Review for first midterm

• Unregistered clickers: 90C33261 and A4B91508

• Midterm: Friday at 3-5pm• bring scantrons and calculators!

Scientific notation — the rules

Move the decimal over to the left until it is after the first digit. The number of places that you moved it is the exponent x.

For small numbers (less than 1) then you move the decimal to the right, and the exponent is negative

Significant figures

• The two rules of significant figures:

Your final answer should have the same degree of precision as the least precise of your input values.

When doing calculations that involve several steps, use more significant figures than necessary. Then round your final answer to the correct number of significant figures. This reduces the chance of round-off errors.

• Jon is 7ft tall. How tall is he in meters? Use the fact that 1ft = 0.305m.• There is a simple trick: just multiply by 1!

Unit conversions

1ft = 0.305m

1 =0.305m

1ft

1 = 0.305m

ft

• Jon is 7ft tall. How tall is he in meters? Use the fact that 1ft = 0.305m.• There is a simple trick: just multiply by 1!

Unit conversions

The celestial sphere — a useful picture

Imagine the stars as points of light fixed on a large rotating sphere surrounding the Earth

Polaris, the north star

The plane of the Earth’s orbit is called the ecliptic

The heliocentric model

The heliocentric model

You can see this region of the sky in December

You can see this region of the sky in June

The heliocentric modelPeople in the northern hemisphere only ever see this part of the sky

People in the southern hemisphere only ever see this part of the sky

The Earth’s axis of rotation is not perpendicular to the ecliptic — there is a 23○ axis tilt. This means that the equator is also tilted at 23○ from the ecliptic.

The axis of rotation points in the same direction (approximately towards Polaris) throughout the Earth’s orbit

The cause of the seasons

Th i i iThe reason it is warmer in sis that the sunlight is more cground when the Sun is highg g

h i iummer than in winterconcentrated on theher in the sky.y

When the light from the sun arrives almost perpendicular — i.e. the light is the most concentrated — you expect higher temperatures. This happens during the summer.

The cause of the seasons

• When it’s summer in the northern hemisphere, it’s winter in the southern hemisphere

• The difference between the seasons is minimized near the equator, because the angle that the sunlight impacts the Earth doesn’t vary as much

At summer solstice, the path of the sun is highest on the sky. This is the longest day of the year

The cause of the seasons

At winter solstice, the path of the sun is lowest on the sky. This is the shortest day of the year

The cause of the seasons

At spring equinox and the fall equinox, the path of the sun is in between — and it rises due east, and sets due west

The cause of the seasons

The phases of the Moon

• 02_MoonriseSetVsPhase.htm

The Moon — synchronous rotation

Eclipses

• Lunar eclipse — the Moon goes into the Earth’s shadow

• Solar eclipse — the Moon blocks out the sun, so (part of) the Earth is in the Moon’s shadow

Eclipses — lunar eclipse

Eclipses — solar eclipse

Eclipses — when do they occur?

Notice also that the plane of the Moon’s orbit is actually slightly tilted from the ecliptic. So the moon doesn’t usually pass directly in front (or behind) the Earth — it is a bit lower, or a bit higher.

Eclipses — when do they occur?

The heliocentric model — retrograde motion

Retrograde motion — the planets move across the celestial sphere, but occasionally they change directions for a while

The heliocentric model — retrograde motion

• mars_retrograde_motion.htm

The history of astronomy

We’ve gone from a picture where the Earth is the center of a perfect and unchanging Universe, surrounded by concentric spheres containing the planets and the stars…

…to a picture where we are orbiting the Sun, which is just one of billions of stars in the Milky Way, which is just one of billions of galaxies in an expanding Universe, which originated in a big bang 14 billion years ago.

The ancient Greek understanding of the universe

• Ancient thinkers believed that the heavens must be perfect and unchanging.

• Since the most perfect shape is a circle (obviously…), stars and planets must move on circular orbits and must also have perfectly spherical shapes.

• Geocentric, and the planets are embedded in spheres made of a transparent “fifth element”

The ancient Greek understanding of the universe

• The answer is that the planets were thought to move in epicycles, or circles-on-circles

The ancient Greek understanding of the universe

Ancient Green astronomy culminated in the Ptolemaic model after Claudius Ptolemy

• This model required the planets to move in epicycles, and the main circular orbits had to be slightly off-center from the Earth. So this model got mathematically complicated and was difficult to use.

• Nonetheless, it could make pretty accurate predictions for planetary positions, and was used for the next 1500 years.

Claudius Ptolemy (c. 100-170 A.D.)

Nicolaus Copernicus (1473-1543)

Copernicus made a heliocentric model. But his ultimate model was still pretty complicated (required epicycles) and not completely accurate, in part because he still required that all orbits be circular.

Tycho Brahe (1546-1601)

Brahe brought scientific precision to measurements of the positions of stars and planets.

Johannes Kepler, 1571-1630

• Worked briefly as Tycho’s assistant before his death; afterwards he took Tycho’s measurements back to Germany, and used them to update Copernicus’ heliocentric model

Johannes Kepler, 1571-1630

He concluded that the planets obey three “laws”• The orbit of a planet is an ellipse, with the Sun at one

focus

Johannes Kepler, 1571-1630

He concluded that the planets obey three “laws”• The orbit of a planet is an ellipse, with the Sun at one

focus• A line segment joining a planet and the Sun sweeps

out equal areas during equal intervals of time• The square of the orbital period of a planet is

proportional to the cube of the semi-major axis of its orbit: p2=a3

Galileo Galilei, 1562-1642

• Constructed the first (useful) telescopes, which he used to make several crucial discoveries

Galileo used telescopes to find that:• Jupiter has moons orbiting around it• Venus has phases, just like the Moon — suggesting

that Venus orbits the Sun• The Moon has craters and mountains; it isn’t

“perfect”• The Milky Way is just a collection of stars

He used these arguments, along with Newton’s first law, to argue in favor of heliocentrism

Galileo Galilei, 1562-1642

Two major questions in early 20th centuryastronomy

In 1700s and 1800s it became universally accepted that the planets orbit the Sun, and that the Sun is just one of many stars in the Milky Way galaxy. But what about the rest of the Universe?

• Is the Milky Way galaxy all that there is?

• Does the Universe evolve, or is it eternal and unchanging?

• Edwin Hubble then used Cepheid variable stars to estimate the distances to the “spiral nebulae” — finding that they are very far away indeed, and hence that they are actually separate galaxies!

• He also showed (as did others) that these other galaxies are receding from us, and so the Universe is not unchanging.

(1889-1953)

Two major questions in early 20th centuryastronomy

What is science?

• Seek explanations for observed phenomena that rely solely on natural causes

• The creation and testing of models of nature that explain phenomena as simply as possible

• A scientific model must make testable predictions about natural phenomena that would force us to revise or abandon the model if the predictions do not agree with observations

The progress of science has been largely about learning not to impose our pre-conceived notions on nature; you go where the evidence takes you

Occam’s razor

• Given two competing — and equally successful — explanations for an observed phenomenon, the simpler one is preferred

• This idea has really been taken to heart by scientists. The development of physics has largely been driven by the search for a small number of principles or equations that describe a huge range of phenomena

William of Occam1285-1347

Classical mechanics

The theory of classical mechanics (also often called “Newtonian mechanics”) is concerned with the motion of objects.

Basic concepts in movement

• speed — the distance travelled in a certain amount of time. (For example, a car driving at 60mph.)

• velocity — this refers to both the speed and the direction. (For example, a car driving north at 60mph has a different velocity than a car driving east at 60mph.)

• acceleration — a change in velocity. (So a car that is turning but maintaining a constant speed is accelerating. So is a car that is going in a straight line but is slowing down; this is negative acceleration)

Acceleration

Newton’s first law (originally due to Galileo)

“A body at rest remains at rest and a body in motion continues to move at constant velocity along a straight line unless acted upon by an external force” (this is Galileo’s formulation)

This law may seem contrary to our everyday experience. But that’s because in our everyday experience, there are always the forces of gravity, friction, and air resistance at work

Newton’s second law

Force = mass x acceleration

F=ma

A force is just a push or a pull that causes something to accelerate, i.e. to change its velocity

Newton’s third law

• For every force, there is an equal and opposite force

Newton’s third law

Momentum

• An object at rest will remain at rest, and an object in motion will remain in motion with constant velocity

• Why is this? Because of momentum!

• momentum = mass x velocity

• So we can re-state Newton’s first law as: an object’s momentum will not change unless a force is applied to it

Angular momentum

• There is also a “circling momentum”, or angular momentum

• A version of Newton’s first law also applies to angular momentum — an object’s angular momentum will not change unless some external force is applied. Angular momentum is conserved.

• This is why the Earth continues to orbit around the Sun, even though nothing is pushing it. And it’s why the Earth continues to spin.

Angular momentum

angular momentum = mass x velocity x distance

So, at fixed angular momentum, an object that is further away will move slower

Angular momentum

angular momentum = mass x velocity x distance

So, at fixed angular momentum, an object that is further away will move slower

Energy

• Energy is usually defined as “the ability to do work” or “the ability to make things move”

• There are different kinds of energy• kinetic energy — the energy of movement• thermal energy — heat is a kind of energy• potential energy — energy that is being stored• radiative energy — this is just light• mass-energy — this is how nuclear reactions work

Potential energy

• This is energy that is being stored up, and can be converted into a different kind of energy

This is an example of gravitationalpotential energy, which is a particularly important concept in astronomy

Conservation laws in physics

• Energy is conserved

• Momentum is conserved

• Angular moment is conserved

Conservation laws — angular momentum

The orbital radius is large,so the velocity is small

The orbital radius is small,so the velocity is large

angular momentum = mass x velocity x radius

Conservation laws — energy

The gravitational potential energy is large, so the kinetic energy is small

The gravitational potential energy is small, so the kinetic energy is large

orbital energy = kinetic energy + potential energy

Gravity — an inverse square law

Gravity follows an “inverse-square” law:

This means that if the distance between two objects doubles, the force between them is reduced by a factor of 4. If the distance triples, the force is reduced by a factor of 9.

The gravitational constant:

Gravity — tides

It is a common misconception that the high tide is caused by the Moon “pulling” the oceans toward it. But this can’t be quite true — if it were, how come there’s a high tide on Earth both in the direction of the Moon, but also in the opposite direction?

Gravity — tides

• Gravity from the Moon pulls harder on the near side of the Earth than on the far side, squishing the Earth and creating a bulge on both sides. This causes high tides.

• The Earth rotates, but the locations of the bulges are always approximately in-line with the Moon.

The Sun

• Over 99.8% of solar system's mass• Made mostly of hydrogen and helium• Converts 4 million tons of mass into energy each second

(E=mc2)

Some basic facts about the solar system:similar spin and orbital rotation among the planetsAll of the major planets:

• lie approximately in the same plane• follow ~circular orbits• orbit in the same direction, and in the same direction as

the Sun’s rotation• rotate in the same direction and have axes of rotation

that are ~perpendicular to the orbital plane (with the exceptions of Venus and Uranus)

Some basic facts about the solar system:similar spin and orbital rotation among the planetsAll of the major planets:

• lie approximately in the same plane• follow ~circular orbits• orbit in the same direction, and in the same direction as

the Sun’s rotation• rotate in the same direction and have axes of rotation

that are ~perpendicular to the orbital plane (with the exceptions of Venus and Uranus)

• have moons that also orbit in the same direction (with the exceptions of Mercury and Venus which don’t have moons, and Neptune that has a moon orbiting in the opposite direction)

Some basic facts about the solar system:two kinds of (major) planets

• The four inner planets (Mercury, Venus, Earth, and Mars) are relatively small, and are rocky. These are the terrestrial planets

• The four outer planets — Jupiter, Saturn, Uranus, and Neptune — are larger, more massive, and made primarily of hydrogen and helium gas. These are the jovian planets (aka gas giants)

• Asteroids are small rocky objects, and most are orbiting in the asteroid belt between Jupiter and Mars.

Some basic facts about the solar system:asteroids

Some basic facts about the solar system:comets• Comets are small objects of rock and ice (dirty snowballs)

• 1000s exist out in the Kuiper belt beyond Neptune, and many more than that exist out in the spherical Oort cloud surrounding our solar system.

• Some have highly elliptical orbits, and come into the inner regions of the solar system and heat up. Then we can see their tails of melted material.

How did the solar system form?

06_CollapseSolarNebula

The nebular theory of solar system formation

• A nebula becomes dense and massive enough to start self-gravitating. It begins collapsing.

• But because it has some net angular momentum, it begins rotating very rapidly. So there is a limit to how much it can collapse in the plane of rotation, but it can fully collapse in the vertical direction, leaving a spinning protoplanetary disk with a dense protostellar core

• The gas heats up as it contracts; gravitational potential energy -> thermal energy

Planets condense out of the protoplanetary disk

The nebular theory of solar system formation

Inside of the frost line, only rocks and metals can condense into planetesimals, and hydrogen compounds (including water) remain in a gaseous state. But beyond the frost line hydrogen compounds can also condense into ices.

• But there simply isn’t much rocky material and metals around, so the planetesimals inside the frost line never become very big -> so the inner terrestrial planets are relatively small and rocky

• However the outer planetesimals have a lot more hydrogen, helium, and hydrogen compounds that can condense. So they can become much more massive, and due to their gravitational force can start attracting even more material -> so the outer jovian planets are large and contain mostly hydrogen, helium, and hydrogen compounds

The nebular theory of solar system formation

Eventually the protostellar core becomes so hot, and so dense, that it begins to undergo nuclear reactions. It becomes a star.

The nebular theory of solar system formation

The protoplanetary disk begins to clear out because:• The planets accrete the material around them• The planets eject material that they don’t accrete by

gravitational perturbations• After the star begins to shine brightly it eject winds,

blowing out remaining diffuse material

The nebular theory of solar system formation

• The remaining planetesimals within the frost line are asteroids in the asteroid belt.

• The remaining planetesimals outside of the frost line are comets in the Kuiper belt

How does this theory explain the asteroids and comets?

How do planets get their moons? Three ways…

The moons of the jovian planets probably formed out of disks in the same way that the planets formed out of the protoplanetary disk

There may not have been enough rocks/metals around for the terrestrial planets to form significant disks, and therefore moons. Mercury and Venus don’t have moons. However Mars has two moons which may have been asteroids that were gravitationally captured

How do planets get their moons? Three ways…

There may not have been enough rocks/metals around for the terrestrial planets to form significant disks, and therefore moons. Mercury and Venus don’t have moons. However Mars has two moons which may have been asteroids that were gravitationally captured

How do planets get their moons? Three ways…

Neptune also has a moon that orbits in the “wrong” direction, so that may also have been gravitationally captured.

So then how do we explain our Moon? It is too large for gravitational capture. It also has a significantly lower density than the Earth, suggesting that it did not form in a disk of the same material…

How do planets get their moons? Three ways…

The giant impact hypothesis — when the Earth was young it may have collided with another smaller (Mars-sized) planet, spewing debris out into a ring which eventually condensed into the Moon

How do planets get their moons? Three ways…

Age of the solar system

The most accurate way to determine the age of the solar system is through radiometric dating (also called radioactive dating)

• The atoms of certain elements can spontaneously decay into other kinds of elements

• Example: potassium-40 (19 protons and 21 neutrons) decays into argon-40 (18 protons and 22 neutrons) with a half-life of 1.25 billion years. This means that after 1.25 billion years, half of the potassium has become argon. After another 1.25 billion years, half the remaining potassium also becomes argon. And so on.

Age of the solar system