by a distance \(r\): one of mass \(m\) and charge \(q\), while the other is of mass \(M\) and charge \(Q\).
The magnitude of the acceleration \(a_g\) of mass \(m\) due to the gravitational force is found from $\(ma_g=\frac{GMm}{r^2}\)\( while the magnitude of the acceleration \)a_e\( due to the electrostatic force is found from \)\(ma_e=\frac{qQ}{4\pi\epsilon_or^2}\)$
The mass \(m\) is an inertial mass and measures the object’s resistance to being accelerated.
The masses and charges on the right are numbers that couple the masses or charges to their respective forces and determine the strength of these forces.
To distinguish between the inertial and gravitational mass of each object \(ma_g\) should be written as $\(m_ia_g=\frac{GM_g}{r^2}m_g\)\( \)\(a_g=\frac{GM_g}{r^2}\frac{m_g}{m_i}\)\( and \)ma_e\( should be written as \)\(a_e=\frac{qQ}{4\pi\epsilon_or^2}\frac{1}{m_i}\)$.
If the gravitational mass were chosen to be twice the inertial mass, the laws of physics would be unchanged because the gravitational constant \(G\) would be assigned a new value that is only one-fourth as large.
At a given location, all objects experience the same gravitational acceleration.
Einstein realized that if an entire laboratory were in free-fall, with all of its contents together, there would be no way to detect its acceleration.
An inertial frame cannot even be defined in the presence of gravity because it is equivalent to an accelerating laboratory.
The Principle of Equivalence: All local, freely falling, nonrotating laboratories are fully equivalent for the performance of all physical experiments.
Nonrotating laboratories are called local inertial reference frames.
Jan 31 (Scarlett & Leonard)#
General Relativity II#
3.13. The Bending of light
Imagine an elevator suspended above the ground byt a cable. A ptoon leaves a horizontal flasligt at the same isntasnt the cable holding the elevator is cut.
The area is now in free fall, and it is now a local inertail refrence frame.
Using the Equivalence principle, The ligts path across the elevator to be a straight horizontal line
in the refrence frame of the ground the elevator is falling under the influecen of gracvity
This means from this frame of refrence the light would be moving through a curved path, becasue that is the fasted point.
Important formulas
The angle of deflection of the photon is very slight. The photon does not follow a circular path, but we can use a best-fitting circle of radius as an approximation of the path measured by the ground observer. DO the math later
3.14. Gravitational Redshift and Time Dilation
Important formulas
Slow free-fall speeds involved, expected increase in frequency. \( \frac{\Delta v}{v_{o}}=\frac{v}{c}=\frac{gh}{c^2}\)
Gravititaonal Redshift \(\frac{\Delta v}{v_{o}}=-\frac{v}{c}=-\frac{gh}{c^2}\)
Exactly result for the gravitational redshift
Time in Gravity
In a strong field
In a weak field
We must conclude that time passes more slowly as the surrounding spacetime becomes more curved, which is called gravitational time dilation.
3.2. Intervals and Geodesics
An event in spacetime unifies the concepts of space and time because it is expressed using four coordinates \((x,y,z,t)\).
The appearance of the gravitational constant G and the speed of light c symbolizes the extension of special relativity to include gravity. It is beyond the scope of this course to explicitly evaluate this equation.
3.2.1. Worldlines and Light Cones
Spacetime diagrams represent a 2D physical plane with \((x,y)\) a perpendicular axis denoting time \(t\) . The third spatial dimension \((z)\) is ignored for simplicity.
The path of an object as it moves through spacetime is called its Worldline
In principle, every event in spacetime has a pair of light cones extending from it.
Two events can influence each other only if their light cones overlap in the past or future.
Your entire future worldline (i.e., your destiny) must lie within your future light cone at every instant. Light cones represent horizons in space that separate the knowable from the unknowable.
3.2.2 Spacetime intervals, Propoer Time, and Proper Distance
the distance \(Δ\ell\) measured along the straight line between two points in flat space is defined by
The analogous measure of “distance” in spacetime is called a Spacetime interval
the interval \(Δs\) measured along the straight worldline between two events in flat spacetime is defined by
The definition of the interval is very useful becasue \((Δs)^2\) is invariant under a Lorentz transformation.
if \((Δs)^2 > 0\), then the interval is timelike
light has enough time to travel bewteen the two events.
the time is measured is called proper time \(Δτ\) and is calculated by $\(Δτ=\frac{Δs}{c}\)$
if \((Δs)^2=0\), then the interval is lightlike or null
light has just enough time to travel bewteen events A and B
if \((Δs)^2 < 0\) , ten the interval is spacelike.
Light doesnt have enough time to travel
but it has a Proper distance \(\mathcal{L}\)
3.2.4. Curvred Spacetime and Schwarzchild Metric
Feb 2 (Kennedy & Freeman)#
Black Holes I#
3.2.4 Curved Spacetime and the Schwarzschild Metric
In spacetime curved by mass, even straightest worldlines curve
Geodesics- Straightest possible worldlines
Massless particles follow null geodesics
Einstein realized: paths followed by freely falling objects are geodesics
3 fundamental features developed:
Mass acts on spacetime, telling it how to curve.
Spacetime acts on mass, telling it how to move.
Any freely falling particle follows the straightest possible worldline through spacetime.
For symmetry, use spherical coordinates:
Metric b/w 2 nearby points in flat space-
Metric in flat spacetime-
Step one- calculate how this object acts on spacetime
In curved spacetime around a massive sphere:
The origin (which is inside the sphere) should not be used as a point of reference… Instead imagine a series of nested concentric spheres centered at the origin.
The surface area of a sphere can be measured w/o approaching the origin, so the coordinate \(r\) will be defined by the surface of that sphere having an area \(4\pi r^2\).
As an object moves through the curved spacetime, its coordinate speed is just the rate of change of the spatial coordinates.
At large distances spacetime is “flat”
Gravitational time dialation is given by: (3.12)
Karl Schwarzschild- solved Einstein’s field equations
His solution is ONLY VALID in empty space outside the object
Contains the “curvature of space”
Distance \(d\mathcal{L}\) b/w two points on the same radial line is GREATER than the coordinate difference \(dr\)
Factor \(1/\sqrt{1-2GM/(rc^2)}\) must be included in any calculation of spatial distances *Also incorporates time dilation and gravitational redshift
If a clock is at rest at the radial coordinate \(r\), then the proper time \(d\tau\) it records is related to the time that elapses at an infinite distance by:
B/c \(d\tau < dt\), time passes more slowly closer to the massive sphere.
3.2.5 The Orbit of a Satellite
According to Newton a satellite orbiting Earth can be found by:
where \(v\) is the orbital speed.
Determining MAX interval b/w fixed events requires conservation of energy, momentum, and angular momentum
Assume the satellite travels around the equator of the host w/ an angular speed \(\omega = v/r\).
For a closed, unperturbed orbit, we find \(dr = 0\), \(d\theta = 0\), and \(d\phi = \omega dt\).
Apply to Schwarzschild metric
*We assumed that the orbit is closed and unperturbed
For this to be true -> the satellites must begin and end at the same position \(r_o\) for all worldlines
It can move rapidly to a radius \(r\), as long as it returns at the same speed
To find extremum:
The derivative may be taken inside the integral-
Using \(v=r\omega\)-
This is the COORDINATE SPEED of the satellite
3.3 Black Holes
John Michell- believed that stream of particles should be influenced by gravity *A star with 500 times more gravity than the Sun would be strong enough to prevent light from escaping -> Escape Velocity = Speed of Light
Newtonian Formula shows: \( R = \frac{2GM}{c^2}\) is the radius of a star whose escape velocity is the speed of light
Oppenheimer and Snyder described the gravitational collapse of a massive star that exhausted its sources of nuclear fusion
3.3.1 The Schwarzschild Radius
Schwarzschild radius, also called gravitational radius, the radius below which the gravitational attraction between the particles of a body must cause it to undergo irreversible gravitational collapse.
The speed of light measured by an observed suspended above the collapsed star must always be. A distant observer can determine that the light is delayed as it moves through curved spacetime. The apparent speed of light (i.e., the rate of change for the spatial coordinates of a photon) is called the coordinate speed of light.
3.3.2 A trip into a black Hole
The coordinate speed of light becomes slower as the astronomer approaches the black hole, so the signals travel back to us more slowly.
The gravitational pull on the astronomer’s feet (nearer to the black hole) is stronger than on the astronomer’s head.
The astronomer need be indestructible because the tidal force would tear the astronomer apart at a distance of several hundred kilometers from the black hole.
If the photons are pulled toward the center, then they can’t make the return trip for you to detect them. This means that the astronomer never has an opportunity to glimpse the singularity.
3.3.3 Mass ranges of Black Holes
Stellar-mass black holes may from directly or indirectly as a consequence of the core-collapse of a sufficiently massive supergiant star.
It is not clear how intermediate-mass objects might form, although correlation of IMBHs with the cores of globular clusters and low-mass galaxies suggests that they develop through mergers of stars (to form supermassive stars that undergo core-collapse), or by the merger of stellar-mass black holes.
Supermassive black holes (SMBHs) lie at the centers of many galaxies (probably most). Our own Milky Way Galaxy has a central black hole.
Primordial black holes may have been manufactured in the instants of the universe.
3.3.4 Black Holes have no hair!
Once the surface of collapsing star reaches the event horizon, the exterior spacetime horizon is spherically symmetric and described by the Schwarzschild metric.
If the angular momentum of a rotating black hole were to exceed this limit, there would be no event horizon and a naked singularity would appear (in violation of the Law of Cosmic Censorship).
Feb 7 (Carey & Leonard)#
Black Holes II#
Feb 9 (Freeman & Norlund)#
Nature of Galaxies I#
4.1. The Hubble Sequence
Immanuel Kant and Thomas Wright suggested that the Milky Way was a finite-sized system of stars in the mid-1700s.
Kent proposed that if the Milky Way was limited, then the elliptical nebulae might be in distant galaxies called island universes.
4.1.1. Cataloging the Island Universes
Charles Messier made the first catalogue of objects that could be confused as comets.
This catalogue was of 103 nebulae, star clusters, and galaxies, and seven more objects were added onto the catalogue by other people.
The objects on the catalogue would have an M in front of their object number. (ex. M41, M35, etc.)
However this catalogue only pertained to objects visible in the northern hemisphere.
William Herschel and his son, John Herschel also made a catalog and it included objects in the southern hemisphere.
The New General Catalog was published by John Dreyer, and it was based off the work of the Herschel’s and contained around 8000 objects.
However, instead of having an M in front of the object number, they would have the letters NGC out front.
In 1845, William Parsons built the 72-inch telescope called the Leviathan and was able to discern that spiral nebulae were rotating.
4.1.2. The Great Shapley-Curtis Debate
On April 26, 1920 there was a debate over where nebulae were in relation to our galaxy.
Shapley thought that nebulae were in our own galaxy, and Curtis thought that they were outside of our galaxy.
Shapley used the apparent magnitude of novae observations in M31 and their relation to the angular size of M31 as his argument for nebulae being within our galaxy.
He measured the Milky Way to be around 100 kpc.
Shapley relied heavily on measurements that were taken by Adrian van Maanen who had later shown that his measurements had been incorrect.
Curtis concluded that the novae observed in the spiral novae must be at least 150 kpc away and therefore the distance of M31 would be smaller than Shapley’s estimate.
The debate was ended in 1923 when Cepheid variable stars in M31 were detected by Edwin Hubble with the 100 inch telescope at mount Wilson.
4.1.3. The Classification of Galaxies
There are three primary ctaegories: elliptical, spiral, and irregulars.
The giant elliptical galaxies are some of the largest objects in the universe, while dwarf galaxies can be as small as a globular cluster. The lenticular galaxies have masses and luminosities comparable to the larger ellipticals. The dwarfs are the most numerous, even though the giant ellipticals and lenticulars are easier to observe.
An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.
Hubble split the remaining category of irregulars into Irr I if there was at least some hint of an organized structure (e.g., spiral arms) and Irr II for the most extremely disorganized structures.
The Large and Small Magellanic Clouds are examples of Irr I galaxies, while M82 (NGC 3034) is an example of an Irr II.
While some galaxies have spiral arms that can be followed nearly all the way into the center, others have arms that appear to terminate at the location of an inner ring.
Feb 14 (Scarlett & Kennedy)#
Nature of Galaxies II#
4.2 Spiral and Irregular Galaxies
Hubble’s classification scheme for late-type galaxies is very successful in organizing our study of these objects.
the tightness of the spiral arms
Ability to resolve te arms into stars
Bulg-to-disk ratios
H II regions
All of these correlate well with hubble type
Larger B - V values indicated redder stars.
4.2.1 The K -Correction
when you take into account things that could’ve changed the apparent magnitude
these corrections must take properly account for the effects of extinction
4.2.2. The Brightness of the Background Sky
The dimly glowing night sky has an average brightness of \( μ_{sky} = 22\)
Sources of this background light include
light pollution from nearby cities,
photochemical reactions in Earth’s upper atmosphere,
the zodiacal light,
unresolved stars in the Milky Way, and
unresolved galaxies.
In modern photometric studies using CCDs, the surface brightnesses of galaxies can be measured down to levels of 29 arc seconds or fainter
4.2.3. Isophotes and the de Vaucouleurs Profile
Isophotes
Maps of constant brightness
In specifying the “radius” of a galaxy, it is necessary to define the surface brightness of the isophote begin used to determine that radius.
One commonly used radius is the Holmberg radius \(r_{H}\)
which is defined to be the projected length of the semimajor axis of an ellipsoid having an isophotal surface brightness of \(μ_{H}=26.5\) arcsecs^2
A second standard radius in frequent use is the effective radius \(r_{e}\)
Surface brightness distribution for bulges of spiral galaxies
4.2.4 The Rotation Curves of Galaxies
rotation curve
Provides a direct means to determine the distribution of all matter
When rotation curves are compared with either luminosity (or Hubble type), a number of correlations are found.
With increasing luminosity in the band
the rotation curves tend to rise more rapidly with distance from the center and peak at higher maximum velocities .
For galaxies of equal band luminosities, spirals of an earlier type have larger maximum velocities.
Within a given Hubble type, galaxies that are more luminous have larger maximum velocities.
For a given maximum velocity , the rotation curves tend to rise more rapidly with radius for galaxies of progressively earlier type.
4.2.5. The Tully-Fisher Relation
Tully Fisher Relation
The correlation that accounts for the relationship between the luminosity of a spiral galaxy and its maximum rotation velocity.
The average radial velocity of the galaxy relative to the observer is the midpoint value between the two peaks
Shift of Δλ
\[\frac{Δλ}{λ_{rest}}≈\frac{v_{r}}{c}=\frac{v \sin i}{c}\]radial velocity and inclination angle, and direction perpendicular to galactic plane effect the measurement of the shift.
The Tully-Fisher relation can be further refined and tightened if observations are made at infrared wavelengths. This offers two advantages
Observing at dust-penetrating IR reduces extinction by a factor of 10.
The IR light come from primarily late-type giant stars that are good tracers of the galaxy’s overall luminous matter distribution; the band tends to emphasize you, hot stars in regions of recent star formation.
4.2.6. Radius-Luminosity Relation
Radius increases with increasing luminosity, independent of Hubble type. At the disk radius \(R_{25}\) corresponding to a surface-brightness, is represented by
\(R_{25}\) is measured in kpc.
4.2.7. Colors and the Abundance of Gas and Dust
Sc galaxies tend to have a greater fraction of massive main sequence stars relatve to earlier spirals.
Sc galaxies are bluer than Sa and Sb galaxies
Mean value of B - V decrease with later hubble types
0.75 for Sa,
0.64 for Sb, and
0.52 for Sc
For successively later-type galaxies, progressively greater portions of the overall light from spirals is emitted in bluer wavelength regions, implying an increasingly greater fraction of younger, more massive, main-sequence stars
Irregulars tend to be the bluest of all galaxies.
Blue main-sequence stars are short lived
They must have formed recently
The relative amounts of atomic and molecular hydrogen also change with Hubble type.
Feb 16 (Leonard & Freeman)#
Nature of Galaxies III#
4.2.8 Metallicity and Color Gradients of Spirals
Individual spiral galaxies also exhibit color gradients with their bulges generally being redder than their disks. This arises for two reasons: metallicity gradients and star formation activity.
Star formation (the second major cause of color gradients) implies that the disks of spiral galaxies are more actively involved in star-making than are their bulges. This is consistent with the distribution of gas and dust in the galaxies.
Metallicity correlates with the absolute magnitude of galaxies
4.2.9 X-ray Luminosity
The X-ray luminosities of galaxies also provide some information concerning their evolution.
X-rays are due to a class of objects that constitutes an approximately constant fraction of the population of all objects in spirals. The suspected sources are X-ray binaries.
It is probable that supernova remnants also contribute to the X-ray emission.
4.2.10 Supermassive black holes
How do we know that there is a SMBH in the middle of the milky way? Observations of stellar and gas motions near the centers of some spirals suggest the presence of supermassive black holes.
To determine an accurate value for the mass, an appropriate choice of must be made. As the observations move farther from the black hole, contributions to the total mass increase from surrounding stars and gas.
4.2.11 Specific Frequency of globular clusters
Galaxies that are more spheroidally dominant (i.e., earlier Hubble types) were more efficient at forming globular clusters during their histories.
4.3 Spiral Structure
Galaxies exhibit a rich variety of spiral structure, which vary in
the number of arms and how tightly wound they are,
degree of smoothness in the distribution of stars and gas
surface brightness, and
the existence (or lack) of bars.
Not all spirals are grand designs with two distinct arms. For instance, M101 has four arms, and NGC 2841 has a series of partial arm fragments. Galaxies like NGC 2841, which do not possess well-defined spiral arms that are traceable over a significant angular distance are called flocculent spirals.
The optical images of spiral galaxies are dominated by their arms because very luminous O and B main-sequence stars and H II regions are found preferentially in the arms.
When spiral galaxies are observed in red light, the arms become much broader and less pronounced. Although they still remain detectable. Observations at red wavelengths emphasize the emission of long-lived, lower-mass main-sequence stars and red giants, which implies that the bulk of the disk is dominated by older stars.
4.3.1 Trailing and Leading Spiral Arms
The general appearance of spiral galaxies suggests that their arms are trailing, where the tips of the arms point in the opposite direction relative to the direction of rotation.
What is the distinction between trailing and leading? Distinguishing between trailing and leading arms (along the direction of rotation) requires determination of the orientation of the plane of the galaxy relative to our line-of-sight
4.3.2 The winding problem
One problem arises when the nature of the spiral structure is considered: material arms composed of a fixed set of identifiable stars and gas clouds would necessarily “wind up” on a timescale that is short compared to the age of the galaxy.
The simplest idea for the origin of spiral arms is that somehow the material in the galaxy condensed into its spiral pattern from the very start, and that pattern has remained fixed since then. Unfortunately, this idea immediately runs into trouble because galaxies exhibit differential rotation.
4.3.3 The Lin-Shu Density Wave Theory
C.C. Lin and Frank Shu proposed that spiral structure arises because of the presence of long-lived quasistatic density waves.
Less massive, redder stars will be able to live much longer (some much longer than the current age of the galaxy) and so will continue through the density wave and become distributed throughout the disk.
It is likely that less dust and gas will be found in these outer regions of the galaxy.
Feb 21 (Scarlett)#
Nature of Galaxies IV#
4.4.3 Metallicity and Color Gradients
The metallicity of elliptical galaxies is well-correlated with luminosity, which means that brighter galaxies have a higher overall metal content.
The central regions of E-type galaxies are redder and more metal-rich than are regions at larger radii.
Any successful theory of galaxy formation must incorporate the available observations concerning chemical enrichment.
4.4.4. The Faber-Jackson Relation
relates dE, dSph, normal E, and the bulges of spiral galaxies
Has correlations between their central radial-velocity dispersion and their absolute magnitude in the B band.
Do math Later
This relationship was first identified by Sandra Faber and Robert Jackson and is now referred to as the Faber-Jackson relation.
##4.4.5. The Fundamental Plane**
There lies scatter in the data of Faber-Jackson relation
Astronomers have introduced a second parameter into the expression to find a tighter fit to the data, using the effective radius \(r_e\) . One representation of this fit is
Here galaxies are visualized as residing on a 2D “surface” in the 3D “space”
The above equation is known as the fundamental plane
combines galaxies velocity dispersion and its radius and luminosity
4.4.6. The Effect of Rotation
It is evident that most ellipticals are not purely oblate or prolate rotators with two axes, but are triaxial, which means that there is no single preferred axis of rotation. Evidence exists in the:
randomness of dust lanes found in at lest 50% of all ellipticals, -observations of counter-rotating stellar cores in as many as 25% of larger ellipticals.
It appears that (in at least some cases) material in the form of gas, dust, globular clusters, or dwarf galaxies has been captured sometime since the galaxy’s formation.
The shape of the galaxies aren’t do the to the rotation
not caused by anisotropic velocity
Bright E- and gE-type galaxies have typical values of approximately .4 and are Pressure-supported
shapes are due to random stellar motions
4.4.7. Correlations with Boxiness or Diskiness
elliptical galaxies can be understood in terms of the degree of boxiness or diskiness that there isophotal surfaces exhibit.
isophotal just means constant brightness
The shape of an isophotal contour is written in polar coordinates (as a Fourier series) of the form
do math later
The terms of the expansion represent
THe shape of a perfect circle in the first term
the amount of ellipiticity in the second term
the degree of boxiness in the third term
if \(a_4\) is less than 0
surface is boxy appearance if \(a_4\) greater than 0
surface tends being disky
4.4.8 The Relative Numbers of Galaxies of Various Hubble Types
The relative numbers of galaxies of various Hubble types is usually represented by the luminosity function \(\phi(M)dM\) .
Although spirals represent the largest fraction of bright galaxies in each case, there is a somewhat higher proportion of ellipticals in the Virgo cluster.
When compared to much larger coma cluster relative members of spirals and ellipiticals change
In the Virgo cluster: 12% E, 26%S0, 62% S+Ir
in Coma cluster: 44% E, 49% S0, and only 7% S+IR
This is evidence that environment plays a role in galaxy formation and/or evolution.
Feb 23 (Exam I)#
Feb 28 (Kennedy & Carey)#
Milky Way Galaxy I#
Mar 2 (Norlund & Scarlett)#
Milky Way Galaxy II#
Mar 7 (Freeman & Kennedy)#
Milky Way Galaxy III#
Mar 9 (Scarlett & Leonard)#
Milky Way Galaxy IV#
Astropy (Teaching) Presentations#
Mar 14 & 16 (Spring Break)#
Mar 21 (Carey & Norlund)#
Galactic Evolution II#
Mar 23 (Kennedy & Scarlett)#
Milky Way Galaxy V#
Mar 28 (Norlund & Freeman)#
Galactic Evolution I#
Chapter 6
6.1 Interactions of Galaxies
6.1.1 Evidence of Interactions
Nearly all galaxies belong to clusters. Densely populated clusters (e.g., the Coma cluster) have a higher proportion of elliptical galaxies (i.e., early-type) in their center than they do in their outer, less dense regions.
The central regions of these rich, regularly shaped clusters also have a higher proportions of ellipticals than the centers of less populated, amorphous irregular clusters (e.g., Hercules cluster).
At least of all disk galaxies display warped disks. Due to observations, we suggest that hot, X-ray emitting gas occupies much of the space between galaxies in rich clusters and has a mass equal to (or exceeding) the mass of all the cluster’s stars. It could be due to the grtavitational influence.
6.1.2 Dynamical Friction
The interactions between stars are gravitational in nature.
The dynamical friction is the opposing force that comes into play when one body is actually moving over the surface of another body. It is the friction related with motion or with simple term a body slides over another body and experiences a opposing force know dynamic friction.
The expression for the force of dynamical friction will have the following form,
where \(C\) is dimensionless, but not a constant. It is a function that depends on how \(v_M\) compares with the velocity dispersion \(\sigma\) of the surrounding medium.
Satellite galaxies are also affected by dynamical friction.
6.1.3 Rapid Encounters
In impulse approximation, the stars barely have time to alter their positions. As a result, the internal potential energy of each galaxy is unchanged by the collision. However, the gravitational work that each galaxy performs on the other has increased the internal kinetic energies of both galaxies in a random way.
The equation explains hpow the equilibrium can be re-established after the encounter caused the galaxy’s internal kinetic energy to increase.
Tidal stripping occurs when a larger galaxy pulls stars and other stellar material from a smaller galaxy because of strong tidal forces.
Polar-ring galaxies and dust-lane ellipticals are normal galaxies that are orbited by rings of gas, dust, and stars that were stripped from other galaxies as they passed by or merged.