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.
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.
1.2.3. Aristotelian Cosmology
1.2.4. Ancient China
1.2.5. Pre-Columbian America
1.2.6 Medieval and Early Renaissance Europe
1.2.7 Galileo, the Telescope, and Cosmology
1.2.8 Cosmology in Bentley and Newton
1.2.9 The Renewal of Cosmology
1.3 Differences in Historical, Scientific, and Mathematical Cosmology
1.3.1 Connections between Astronomy and the State
If A only happens when B is present, we might infer that B causes A to occur.
1.3.2 Models of the Universe
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
2.3.4 The Relativistic Doppler Shift
2.3.5 The Relativistic Velocity Transformation
2.4 Relativistic Momentum and Energy
2.4.1 The Derivation of $E=mc^2$
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.
3. General Relativity and Black Holes
3.1. The General Theory of Relativity
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:
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.
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
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
Important formulas
$$\text{radius of curvature} = r_{c}$$$$r_{c}=\frac{c^2}{g}=9.17 * 10^{-15}\text{rad}$$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
$$\frac{v_{∞}}{v_{o}}=(1-\frac{2GM}{r_{o}c^2})^\frac{1}{2}$$$$ Z=\frac{Δλ}{λ_{o}}=\frac{v_{o}}{v_{∞}}-1$$
$$≈\frac{2GM}{r_{o}{c^2}}$$
Time in Gravity
In a strong field
$$ \frac{Δt_{o}}{Δt_{∞}}=\frac{v_{∞}}{v_{o}}=(1-\frac{2GM}{r_{o}c^2})^\frac{1}{2}$$
In a weak field
$$\frac{Δt_{o}}{Δt_{∞}}≈1-\frac{GM}{r_{o}{c^2}}$$3.2. Intervals and Geodesics
$$ \mathcal{G} = -\frac{8πG}{c^4}Τ$$ $$ G=\text{Gravitational Constant}$$ $$Τ=\text{Stress-energy-tensor}$$
3.2.1. Worldlines and Light Cones
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
$$(Δ\ell)^2=(x_{2}-x_{1})^2+(y_{2}-y_{1})^2+(z_{2}-z_{1})^2$$
$$Δτ=\frac{Δs}{c}$$
$$\mathcal{L}=\sqrt{-(Δs)^2}$$
3.2.4. Curvred Spacetime and Schwarzchild Metric
3.2.4 Curved Spacetime and the Schwarzschild Metric
Metric b/w 2 nearby points in flat space-
\begin{align} (d\ell)^2 = (dr)^2 + (r\ d\theta)^2 + (r\ \sin{\theta}\ d\phi)^2, \end{align}Metric in flat spacetime-
\begin{align} (ds)^2 = (c\ dt)^2 - (dr)^2 - (r\ d\theta)^2 - (r\ \sin{\theta}\ d\phi)^2. \end{align}In curved spacetime around a massive sphere:
{eq}grav_time_dilation
*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: \begin{align} d\tau = \frac{ds}{c} = dt \sqrt{1-2GM/(rc^2)}. \end{align}
3.2.5 The Orbit of a Satellite
where $v$ is the orbital speed.
Assume the satellite travels around the equator of the host w/ an angular speed $\omega = v/r$.
*We assumed that the orbit is closed and unperturbed
To find extremum:
$$ \frac{d}{dr}(\Delta s) = \frac{d}{dr} \left(\int_0^T \sqrt{c^2-\frac{2GM}{r} - r^2\omega^2 }dt \right) = 0. $$The derivative may be taken inside the integral- \begin{align*} \frac{d}{dr} \sqrt{c^2-\frac{2GM}{r} - r^2\omega^2 } &= 0,\\ \frac{d}{dr} \left( \frac{2GM}{r} - r^2\omega^2 \right) &= 0,\\ \frac{2GM}{r^2} - 2r\omega^2 &=0. \end{align*} Using $v=r\omega$- \begin{align} v = r \omega = \sqrt{\frac{GM}{r}}, \end{align}
3.3 Black Holes
*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
3.3.1 The Schwarzschild Radius
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!
4.1. The Hubble Sequence
4.1.1. Cataloging the Island Universes
4.1.2. The Great Shapley-Curtis Debate
4.1.3. The Classification of Galaxies
4.2 Spiral and Irregular Galaxies
4.2.1 The K -Correction
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
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 $r_{H}$
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
band
.
band luminosities, spirals of an earlier type 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 average radial velocity of the galaxy relative to the observer is the midpoint value between the two peaks
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
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
4.2.7. Colors and the Abundance of Gas and Dust
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
The relative amounts of atomic and molecular hydrogen also change with Hubble type.
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
4.2.10 Supermassive black holes
4.2.11 Specific Frequency of globular clusters
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
4.3.2 The winding problem
4.3.3 The Lin-Shu Density Wave Theory
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
. One representation of this fit is
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:
-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
Bright E- and gE-type galaxies have typical values of approximately .4 and are Pressure-supported
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.
The shape of an isophotal contour is written in polar coordinates (as a Fourier series) of the form
The terms of the expansion represent
if $a_4$ is less than 0
if $a_4$ greater than 0
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.
This is evidence that environment plays a role in galaxy formation and/or evolution.
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.
6.1.3 Rapid Encounters
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.