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Chapter 2 Motions of Earth. The material of Chapter 2 is covered in ASTR 1000, but there are a few concepts that are useful in ASTR 1001.
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Chapter 2Motions of Earth The material of Chapter 2 is covered in ASTR 1000, but there are a few concepts that are useful in ASTR 1001. 1. What we see in the sky results from the rotation of the Earth on its axis, the orbital motion of the Earth about the Sun, the orbital motion of the Moon about Earth, and, to a small extent, the gravitational effect of the Sun and the Moon on the Earth’s axis of rotation. 2. The motions of the Earth produce a fundamental frame of reference for stellar observations.
What we see in the sky is the direct result of our perspective from human beings on the surface of a spinning spheroidal planet that is also orbiting the Sun.
Celestial co-ordinates are like the terrestrial co-ordinates longitude and latitude. The celestial equivalent of longitude is right ascension, RA or α, and the celestial equivalent of latitude is declination, Dec or δ. It is possible to measure the co-ordinates of stars and planets relative to one another by measuring angles in the sky.
Two great observers: Hipparchus Tycho Brahe (2nd century BC) (1546-1601 AD)
Angle measuring devices: mural quadrant cross staff
Precession makes the Earth’s axis spin slowly about the ecliptic pole (┴ to orbit).
Astronomical Terminology Zenith. The point in the sky directly overhead. Nadir. The point directly beneath one’s feet. Azimuth. A measurement of angle increasing from north through east. Altitude (astronomical). A measurement of angular distance from the true horizon upwards. Ecliptic. The great circle in the sky along which the Sun appears to move because of Earth’s orbit about it. Right Ascension. A celestial co-ordinate like longitude on Earth, increasing eastwards. Declination. A celestial co-ordinate like latitude on Earth, measured from the celestial equator. Celestial Equator. The projection on the celestial sphere of Earth’s equator. Celestial Sphere. The imaginary sphere centred on the observer upon which the stars appear to be projected.
Astronomical Terminology 2 Diurnal. = daily (once a day). Insolation. The amount of sunlight falling on Earth’s surface. Constellation. A group of conspicuous stars designated by ancient star gazers. Zodiacal Constellation. A constellation lying in the band of sky around the ecliptic, where the Moon and planets are always found. Solstice. Time of greatest or smallest declination for the Sun. Equinox. Time when the Sun crosses the celestial equator. (Vernal = spring) Stellar Aberration. The apparent displacement in a star’s location in the sky of at most 20½ seconds of arc resulting from Earth’s orbital motion about the Sun at a speed of 30 km/s.
Sample Questions 3. Earth has a North Pole, a South Pole, and an equator. What are their equivalents on the celestial sphere? Answer: The equivalent features on the celestial sphere are the north celestial pole (NCP), the south celestial pole (SCP), and the celestial equator (CE).
4. Polaris was used for navigation by seafaring sailors such as Columbus as they sailed from Europe to the New World. When Magellan sailed the South Seas, he could not use Polaris for navigation. Explain why.
Answer. Polaris lies so close to the north celestial pole that it can only be seen from the Northern Hemisphere. Magellan sailed the South Seas where Polaris never rises above the horizon. Thus, he was unable to use it for navigation because it was never visible to him.
Chapter 3Gravity and Orbits: A Celestial Ballet The important concepts of Chapter 3 pertain to orbital motion of two (or more) bodies, central forces, and the nature of orbits. 1. What we see in the sky results from the rotation of the Earth on its axis, the orbital motion of the Earth about the Sun, the orbital motion of the Moon about Earth, and, to a small extent, the gravitational effect of the Sun and the Moon on the Earth’s axis of rotation. 2. The motions of the Earth produce a fundamental frame of reference for stellar observations.
Nicholas Copernicus (1473-1543) revived the heliocentric model for the solar system, where planetary orbits are envisaged as circular for simplicity. Even circular orbits are sufficient for understanding the difference between sidereal (star) period of a planet, Psid = time to orbit the Sun, and its synodic period, Psyn = time to complete a cycle of phases as viewed from Earth. The relationship between the two is best demonstrated by considering the amount by which two planets A and B advance in their orbits over the course of one day.
Over the course of one day, planet A advances through the angle Planet B advances through the angle The difference in the angles is the amount by which planet A has gained on planet B, which is related to its synodic period, i.e.:
In other words, Or: If A is Earth, and B a superior planet (orbits outside Earth’s orbit), then: For Earth P = 365.256363 days (a little more than 365¼ days), i.e.: Given two values, the third can be found !
The same technique can be used to relate a planet’s rotation rate and orbital period to the length of its day. The arrow is a fixed feature on the planet.
For example, the planet Mars has a synodic period of 780 days, which means it returns to opposition from the Sun every 2.14 years. But its true orbital period is 687 days, or 1.88 years, which means it returns to the same point in its orbit about the Sun every 1.88 years. Some further consequences: Mercury: Prot = 58d.67, Psid = 88d.0, Pday = 176d. Venus: Prot = 243d (retrograde rotation), Psid = 224d.7, Pday = 117d. Moon: Prot = 27d.3215, Psid = 365d.2564, Pday = 29d.5306. Earth: Prot = 23h 56m, Psid = 365d.2564, Pday = 24h. Which planet has the longest “day”?
How Kepler triangulated the orbit of Mars.He took Tycho’s observations of Mars separated by the planet’s 687d orbital period (with Earth at different parts of its orbit) and used them to triangulate the location of Mars, which was at the same point of its orbit.
Kepler’s study of the orbit of Mars resulted in his three laws of planetary motion: 1. The orbits of the planets are ellipses with the Sun at one focus. Actually they are conic sections. 2. The line from the Sun to a planet sweeps out equal areas of orbit in equal time periods. 3. The orbital period of a planet is related to the semi-major axis of its orbit by P2 = a3 (Harmonic Law).
Isaac Newton formulated Kepler’s Laws into a model of gravitation, in which: a mass attracts another mass with force inversely proportional to the square of the distance between the two, i.e. F ~ 1/d2. Forces produce acceleration of an object proportional to its mass, i.e. F = m×a, and objects stay at rest or in constant motion in one direction unless acted upon by a force.
Objects in orbit around Earth are constantly falling towards the Earth. They are acted upon by gravity, but are in free-fall towards Earth. They will not “fall” to Earth if their transverse speed is large enough.
The importance of Kepler’s 3rd Law is that, as shown by Newton, the constant of proportionality for a3 = P2 contains two constants, π (pi) and G (the gravitational constant), plus the sum of the masses of the two co-orbiting bodies. If one can determine orbital periods P and semi-major axes a, then one can derive the masses of the objects in the system: either planets or stars ! For example: Jupiter’s mass from the Galilean satellites.
Astronomers try to keep the calculations simple, so they usually omit π and G. Thus, the Newtonian version of Kepler’s 3rd Law is usually written as: where the sum of the masses of the two co-orbiting objects, “M1” and “M2”, is calculated in terms of the Sun’s mass, the orbital semi-major axis (~radius) “a” is calculated in terms of the Earth’s distance from the Sun, the Astronomical Unit, and the orbital period “P” is expressed in Earth years. The point to be emphasized is that a measurement of two of the parameters permits one to calculate a value for the third parameter. Astronomers use the relationship to measure the masses of stars.
Astronomical Terminology Rotation. The act of spinning on an axis. Revolution. The act of orbiting another object. Geocentric. = Earth-centred. Heliocentric. = Sun-centred. Opposition. When a planet is opposite (180° from) the Sun. Conjunction. When a planet is in the same direction as. Typically refers to conjunction with the Sun. Inferior planet. A planet orbiting inside Earth’s orbit. Superior planet. A planet orbiting outside Earth’s orbit. Prograde motion. When a planet’s RA increases nightly. Retrograde motion. When a planet’s RA decreases nightly. Astronomical Unit = A.U. The average distance between Earth and the Sun. Inertia. An object’s resistance to its state of motion. Inertial reference frame. = non-accelerated frame.
Astronomical Terminology 2 Eccentricity. The amount of non-circularity of an orbit, from round (e = 0.0) to very flattened (e = 0.9). Semi-major Axis. Half the length of the long axis of an ellipse, equivalent to the “radius” of an orbit. Orbital Period. The time taken for one object to orbit another object. Synodic Period. The time taken for an object to cycle through its phases as viewed from Earth. Inferior planet. A planet orbiting inside Earth’s orbit. Superior planet. A planet orbiting outside Earth’s orbit. Prograde motion. When a planet’s RA increases nightly. Retrograde motion. When a planet’s RA decreases nightly. Gravity. The force exerted by an object on any other object in the universe. “Zero gravity.” A fictional term referring to the apparent weightlessness of an object in free fall.
Sample Questions 12. Imagine a planet moving in a perfectly circular orbit around the Sun. Because the orbit is circular, the planet is moving at a constant speed. Is the planet experiencing acceleration? Explain your answer. Answer: Yes, it is. The planet experiences acceleration since it is constantly falling towards the Sun.
23. Suppose that astronomers discovered a comet approaching the Sun in a hyperbolic orbit. What would that say about the origin of the planet?
Answer. Objects in hyperbolic orbits are not bound to the object they are orbiting. Astronomers would therefore conclude that the comet is not bound to the solar system and must therefore have originated from “outside” the solar system.
