accelerate/acceleration — to change velocity; to speed up, slow down, or change direction
apparent magnitude — a relative measure of how bright a star looks in the sky (i.e. how bright it appears to be from our perspective on Earth, which could be very different from how bright it actually is, similar to how an atom bomb would appear to be very bright when viewed from just outside its blast radius, but less so when viewed from 300 km away); the larger the number, the dimmer the star appears to us
astrology — the pseudoscience that deals with the supposed influences on human destiny of the configurations and locations in the sky of the Sun, Moon, and planets
celestial equator — a great, imaginary circle projected out onto the celestial sphere 90° from the celestial poles; this is where the plane of Earth’s equator intersects with the celestial sphere midway between the poles, due specifically to Earth’s rotation on its axis
celestial poles — the points in the polar skies about which the celestial sphere appears to rotate; celestial sphere intersections with the Earth’s polar axis
celestial sphere — the apparent sphere of the sky; a sphere of large radius centered on the observer; directions of objects in the sky can be denoted by their position on the celestial sphere (using celestial GPS system coordinates)
circumpolar zone — those portions of the celestial sphere near the celestial poles that are either always above or always below the horizon
cosmology — the study of the organization and evolution of the universe
ecliptic — the apparent annual path of the Sun on the celestial sphere
epicycle — the circular orbit of a body in the Ptolemaic system, the center of which revolves about another circle (the deferent)
geocentric — centered on Earth
heliocentric — centered on the Sun
horizon (astronomical) — a great circle on the celestial sphere 90° from the zenith; more popularly, the circle around us where the dome of the sky meets Earth
horoscope — a chart used by astrologers that shows the positions along the zodiac and in the sky of the Sun, Moon, and planets at some given instant and as seen from a particular place on Earth — usually corresponding to the time and place of a person’s birth
parallax — the apparent displacement of a nearby star that results from the motion of Earth around the Sun
planet — today, any of the larger objects revolving about the Sun or any similar objects that orbit other stars; in ancient times, any object that moved regularly among the fixed stars
precession (of Earth) — the slow, conical motion of Earth’s axis of rotation caused principally by the gravitational pull of the Moon and Sun on Earth’s equatorial bulge the apparent westward motion of a planet on the celestial sphere or with respect to the stars
year — the period of revolution of Earth around the Sun
zenith — the point on the celestial sphere opposite the direction of gravity; point directly above the observer
zodiac — a belt around the sky about 18° wide centered on the ecliptic
Further Reading
Problem Set
Long-Answer
[1] How many ways could you prove to a member of the Flat Earth Society that the Earth is round?
If I were to attempt to prove to a Flat-Earther that the Earth was round, I would teach them (or show them):
How ships going out to sea disappear over the horizon.
Take them in a spacecraft to see the curvature of the Earth.
Show them pictures of the Earth from space or the Moon, and point out the ISS to them.
Use a telescope so they can view the Moons of Jupiter or Saturn, and the planets themselves. (You can see that they are spherical, if you observe enough times to see different faces, which doesn’t prove directly that the Earth is a sphere, but is certainly suggestive of that fact.)
Explain how the altitude of the North Star changes with latitude over the northern horizon when you change your latitude on the surface, and also how the zenith stars change as well. (Ideally showing them with a road trip, and showing them how to calculate their own latitude with their hands.)
Set up a conference call with people in four separate time zones around the Earth, and get them to all look at the sky outside at the same moment, and give either the Sun’s position, or a common constellation plus the zenith constellation.
[2] Make a list of ways in which a belief in astrology (that your destiny and personality are determined by celestial bodies positions) might be harmful to an individual, or society at large?
Frankly, I do not believe that it is “harmful” per se. People are looking for answers, guidance, and some way to make sense of themselves, their world, and their life. Moreover, the evidence against astrology proved unequivocally that people will see what they want to see anyways … so it is almost like a Rorshak test: they will project what is inside of them.
The danger, and potential harm, arises when people contradict themselves in favour of horoscope predictions. She wants to be a doctor, and has always loved medicine, but goes into politics (or sports, or whatever) because her horoscope says so. Or she loves this man … but he is an Ares … and she needs a Gemini.
Its only potential harm is, in essence, allowing unfounded nonsense to dictate major decision, over trying to unravel the nonsense within.
This, in addition to when it promotes general ignorance on an individual or societal level (i.e. unintentionally promoting false flat-earth or geocentric doctrines, or neglecting the influence of genetics, psychology, or the like).
[3] Why do you think that people of today are less familiar with the night sky (specific stars, constellations, celestial bodies etc) compared to people of the ancient world? Discuss reasons why a person today might want to be more familiar with the stars?
The main reasons why a person of the past was more familiar with the sky is:
They needed it: they did not have watches, GPS, even compasses. They used the stars for all of this. The used the stars to determine the latitude of a town, to know the directions, and to tell the right time at night — in addition to the season for planting, harvest, seasonal flooding, and so on.
More farmers, and no electricity: More people were farmers, and spent more time outside (including soldiers, merchants, sailors, pirates, etc). They looked at the stars during the night, and new their names, in part because they had to, partially because they are beautiful, but also because they had no other distractions. Many people would frequently (even normally) lie outside on the grass at night by a campfire, waiting to drift off to sleep, with only the heavens to look upon.
Today, there are not many reasons why the average person needs to know the heavens. The only one that applies (to me specifically) is navigation. If you are going to go hiking, camping, trekking, road-trips, even travelling: the stars can save your life. You better learn them. Basic navigation and orientation. (In addition to this, know your maps too.)
[4] Constellations commemorate great heroes, gods, dangers, monsters, and events from myth, legend, and history. What would you choose today to commemorate in the heavens?
I am not going to answer this one, because it is both entirely subjective, and too challenging to do inclusively for our global culture.
If I were to choose, I would commemorate the Great Flood c.10,500 (maybe with Cassiopeia since she looks like a “W” which could be a wave.
Other than that, you could choose WWI, WWII, maybe the Black Death. Then for people, anyone from Einstein, to Michael Jordan, Malcom X, MLK, Gandhi, Jesus, Muhammad, Adele, or Alan Turing.
Completely subjective.
[5] Although astronomical mythology no longer holds a powerful sway over the modern imagination, we still find proof of the power of astronomical images in the number of products in the marketplace that have astronomical names. How many can your group come up with? (Think of things like Milky Way candy bars, Eclipse and Orbit gum, or Comet cleanser.)
From the equator you can see both poles, and every star between them, over the course of a year. Assuming that we are talking about N-S fraction, 90° from the North Celestial Pole to the Celestial Equator
[2] Give four ways to demonstrate that Earth is spherical.
Set up a conference call in four time zones, and look at the sky to preset constellation positions or the position of the sun (ideally to see four quadrants of the sky at the same time) noting whether they are at zenith, near horizon, etc.
Ships dipping below the horizon.
Pole stars changing elevation as you change latitude (or the zenith stars or equatorial constellation altitude changing with altitude).
In the geocentric cosmology, Ptolemy explained retrograde motion using epicycles (planets as they orbit the Earth periodically perform small orbits around a point called the deferent) that are little sub-orbits on their greater orbital path around Earth, like getting caught in whirlpools.
In the Heliocentric Model of Copernicus, retrograde motion is due to the fact that the Earth overtakes planets in our orbit, passing their position, and making it appear like they are moving backwards temporarily.
[4] In what ways did the work of Copernicus and Galileo differ from the views of the ancient Greeks and their contemporaries?
Discovered the Moons around Jupiter (proving that not everything must orbit the Earth).
The phases of Venus (which contradict the Ptolemaic Model, which predicts the wrong phases at the wrong times and in the wrong order — which the Copernican doctrine explains correctly).
Observations of the lunar surface (which showed what looked like valleys, mountains, and seas) which told us that the Moon, hence other celestial bodies, might not be that different from the Earth.
Observation of far more stars than the naked eye can discern (such as the distinction of the multitude of stars that make up the band of the Milky Way).
[6] Explain the origin of the magnitude designation for determining the brightness of stars. Why does it seem to go backward, with smaller numbers indicating brighter stars?
The magnitude system was originally devised by Hipparchus in the final centuries BCE to organize the stars into categories from brightest to faintest.
This is why Magnitude 1 is brighter than Magnitude 3 — because the brightest stars are the most prominent, and therefore in category 1. Then descending in categories from there.
[7] Ursa Minor contains the pole star, Polaris, and the asterism called the Little Dipper. From most locations in the Northern Hemisphere all of the stars of Ursa Minor are circumpolar. Does these mean they are above the horizon during the day?
Yes, from most locations in the Northern Hemisphere the stars of Ursa Minor are always above the horizon — day or night.
When you watch the stars at night, Polaris never changes its altitude (it is the “North Star” for a reason) because the Earths axis of rotation is pointing in that direction. As the Earth spins on its axis, this is what changes day to night. Even though the stars are invisible during the day, they are still there, hanging in the northern skies.
[8] How many degrees does the Sun move per day relative to the fixed stars? How many days does it take from the Sun to return to its original location relative to the fixed stars?
[9] How many degrees does te Moon move per day relative to the fixed stars? How many days does it take for the Moon to return to the same position against the stars?
The Moon moves about 0.5° every hour against the stars/through the sky. Therefore each day the Moon moves about 12° through the sky (over a 24h period).
Using this value, we can then calculate how long it takes for the Moon to make one complete orbit around the Earth (at least one orbit relative to the background of stars, i.e. returning to the same place in the stars).
\[ \frac{360 \deg}{12 \deg /day} = 30 days \]
From this calculation, we see that the Moon takes almost exactly 30 days for one orbit.
[10] Explain how the zodiacal constellations are different from the other constellations.
The Zodiacal constellations are different from other constellations only in that they lie (roughly) along the celestial equator.
They are a band of constellations (groups of stars) around the circumference of the celestial sphere, directly between the celestial poles (since they are a projection of the Earth’s equator.
All of the planets in our solar system remain within (or in close proximity to) the band of the Zodiac. Moreover, these constellations are also very good for telling the time of day (marking hours at night), and the time of year (season).
[11] The Sun was once thought to be a planet. Explain why.
Well, technically the Sun was — and is — a planetai (πλανηται). This is the rood word of our word “planet”. The ANcient Greek word πλανηται (planetai) mean “wanderers” because there was a group of celestial bodies that moved around the background of fixed stars. These included Mercury, Venus, Mars, Jupiter, Saturn, Moon, and the Sun. By definition, the Sun was one of the “wanderers”.
While by our modern definitions, the Sun is not a “planet” but a “star”. In the terminology of the ancient Greeks, the Sun was absolutely, by definition, a planetai.
[12] Is the ecliptic the same thing as the celestial equator? Explain.
No, they are different. The celestial equator is a projection of the Earth’s equator onto the celestial sphere, and is the result of the Earth revolving on its axis.
The ecliptic is the path that the Sun appears to trace on its yearly journey through our skies, against the stars. This is the result of the Earth orbiting the Sun, and is related to the “equatorial plane” of our solar system — the horizontal plane extending from the Sun related to its magnetic field that all of the planets in our solar system orbit on.
[13] What is an asterism? Can you name an example?
An asterism is a notable, well-known, or recognizable pattern, shape, or group of stars in the sky that is not part of an official constellation. For example, Orion’s Belt (as part of the Orion constellation), the Little Dipper (as part of Ursa Minor), or the Big Dipper (as part of Ursa Major).
[14] Why did Pythagoras believe the Earth should be spherical?
Pythagoras believed that the Earth was a sphere because he was a believer in the idea of a “perfect circle” — that the circle, and sphere, were perfect geometries. It was his belief that given the perfection of these shapes, that they had inherent divinity, and would have been chosen by the gods for the creation of the Earth.
This idea probably has its roots in Egypt or Babylon (both of whose traditions had a significant impact on the Greeks, Pythagoras having been trained in Egypt).
[15] How did Aristotle deduce that the Sun was further away than the Moon?
Aristotle knew that the Sun was further away than the Moon due to solar eclipses — periodically the Moon passed between the Sun and Earth, blocking the Sun from view.
Therefore, it has to be closer to the Earth than the Sun. (This is not so much a deduction per se (in my opinion) as it is an observation.)
[16] What are two ways in which Aristotle deduced that Earth is spherical?
Hipparchus noticed that the position of the North Celestial Pole had changed (meaning that the orientation of the Earth’s axis had shifted. He had access to ancient Greek, Egyptian, and Babylonian star charts and was able to compare them, noticing this difference from centuries before.
He also correctly understood that this is a continuous motion, not a singular, jerky shift.
[18] Why did Ptolemy have to introduce multiple circles of motion for the planets instead of a single, simple circle to represent the planet’s motion around the Sun?
First, the predictions of planetary positions made by the Ptolemaic Model were bcoming increasingly inaccurate. He needed to update them.
He also didn’t assume that the Earth was the center of the Universe (like Aristarchus of Samos millennia before) arguing that the Earth orbiting the Sun could explain observations equally as well).
[20] What are two factors that made it difficult, at first, for astronomers to choose between the Copernican heliocentric model and the Ptolemaic geocentric model?
Both Models Explained The Observed Motions Almost As Well As The Other — While the Ptolemaic Model was becoming inaccurate, with some minor updates, it would have been as accurate. Moreover, if the Sun was moving around the Earth, it would have looked the same.
There was no obvious way to test it, whether the Earth was orbiting, or the Sun — It would be another century before the telescope (developed by Galileo) was able to observe phenomenon that at least indicated that the Earth was in motion, and far longer before we could prove it unequivocally.
[21] What phases would Venus show if the geocentric model were correct?
Observe the constellation above the eastern horizon just before sunrise (or on the western horizon just after sunset) and estimate the position of the Sun measuring degrees.
Look at the constellation/star at the altitude of the Sun at midday exactly midway between sunset and sunrise, and the Sun is exactly 180 degrees longitude on the opposite side of the celestial sphere.
[23] What is a constellation as astronomers define it today? What does it mean when an astronomer says “I saw a comet in Orion last night”?
Constellations are specific regions in the sky, precisely defined according to celestial longitude and celestial latitude. They are used to give position.
When an astronomer says they saw a comet in Orion, they are telling you in exactly which region of the sky they saw the comet, with precision.
[24] Draw a picture that explains why Venus goes through phases the way the Moon does, according to heliocentric cosmology. Does Jupiter also go through phases as seen from Earth? Why?
The Phases of Venus. As the Earth and Venus both orbit the Sun, Venus displays phases of illumination just like the Moon, which change depending on the relative Earth-Venus-Moon geometry.
Technically Jupiter does go through phases, but they are so slight as to be almost undetectable from Earth. This is because Jupiter is so far away from the Earth (and the Sun) that from our perspective, regardless of where Jupiter is in its orbit relative to the Earth, we essentially only see the illuminated face.
Jupiter has an extremely small phase angle — only 11.1° — so Jupiter is always at least 99% full from our perspective at minimum. All of the “superior planets” (those outside of the orbit of the Earth, like Mars, Jupiter, Saturn, Uranus, and Neptune) will never show quarter or crescent phases from the Earth, because that requires a unique relative position to Earth inside our orbit. From Pluto however, even these planets will show phases like the Moon.
[25] The hull of a ship disappears over the horizon before the deck, mast, or masthead when they go out to sea. Why could lookouts on the masthead see further than sailors on deck? Would there be any difference if the Earth was flat?
The reason why a person watching a ship go out to sea from a skyscraper overlooking the ocean can see the ship for longer than a person on shore (which is the same geometrical problem as a lookout on the masthead vs on deck) — even if both have telescopes — is because it is not a matter of how far one can see, but the Earth blocking line of sight.
The further away on object is, the more Earth curves over that distance. Eventually the are literally below the horizon. The reasons why someone higher up can see more distant objects, is because their elevation allows them to see “below the horizon” compared to someone lower down.
A lookout on the masthead will literally be able to see the tip of a mountain on a distant island before it even comes above the relative horizon for someone on deck. The “horizon” is a relative concept that has to do with geometry and the curvature of the Earth.
No, if the Earth was flat, then there would be no difference on how far one could see depending on whether a sailor was on the mast or on the deck.
In fact, being on the deck could actually be more advantageous if the Earth was flat, because their sightline actually covers a slightly shorter distance. If they had equally keen eyes, they would be able to resolve the object slightly earlier.
(P.S. These nautical arguments for a spherical Earth were well-known to sailors from the time of Columbus, and long before too.)
[26] Parallaxes of stars were not observed by ancient astronomers. How can this fact be reconciled with the heliocentric hypothesis?
Parallaxes of stars were not able to be observed by until recently, because the stars are so far away — it is a problem of geometry, optical magnification technology, and precision in measuring celestial positions.
The idea at the time of the ancient Greeks (which we used today to measure stellar parallax) is to observe the shift in position of a star by observing the star when the Earth is on opposite sides of the Sun — i.e. the March Equinox and the September Equinox.
The Greeks used the keenest-eyed soldiers, who were unable to detect any difference in position, and so it was concluded that the Earth did not move around the Sun.
Today, we use space telescopes, and can see billions more stars, and measure their positions with astronomical precision. We can measure differences in position down to probably millionths of a degree, when the Earth is on opposite sides of the Sun.
[27] Why do you think so many people still believe in astrology and spend money on it? What psychological needs does such a belief system satisfy?
It satisfies the need to be heard, and talk about one’s problem, and to think them through in some capacity. Furthermore it provides some degree of order in a chaotic world (who to marry, what job to take, what to major it) which people use as a lens anyways to project their dreams and desires, and so gain some sense of “divine guidance or destiny” in their decision — confidence, however misguided.
[28] Consider three cosmological perspectives — the geocentric perspective, the heliocentric perspective, and the modern perspective — in which the Sun is a minor star on the outskirts of one galaxy among billions. Discuss some of the cultural and philosophical implications of each point of view.
The narcissistic worth of humanity decreases rapidly from the geocentric to the modern perspectives.
In the geocentric model, the Earth and humanity were created by the gods in the center of the universe. However, as time progressed and our scientific understanding developed, it became clear that the hand of gods in the shaping of all things, if it can be seen, must be sought in the shaping of fundamental dynamics of natural law that set the stage for the formation of universes, galaxies, suns, and planets — and ultimately DNA, life, and humans — in an infinite reality.
Moreover, both Earth and Humanity decreased in importance. Not only are we in a tiny planet, around a mediocre star, far away from the center of our galaxy, we are about 13.7 light-years from the center of the universe. Our very existence as a species has been an evolution from other species over billions of years.
I am not saying that the hand of a universal deity cannot be seen … but it is certainly operating very slowly over a cosmic timescale.
For some this cultivates atheism, agnosticism, and pessimism. For me, I see a universal consciousness behind all creation, and so the vastness of space only excites me with the possibility of exploration — and adventure.
[29] (This is a problem for those who live in the Northern Hemisphere) The north celestial pole appears at an altitude above the horizon equal to the observer’s latitude. Identify Polaris (the North Star) and measure its altitude. Compare this with your latitude.
The apparent motion of the Sun against the fixed stars each year can be equally by either the Sun orbiting Earth, or the Earth orbiting the Sun.
That if the Earth were not only spinning on its axis, but hurtling through space around the Sun, that we would certainly be able to feel it.
[31] Although the Copernican system was largely correct to place the Sun at the center of all planetary motion, the model still gave inaccurate predictions for planetary positions. Explain the flaw in the Copernican model that hindered its accuracy.
The major flaw in the Copernican Model is that Copernicus still adhered to the age-old doctrine that the planets had to orbit on circular paths around the Sun.
In fact, the orbit of (most) of the Planets are elliptical in nature (squashed circles) with varying degrees of eccentricity.
[32] During a retrograde loop of Mars, would you expect Mars to be brighter than usual in the sky, about average brightness, or fainter in usual in the sky? Explain.
During a retrograde “loop” of Mars in the sky (I am assuming that the word “loop” implies that we are talking about the Ptolemaic Model) I would expect it to be brighter than average, because in his model, the retrograde portion of the epicycle takes it closer to the Earth and Sun, thus, presumably both brighter and larger.
[33] The Great Pyramid of Giza is believed to have been constructed nearly 5000 years ago. A shaft was discovered inside the pyramid from the central chamber to the bright star Thuban at that time. Thinking about Earth’s precession, explain why Thuban might have been an important star to the ancient Egyptians?
The star Thuban (Alpha Draconis) was bright at that time, and more importantly, it was the North Star from about 3900-1900 BCE).
This star was the center of the cosmos to the Egyptians, or it appeared that way, since all stars revolved around it. From their perspective in might have been viewed as the axis of the cosmos — the perfect symbol for aligning a monument that would last millennia to.
Today Thuban is no longer the North Star, nor is it as bright as it once was.
(Curiously, what is more interesting to me is that the Egyptians aligned their star shaft precisely with the Earth’s axis of rotation.)
[34] Explain why more stars are circumpolar for stars at higher altitudes.
More stars are circumpolar at higher latitudes because the celestial poles are higher in the sky — meaning that the host of stars that clearly revolve around the celestial poles are clearly visible each night, constantly above the horizon.
Compare this to regions at lower latitudes near the equator. From this perspective, the celestial pole(s) are only a few degrees altitude above the horizon, meaning there are fewer stars that never set, thus few circumpolar stars.
[35] What is the altitude of the North/South Celestial Pole in the sky from your latitude?
The further south that I drive, the lower the altitude of the North Celestial Pole would be. As I drive south, my latitude decreases as I drive around the curvature of the Earth, and so the North Star would get lower in the sky as well, closer to the horizon.
[37] Hipparchus could have warned us that the dates associated with each of the natal astrology sun signs would eventually be wrong. Explain why?
Hipparchus discovered precession (the slow wobble of the Earth’s axis, resulting in changes between the pole stars).
A consequence of this wobble that is precession, is a phenomenon called precession of the equinoxes — a continuous retrograde shift of the zodiacal constellations in how they align with times of the year.
For example, the constellation that the sun rises in at dawn on the March Equinox (June Solstice, September Equinox, or December Solstice) shifts by about 1 degrees every 72 years, or by a full sign every 2000 years or so.
Ancient star records (ancient even by the time of Hipparchus from the Greeks, Babylonians, and Egyptians centuries — or millennia — before Hipparchus’ time) would have recorded the constellation the Sun rose in in these important dates.
He would have known. He did warn us in fact, technically, because he told us about precession.
[38] Explain three lines of evidence that argue against the validity of astrology.
Precession and Precession of the Equinoxes — the “sun sign” no longer aligns with the constellation after which it is named that is said to give off the energy that influences personality, destiny, etc.
Statistical tests show zero correlation between real people and sun-sign archetypes — reductive Sun-Sign Astrology occasionally makes predictions like one sign is likely to become a politician, soldier, athlete, famous, wealthy, and so on, which is not statistically evident.
Other experiments show that people will see what they want to see — regardless of what is written in a horoscope, people have a tendency to see what they want to see.
[39] What did Galileo discover about the planet Jupiter that cast doubt on exclusive geocentrism?
He discovered that Jupiter has its own moons that orbit it — not Earth.
The significance of this is that geocentrism held that all things — the entire universe in fact — orbited the Earth. For Galileo to find definitive proof that objects can orbit bodies other than the Earth, this was like a crack in the foundation of ancient doctrine that eroded all the way down to bedrock.
[40] What did Galileo discover about Venus that cast doubt on geocentrism?
He discovered that Venus displayed phases in the wrong order, and at the wrong times, than predicted by geocentrism. However, the phase order was explained correctly by the heliocentric model.
Venus displaying phases alone fit into both models, but the order of phases supported heliocentrism, while contradicting heliocentrism.
Figuring For Yourself
[41] Suppose Eratosthenes had found that, in Alexandria, at noon on the first day of summer, the line to the Sun makes an angle 30° with the vertical. What, then, would he have found for Earth’s circumference?
Using the circumference of 60,000 stadia I calculated above, the length of his stadium in kilometers is about 0.212 km (212m). This is equal to about 1/5th of a km.
Using his original calculations of 1/50th of a circle (rather than 1/12th above) the measure of his stadium is
\[ \frac{12,740 km}{50} = 254.8 km = 5000 stadia \]
Therefore
\[ \frac{254.8km}{5000std} = 0.05096 \]
Each stadia is equal to about 0.051km (51m). Equal to bout 1/20th of a km.
[43] Suppose you are on a strange planet and observe, at night, that the stars do not rise and set, but circle parallel to the horizon. Next, you walk in a constant direction for 8000 miles, and at your new location on the planet, you find that all stars rise straight up in the east and set straight down in the west, perpendicular to the horizon. How could you determine the circumference of the planet without any further observations? What is the circumference, in miles, of the planet?
In this problem, you know that the planet has a 1/4 circumference of 8000 miles, since you walk that distance which resulted in a change of the planet’s axial tilt by 90 degrees.
Therefore the planet has a circumpolar circumference of about 32,000 miles.
