Explain the sky: days, seasons and phases
I can name the parts of the celestial sphere, explain the seasons by the Earth's tilt with labelled diagrams, calculate the Sun's noon height and the length of the day at a stated latitude, and explain the Moon's phases and why eclipses are not monthly.
- Round IExplain the sky: days, seasons and phases (this page)
- Round IILocate and predict with sky coordinatesComing
- Round IIILong cycles: precession, eclipses and the calendarComing
1 Learn
You will be able to
- Name the poles and the great circles of the celestial sphere (the celestial equator, the horizon and the ecliptic), give the 23.44 degree angle between the ecliptic and the celestial equator, and give Aristotle's evidence from the stars and from lunar eclipses that the Earth is a sphere.
- Explain the seasons by the tilt of the Earth's axis, and refute the distance explanation with the date of perihelion (early January) and the opposite seasons of the two hemispheres.
- Calculate the Sun's noon altitude from latitude and declination, as 90 degrees minus the size of their difference, and the length of the day from cos H = −tan(latitude) × tan(declination).
- Explain the phases as the changing view of the Moon's lit half and not the Earth's shadow, say why the cycle of phases (29.53 days) is longer than the Moon's 27.32-day orbit, and say why the Moon's orbit, tilted about 5.1 degrees, prevents an eclipse every month.
- LibraryThe Sky as a Sphere: Days, Seasons and the Phases of the Moon · The celestial sphere as a modelThe sphere, its great circles and the 23.44 degree obliquity, with Aristotle's evidence from the stars that the Earth is round.2 min glance · 7 min
- LibraryThe Sky as a Sphere: Days, Seasons and the Phases of the Moon · The seasons come from the tiltThe noon-altitude rule, and the evidence that the Earth's distance from the Sun cannot be the cause of the seasons.2 min glance · 7 min
- LibraryThe Sky as a Sphere: Days, Seasons and the Phases of the Moon · Day length depends on latitudeThe sunrise equation, day lengths at 45 and 60 degrees north, and why the Sun does not rise or set beyond the polar circles at the solstices.2 min glance · 7 min
- LibraryThe Sky as a Sphere: Days, Seasons and the Phases of the Moon · The phases of the MoonWhy the phases follow the angle between the Sun and the Moon and not the Earth's shadow, and why the cycle takes 29.53 days.2 min glance · 7 min
- LibraryThe Sky as a Sphere: Days, Seasons and the Phases of the Moon · Why eclipses are not monthlyHow the Moon's tilted orbit and its nodes gather eclipses into eclipse seasons, usually two a year.2 min glance · 7 min
- LabThe sky and the seasonsStates the noon-altitude and day-length formulas and the Earth-Sun distances, with sliders for latitude and date and readouts for testing your predictions.
- LabThe Moon's phasesA day-of-the-month slider that shows the lit fraction, the elongation and the time the Moon stands highest, and how fast the Moon gains on the Sun.
2 Practise
Answer each card from memory, then grade yourself honestly. The cards join your review deck and come back just before you would forget them.
-
recall
What is the obliquity of the ecliptic, and how large is it?
Show answer
It is the angle between the ecliptic (the Sun's apparent yearly path) and the celestial equator. It equals the tilt of the Earth's axis from the perpendicular to its orbit, 23.44 degrees, so the Sun's declination swings between plus and minus 23.44 degrees.
-
explain
Why does the date of perihelion show that distance from the Sun cannot be the cause of the seasons?
Show answer
The Earth is nearest the Sun (perihelion, 147.095 million km) in early January and farthest (aphelion, 152.100 million km) in early July. If distance caused the seasons, the north would have summer in January, when sunlight is about 6.9 per cent stronger. Instead January is northern winter. A distance cause would also give both hemispheres summer at once, but their seasons are opposite, as the tilt predicts.
-
apply
A place lies at latitude 40 degrees north. How high is the Sun at local noon at the equinoxes and at the June solstice?
Show answer
The noon altitude is 90 degrees minus the size of the difference between latitude and declination. At the equinoxes the Sun is on the celestial equator, so the declination is zero and the altitude is 90 - 40 = 50 degrees. At the June solstice the declination is +23.44 degrees, so the difference is 40 - 23.44 = 16.56 degrees and the altitude is 73.44 degrees.
-
connect
The lesson 'Eratosthenes Measures the Earth' depends on the Sun's height at noon. Where on Earth can the Sun stand directly overhead at noon, and when does that happen at latitude 23.44 degrees north?
Show answer
Only between 23.44 degrees south and 23.44 degrees north, because the noon altitude reaches 90 degrees only when latitude equals declination, and the declination never goes beyond plus or minus 23.44 degrees. At 23.44 degrees north it happens at the June solstice, when the declination reaches +23.44 degrees.
-
recall
What is the Earth's axial tilt, and how is the Sun's noon altitude found from latitude and declination?
Show answer
The tilt is about 23.44 degrees. The noon altitude is 90° − |latitude − declination|.
-
apply
At latitude 40 degrees north the Sun's declination is +23.44 degrees at the June solstice and −23.44 degrees at the December solstice. What is the noon altitude on each date?
Show answer
June: 90 − |40 − 23.44| = 73.44 degrees. December: 90 − |40 + 23.44| = 26.56 degrees.
-
recall
How long is the synodic month, and why is it longer than the Moon's orbit against the stars?
Show answer
It is 29.53 days. The Moon circles the Earth in 27.32 days against the stars. In that time the Earth has moved about 27 degrees round the Sun, so the Moon needs about two more days to return to the same position relative to the Sun.
-
explain
Why are the Moon's phases not caused by the Earth's shadow?
Show answer
The Sun always lights half the Moon, and a phase is how much of that half faces us. The Earth's shadow falls on the Moon only in a lunar eclipse, at full Moon.
-
apply
The Moon is 22.1 days past new Moon, in a month of 29.53 days. What, roughly, are its elongation from the Sun, the fraction of its disc that is lit, and the local time when it is highest in the sky?
Show answer
22.1 days is about three quarters of the month, so the Moon is about 270 degrees round its orbit from new Moon. The elongation is about 90 degrees, about 50 per cent is lit, and it is highest at about 6 am. This is the last quarter.
-
explain
Why is there not an eclipse at every new Moon and every full Moon?
Show answer
The Moon's orbit is tilted about 5.1 degrees to the plane of the Earth's orbit, so at most new and full Moons it passes above or below the line through the Sun and the Earth. An eclipse needs a new or full Moon near a node, which happens only in eclipse seasons about 173 days apart.
3 Prove it: the mastery check
5 questions drawn from a pool of 12. The pass mark is 80%. There is no time limit, and you can retake it with new questions; your best result counts.
4 Prove it: the performance task
Predict the Sun's noon height, then test the distance explanation
Part A (predict). At latitude 45 degrees north (any place at that latitude will do), use the rule 'noon altitude = 90 degrees minus the size of the difference between latitude and declination' to predict the Sun's noon altitude at the March equinox, the June solstice and the December solstice. Write the declination you use for each date. Do this before you open the sky lab. Part B (test). In the sky lab (/astronomy/#sky) set the latitude to 45 and press the March equinox, June solstice and December solstice buttons. Record the lab's declination, noon altitude and day length beside each prediction. If a prediction and the lab differ, find the step that caused the difference. Part C (explain). Draw two side-view sketches of the Earth's axis and parallel sunlight, one for the June solstice and one for the December solstice, marking the 23.44 degree tilt. Under them write four to six sentences. Say why the tilt makes the seasons, why distance does not (use the date of perihelion and the opposite seasons of the two hemispheres), and one assumption or limit of the celestial-sphere model.
What to hand in: A one-page worksheet: a table of predicted and lab values for the three dates, two labelled sketches (or a written description of each), and a paragraph of four to six sentences. Save it on this page or download it.
Saved only in this browser. To keep a copy, download it or export your progress.
5 Discuss: the seminar
Read the text, then think, write or talk through the question with someone. There is no answer key: the aim is a better question.
Astronomy · Round 1
- Aristotle, On the Heavens (De Caelo), II.14 (Bekker 296a24-298a20; the arguments for a spherical Earth run from 297a8)J. L. Stocks's Oxford translation on the MIT Internet Classics Archive. The page carries Book II in full without Bekker numbers; chapter 14 is 'Part 14', the last on the page. Aristotle argues for a spherical Earth from the curved shadow in lunar eclipses and from the stars that change as one travels north or south, and cites the mathematicians' circumference of 400,000 stades.
- Cleomedes, On the Circular Motion of the Celestial Bodies (Caelestia), Book I, chapter 10 in Ziegler's numbering (chapter 7 in Todd's numbering)T. L. Heath's English translation of the chapter, reproduced on a freely readable page; it also gives Posidonius's method. Cleomedes reports the measurement of Eratosthenes (about 276 to 194 BC). The length of the stade is uncertain, so the result in kilometres is too. Bowen and Todd (2004) is the standard translation but is not free.
Aristotle argues that the Earth is a sphere from the curved shadow in an eclipse and from the stars that change as you travel, and Cleomedes reports that Eratosthenes found its size from one shadow angle and one distance. Which of these arguments depends most on what we already assume, and which could you check yourself this week?
- Eratosthenes needs Syene and Alexandria on one meridian and the two lines to the Sun to be parallel. Cleomedes states both. Does his reason for the parallel lines convince you, and what would go wrong if they were not parallel?
- Cleomedes also reports Posidonius, who used the star Canopus, on the horizon at Rhodes and one forty-eighth of a circle above it at Alexandria. Compare the two methods: which hypotheses do they share, and which does each add?
- Aristotle gives 400,000 stades as the figure of unnamed mathematicians. How much weight can a figure carry when neither its method nor its unit is given?
Where astronomy is used
- LibraryNutrient Coverage on a Plant-Based Diet · Nutrient by nutrientThe vitamin D entry names latitude and season among the factors in sunshine; astronomy explains both through the Sun's height in the sky.2 min glance · 5 min
- LibrarySystems Thinking: Feedback, Queues, Information and Dynamics · Time series and signalsThe yearly cycle is the model of seasonality: separate it from trend before you call a change an anomaly.2 min glance · 7 min
Self-administered checks and self-assessed tasks: no credential is awarded. The design borrows from Khan Academy (mastery levels), WGU (competencies proved by assessment) and St. John's College (seminars on primary texts); this site is not affiliated with any of them.