Introduction to cosmology and astrophysics

A light year ($\text{ly}$) is the distance light travels in a vacuum in one year. Data: $c = 3\times10^{8}\ \text{m/s}$; $1\ \text{year} = 3.156\times10^{7}\ \text{s}$.

1. a) Calculate the value of $1\ \text{ly}$ in meters, in scientific notation and to three significant figures.
2. b) Express the result in kilometers.

Order the following distances from smallest to largest and justify the ordering using scientific notation:

1. a) the mean distance from Earth to the Moon, $3.84\times10^{8}\ \text{m}$
2. b) the astronomical unit, $1\ \text{UA} = 1.5\times10^{11}\ \text{m}$
3. c) the radius of Earth, $6.37\times10^{6}\ \text{m}$
4. d) one light year, $1\ \text{ly} = 9.46\times10^{15}\ \text{m}$
5. e) one parsec, $1\ \text{pc} = 3.086\times10^{16}\ \text{m}$

Proxima Centauri is the nearest star to the Sun: it is located at $4.24\ \text{ly}$. Data: $1\ \text{ly} = 9.46\times10^{12}\ \text{km}$.

1. a) Express this distance in kilometers, in scientific notation.
2. b) How long does its light take to reach Earth?
3. c) If a spectacular flare occurred on that star today, could we observe it tonight? Explain your reasoning.

The parsec is defined from annual parallax: a star is at $1\ \text{pc}$ if its parallax is one arcsecond. In practice, the relation $d\ (\text{pc}) = \frac{1}{p}$ is used, with the parallax $p$ measured in arcseconds. The Gaia mission measured a parallax of $0.02''$ for a star. Data: $1\ \text{pc} = 3.26\ \text{ly}$. The diagram shows the geometry of annual parallax (the figure is not to scale).

1. a) Calculate the distance to that star in parsecs.
2. b) Convert the result to light-years.
3. c) Explain why this method only works for relatively nearby stars: what happens to the parallax when the star is very far away?

Sirius, the brightest star in the night sky, is located $2.64\ \text{pc}$ from the Sun. Data: $1\ \text{pc} = 3.26\ \text{ly}$; $1\ \text{ly} = 9.46\times10^{12}\ \text{km}$.

1. a) Express the distance to Sirius in light years.
2. b) Express it in kilometers, in scientific notation.
3. c) How long would a radio signal sent from Earth today take to reach Sirius?

The Andromeda Galaxy (M31) is located about $2.5\times10^{6}\ \text{ly}$ from the Milky Way. Data: $1\ \text{pc} = 3.26\ \text{ly}$.

1. a) Express this distance in parsecs and in megaparsecs.
2. b) How long has the light with which we observe Andromeda today taken to reach us?
3. c) Explain what we mean when we say that in astronomy "looking far away is looking into the past."

The Hubble–Lemaître law relates the recessional velocity $v$ of a galaxy to its distance $d$ by means of $v = H_0 \times d$, where $H_0 = 70\ \text{km}\times\text{s}^{-1}\times\text{Mpc}^{-1}$. The graph represents this law and marks the galaxy from the question.

1. a) Calculate the recessional velocity of a galaxy located $20\ \text{Mpc}$ away from us.
2. b) Explain what it means for this velocity to be positive: is the galaxy approaching or moving away?

A galaxy in the Virgo Cluster is located $150\ \text{Mpc}$ from the Milky Way. Data: $H_0 = 70\ \text{km}\times\text{s}^{-1}\times\text{Mpc}^{-1}$; $c = 3\times10^{8}\ \text{m/s}$.

1. a) Apply the Hubble–Lemaître law, $v = H_0 \times d$, to calculate its recessional velocity, in $\text{km/s}$.
2. b) Express that velocity as a fraction of the speed of light and comment on whether relativistic effects would be noticeable at that speed.

The redshift of the spectrum of a galaxy indicates that it is moving away from us at $2100\ \text{km/s}$. Data: $H_0 = 70\ \text{km}\times\text{s}^{-1}\times\text{Mpc}^{-1}$; $1\ \text{pc} = 3.26\ \text{ly}$. The figure compares the reference spectrum, measured in the lab, with that observed in that galaxy.

1. a) Use the Hubble–Lemaître law to calculate how far away it is, in $\text{Mpc}$.
2. b) Express that distance in light-years.
3. c) Explain what happens to the wavelength of light when the source moves away from the observer and why we say the spectrum is "redshifted".

Two galaxies are located $100\ \text{Mpc}$ and $300\ \text{Mpc}$ from us, respectively. Data: $H_0 = 70\ \text{km}\times\text{s}^{-1}\times\text{Mpc}^{-1}$; $c = 3\times10^{8}\ \text{m/s}$.

1. a) Calculate the recessional velocity of each according to the Hubble–Lemaître law.
2. b) At what distance would a galaxy be located whose recessional velocity was $\frac{c}{4} = 7.5\times10^{4}\ \text{km/s}$?
3. c) In 1998, observations of distant supernovae showed that these galaxies are farther away than predicted by a constant-rate expansion. Explain what conclusion was drawn about the expansion of the universe and what mysterious component was proposed to explain it.

A group of students analyzes data from three galaxies obtained using a school telescope:

• galaxy 1: distance $10\ \text{Mpc}$, recessional velocity $700\ \text{km/s}$

• galaxy 2: distance $25\ \text{Mpc}$, recessional velocity $1750\ \text{km/s}$

• galaxy 3: distance $40\ \text{Mpc}$, recessional velocity $2800\ \text{km/s}$

The graph represents the data for the three galaxies.

1. a) Calculate the ratio $\frac{v}{d}$ for each galaxy and check that it is approximately constant.
2. b) Identify what quantity this ratio represents and give its value with its units.
3. c) Predict the recessional velocity of a fourth galaxy located $60\ \text{Mpc}$ away.
4. d) Explain why this proportionality is evidence for the expansion of the universe and not for galaxies "fleeing" from us by moving through space.

You can do it on paper and send a photo with your answer.

The Big Bang model is the accepted scientific framework for describing the origin and evolution of the universe. The diagram represents the expansion of the universe from the Big Bang to the present day.

1. a) Explain in your own words what this model asserts about the origin of the universe.
2. b) State the approximate age of the universe and express it in scientific notation, in years.
3. c) Cite two experimental observations supporting the Big Bang model.

A simple estimate of the age of the universe is the so-called "Hubble time", $t_0 \approx \frac{1}{H_0}$. Data: $H_0 = 70\ \text{km}\times\text{s}^{-1}\times\text{Mpc}^{-1}$; $1\ \text{Mpc} = 3.086\times10^{19}\ \text{km}$; $1\ \text{year} = 3.156\times10^{7}\ \text{s}$.

1. a) Express $H_0$ in units of $\text{s}^{-1}$.
2. b) Calculate $t_0 = \frac{1}{H_0}$ in seconds.
3. c) Express the result in years and compare it with the accepted age of the universe, $1.38\times10^{10}\ \text{years}$.

Current measurements of the Hubble constant do not agree with each other: the cosmic microwave background (Planck satellite) suggests $H_0 = 67\ \text{km}\times\text{s}^{-1}\times\text{Mpc}^{-1}$, whereas supernovae in nearby galaxies (Hubble and James Webb telescopes) give $H_0 = 73\ \text{km}\times\text{s}^{-1}\times\text{Mpc}^{-1}$. Data: $1\ \text{Mpc} = 3.086\times10^{19}\ \text{km}$; $1\ \text{year} = 3.156\times10^{7}\ \text{s}$. The graph schematically shows the two measurements of $H_0$.

1. a) Calculate the "Hubble time", $t_0 \approx \frac{1}{H_0}$, in years, for each of the two values.
2. b) Find the difference between the two estimated ages of the universe.
3. c) This discrepancy is known as "Hubble tension". Explain why it worries cosmologists and what it might imply about the standard cosmological model.

The Solar System is our nearest cosmic neighborhood. The diagram shows its structure, with the eight unnamed planets.

1. a) Write down the eight planets in order of increasing distance from the Sun.
2. b) Classify them into rocky planets and gas giants, and state between which planets the asteroid belt lies.
3. c) What is the star of the Solar System and approximately what fraction of the total mass of the system does it contain?

Our Solar System is part of a galaxy, the Milky Way.

1. a) State what type of galaxy the Milky Way is and approximately how many stars it contains, expressed as a power of ten.
2. b) Describe where the Sun is located within the galaxy: in the center, in an arm or at the edge?
3. c) Explain in your own words the difference between the Solar System and the Milky Way.

The ultimate fate of a star depends mainly on its initial mass.

1. a) Put the life stages of a star like the Sun in order: red giant, nebula, white dwarf, main sequence.
2. b) State what happens instead to a star much more massive than the Sun at the end of its life.
3. c) Name the two possible remnant end-states of such very massive stars.

The Milky Way has a diameter of about $10^{5}\ \text{ly}$ and the Andromeda Galaxy is located $2.5\times10^{6}\ \text{ly}$ from us. Data: $1\ \text{ly} = 9.46\times10^{15}\ \text{m}$; diameter of the Sun $d_S = 1.4\times10^{9}\ \text{m}$.

1. a) Calculate how many "galactic diameters" fit into the distance separating the Milky Way from Andromeda.
2. b) Build a scale model: if we were to represent the Milky Way as a plate of diameter $20\ \text{cm}$, at what distance would you place the plate representing Andromeda?
3. c) Calculate what size the Sun would be in that same model, express it in scientific notation, and comment on what you conclude about how "empty" the universe is.

Research in cosmology and astrophysics has driven technologies that we use every day.

1. a) The CCD sensors in cell phone cameras were originally developed for astronomy. Cite another everyday technology related to space research or fundamental physics and explain the connection.
2. b) Explain why countries invest in telescopes and observatories even if they do not produce an immediate economic benefit.

The Voyager 1 probe, launched in 1977, is the farthest human-made object from Earth: it is located at about $160\ \text{UA}$ from the Sun and is moving away at $17\ \text{km/s}$. Data: $1\ \text{UA} = 1.5\times10^{8}\ \text{km}$; $1\ \text{ly} = 9.46\times10^{12}\ \text{km}$; $1\ \text{year} = 3.156\times10^{7}\ \text{s}$.

1. a) Express its current distance in kilometers and in light-years.
2. b) Assuming it maintained a constant speed, calculate how many years it would take to reach a distance of $1\ \text{ly}$ from the Sun.
3. c) In light of the result, justify why crewed interstellar travel is not feasible with current technology.