Week of November 26,2018

Astronomy

What is a planet?

The planets in our solar system didn’t appear out of nowhere. Neither did the sun. They were all part of a big cloud of gas and dust. Gravity collected lots of material in the center to create the sun. The left over stuff swirled around the forming sun, colliding and collecting together. Some would have enough gravity to attract even more gas and dust, eventually forming planets. Watch this to learn more.

Scientists spent a lot of time arguing over what a planet actually is. In 2006, they came up with a definition. They said a planet must do three things. The first thing might seem obvious—it has to orbit around the sun. Second, it must be big enough to have enough gravity to force it into a spherical shape . And third, it must be big enough that its gravity cleared away any other objects of a similar size near its orbit around the Sun.

What about planets in other places?

This definition is very much focused on our own solar system. But there are also planets in places that are not our solar system. These planets are called exoplanets. They can be found circling around stars, just like the planets here in our own solar system. Does that mean that all planets form the same way? Are all planets made from a star’s leftovers?

That depends on who you talk to. What happens if a small cloud of gas floating out in the middle of nowhere forms a sphere because its gravity? Is that a planet, too? After all, Jupiter is a big sphere of gas. And both are just a mass of stuff that wasn’t quite big enough to form a bright, fiery star.

Big planet or tiny star?

Clouds of gas that don’t have enough material to form a bright star collect into spheres all the time. Most of the time these clouds form a type of star called a brown dwarf. They are pretty big compared to most planets, but they are not big enough to turn into the kind of star that makes lots of energy and gives off light.

https://spaceplace.nasa.gov/planet-what-is/en/

Relative sizes of different planets, our sun, and things in between.

But scientists recently discovered an even smaller gassy object in the middle of nowhere (read more about it here). It appears redder than most brown dwarfs, and is likely much younger than most, too. This object could have formed just like a brown dwarf—from a small cloud of gas. Or maybe it was created around a star and it somehow got flung off into space.

Some scientists are calling this object a planet. Others think that it can only be a planet if it formed around a star. They think that if it just formed from a cloud of gas, then it’s nothing more than a not-quite-star.

Foundations of Algebra

Arithmetic Sequences and Sums

Sequence

A Sequence is a set of things (usually numbers) that are in order.

Each number in the sequence is called a term (or sometimes "element" or "member"), read Sequences and Series for more details.

Arithmetic Sequence

In an Arithmetic Sequence the difference between one term and the next is a constant.

In other words, we just add the same value each time ... infinitely.

Example:

1, 4, 7, 10, 13, 16, 19, 22, 25, ...

This sequence has a difference of 3 between each number.
The pattern is continued by adding 3 to the last number each time, like this:

In General we could write an arithmetic sequence like this:

{a, a+d, a+2d, a+3d, ... }

where:

• a is the first term, and
• d is the difference between the terms (called the "common difference")

Example: (continued)

1, 4, 7, 10, 13, 16, 19, 22, 25, ...

Has:

• a = 1 (the first term)
• d = 3 (the "common difference" between terms)

And we get:

{a, a+d, a+2d, a+3d, ... }

{1, 1+3, 1+2×3, 1+3×3, ... }

{1, 4, 7, 10, ... }

Rule

We can write an Arithmetic Sequence as a rule:

x_n = a + d(n−1)

(We use "n−1" because d is not used in the 1st term).

Example: Write a rule, and calculate the 9th term, for this Arithmetic Sequence:

3, 8, 13, 18, 23, 28, 33, 38, ...

This sequence has a difference of 5 between each number.

The values of a and d are:

• a = 3 (the first term)
• d = 5 (the "common difference")

Using the Arithmetic Sequence rule:

x_n = a + d(n−1)

= 3 + 5(n−1)

= 3 + 5n − 5

= 5n − 2

So the 9th term is:

x₉ = 5×9 − 2
= 43

Is that right? Check for yourself!

Arithmetic Sequences are sometimes called Arithmetic Progressions (A.P.’s)

Advanced Topic: Summing an Arithmetic Series

To sum up the terms of this arithmetic sequence:

a + (a+d) + (a+2d) + (a+3d) + ...

use this formula:

What is that funny symbol? It is called Sigma Notation

(called Sigma) means "sum up"

And below and above it are shown the starting and ending values:

It says "Sum up n where n goes from 1 to 4. Answer=10

Here is how to use it:

Example: Add up the first 10 terms of the arithmetic sequence:

{ 1, 4, 7, 10, 13, ... }

The values of a, d and n are:

• a = 1 (the first term)
• d = 3 (the "common difference" between terms)
• n = 10 (how many terms to add up)

So:

Becomes:

= 5(2+9·3) = 5(29) = 145

Check: why don't you add up the terms yourself, and see if it comes to 145

Footnote: Why Does the Formula Work?

Let's see why the formula works, because we get to use an interesting "trick" which is worth knowing.

First, we will call the whole sum "S":

S = a + (a + d) + ... + (a + (n−2)d) + (a + (n−1)d)

Next, rewrite S in reverse order:

S = (a + (n−1)d) + (a + (n−2)d) + ... + (a + d) + a

Now add those two, term by term:

S	=	a	+	(a+d)	+	...	+	(a + (n-2)d)	+	(a + (n-1)d)
S	=	(a + (n-1)d)	+	(a + (n-2)d)	+	...	+	(a + d)	+	a
2S	=	(2a + (n-1)d)	+	(2a + (n-1)d)	+	...	+	(2a + (n-1)d)	+	(2a + (n-1)d)

Each term is the same! And there are "n" of them so ...

2S = n × (2a + (n−1)d)

Now, just divide by 2 and we get:

S = (n/2) × (2a + (n−1)d)

Which is our formula:

https://www.mathsisfun.com/algebra/sequences-sums-arithmetic.html

Week of November 12, 2018

Astronomy

Planet Earth: Facts About Its Orbit, Atmosphere & Size

Earth is the fifth largest of the planets in the solar system. It's smaller than the four gas giants —Jupiter, Saturn, Uranus and Neptune — but larger than the three other rocky planets, Mercury, Mars and Venus.

Earth has a diameter of roughly 8,000 miles (13,000 kilometers) and is round because gravity pulls matter into a ball. But, it's not perfectly round. Earth is really an "oblate spheroid," because its spin causes it to be squashed at its poles and swollen at the equator.

Water covers roughly 71 percent of Earth's surface, and most of that is in the oceans. About a fifth of Earth's atmosphere consists of oxygen, produced by plants. While scientists have been studying our planet for centuries, much has been learned in recent decades by studying pictures of Earth from space.

Earth's orbit

While Earth orbits the sun, the planet is simultaneously spinning on an imaginary line called an axis that runs from the North Pole to the South Pole. It takes Earth 23.934 hours to complete a rotation on its axis and 365.26 days to complete an orbit around the sun.

Earth's axis of rotation is tilted in relation to the ecliptic plane, an imaginary surface through the planet's orbit around the sun. This means the Northern and Southern hemispheres will sometimes point toward or away from the sun depending on the time of year, and this changes the amount of light the hemispheres receive, resulting in the seasons.

Earth's orbit is not a perfect circle, but rather an oval-shaped ellipse, similar to the orbits of all the other planets. Our planet is a bit closer to the sun in early January and farther away in July, although this variation has a much smaller effect than the heating and cooling caused by the tilt of Earth's axis. Earth happens to lie within the so-called "Goldilocks zone" around the sun, where temperatures are just right to maintain liquid water on our planet's surface.

Statistics about Earth's orbit, according to NASA:

• Average distance from the sun: 92,956,050 miles (149,598,262 km)
• Perihelion (closest approach to the sun): 91,402,640 miles (147,098,291 km)
• Aphelion (farthest distance from the sun): 94,509,460 miles (152,098,233 km)
• Length of solar day (single rotation on its axis): 23.934 hours
• Length of year (single revolution around the sun): 365.26 days
• Equatorial inclination to orbit: 23.4393 degrees

Earth's formation and evolution

Scientists think Earth was formed at roughly the same time as the sun and other planets some 4.6 billion years ago, when the solar system coalesced from a giant, rotating cloud of gas and dust known as the solar nebula. As the nebula collapsed because of its gravity, it spun faster and flattened into a disk. Most of the material was pulled toward the center to form the sun.

Other particles within the disk collided and stuck together to form ever-larger bodies, including Earth. Scientists think Earth started off as a waterless mass of rock.

"It was thought that because of these asteroids and comets flying around colliding with Earth, conditions on early Earth may have been hellish," Simone Marchi, a planetary scientist at the Southwest Research Institute in Boulder, Colorado, previously told Space.com. But in recent years, new analyses of minerals trapped within ancient microscopic crystals suggests that there was liquid water already present on Earth during its first 500 million years, Marchi said.

Radioactive materials in the rock and increasing pressure deep within the Earth generated enough heat to melt the planet's interior, causing some chemicals to rise to the surface and form water, while others became the gases of the atmosphere. Recent evidence suggests that Earth's crust and oceans may have formed within about 200 million years after the planet took shape.

Internal structure

Earth's core is about 4,400 miles (7,100 km) wide, slightly larger than half the Earth's diameter and about the same size as Mars' diameter. The outermost 1,400 miles (2,250 km) of the core are liquid, while the inner core is solid; it's about four-fifths as big as Earth's moon, at some 1,600 miles (2,600 km) in diameter. The core is responsible for the planet's magnetic field, which helps to deflect harmful charged particles shot from the sun.

Above the core is Earth's mantle, which is about 1,800 miles (2,900 km) thick. The mantle is not completely stiff but can flow slowly. Earth's crust floats on the mantle much as a piece of wood floats on water. The slow motion of rock in the mantle shuffles continents around and causes earthquakes, volcanoes and the formation of mountain ranges.

Above the mantle, Earth has two kinds of crust. The dry land of the continents consists mostly of granite and other light silicate minerals, while the ocean floors are made up mostly of a dark, dense volcanic rock called basalt. Continental crust averages some 25 miles (40 km) thick, although it can be thinner or thicker in some areas. Oceanic crust is usually only about 5 miles (8 km) thick. Water fills in low areas of the basalt crust to form the world's oceans.

Earth gets warmer toward its core. At the bottom of the continental crust, temperatures reach about 1,800 degrees Fahrenheit (1,000 degrees Celsius), increasing about 3 degrees F per mile (1 degree C per km) below the crust. Geologists think the temperature of Earth's outer core is about 6,700 to 7,800 degrees F (3,700 to 4,300 degrees C) and that the inner core may reach 12,600 degrees F (7,000 degrees C) — hotter than the surface of the sun.

Magnetic field

Earth's magnetic field is generated by currents flowing in Earth's outer core. The magnetic poles are always on the move, with the magnetic North Poleaccelerating its northward motion to 24 miles (40 km) annually since tracking began in the 1830s. It will likely exit North America and reach Siberia in a matter of decades.

Earth's magnetic field is changing in other ways, too. Globally, the magnetic field has weakened 10 percent since the 19th century, according to NASA. These changes are mild compared to what Earth's magnetic field has done in the past. A few times every million years or so, the field completely flips, with the North and the South poles swapping places. The magnetic field can take anywhere from 100 to 3,000 years to complete the flip.

The strength of Earth's magnetic field decreased by about 90 percent when a field reversal occurred in ancient past, according to Andrew Roberts, a professor at the Australian National University. The drop makes the planet more vulnerable to solar storms and radiation, which can could significantly damage satellites and communication and electrical infrastructure.

"Hopefully, such an event is a long way in the future and we can develop future technologies to avoid huge damage," Roberts said in a statement.

When charged particles from the sun get trapped in Earth's magnetic field, they smash into air molecules above the magnetic poles, causing them to glow. This phenomenon is known as the aurorae, the northern and southern lights.

Earth's atmosphere

Earth's atmosphere is roughly 78 percent nitrogen and 21 percent oxygen, with trace amounts of water, argon, carbon dioxide and other gases. Nowhere else in the solar system is there an atmosphere loaded with free oxygen, which is vital to one of the other unique features of Earth: life.

Air surrounds Earth and becomes thinner farther from the surface. Roughly 100 miles (160 km) above Earth, the air is so thin that satellites can zip through the atmosphere with little resistance. Still, traces of atmosphere can be found as high as 370 miles (600 km) above the planet's surface.

The lowest layer of the atmosphere is known as the troposphere, which is constantly in motion and why we have weather. Sunlight heats the planet's surface, causing warm air to rise into the troposphere. This air expands and cools as air pressure decreases, and because this cool air is denser than its surroundings, it then sinks and gets warmed by the Earth again.

Above the troposphere, some 30 miles (48 km) above the Earth's surface, is the stratosphere. The still air of the stratosphere contains the ozone layer, which was created when ultraviolet light caused trios of oxygen atoms to bind together into ozone molecules. Ozone prevents most of the sun's harmful ultraviolet radiation from reaching Earth's surface, where it can damage and mutate life.

Water vapor, carbon dioxide and other gases in the atmosphere trap heat from the sun, warming Earth. Without this so-called "greenhouse effect," Earth would probably be too cold for life to exist, although a runaway greenhouse effect led to the hellish conditions now seen on Venus.

Earth-orbiting satellites have shown that the upper atmosphere actually expands during the day and contracts at night due to heating and cooling.

Chemical composition

Oxygen is the most abundant element in rocks in Earth's crust, composing roughly 47 percent of the weight of all rock. The second most abundant element is silicon, at 27 percent, followed by aluminum, at 8 percent; iron, at 5 percent; calcium, at 4 percent; and sodium, potassium and magnesium, at about 2 percent each.

Earth's core consists mostly of iron and nickel and potentially smaller amounts of lighter elements, such as sulfur and oxygen. The mantle is made of iron and magnesium-rich silicate rocks. (The combination of silicon and oxygen is known as silica, and minerals that contain silica are known as silicate minerals.)

Earth's moon

Earth's moon is 2,159 miles (3,474 km) wide, about one-fourth of Earth's diameter. Our planet has one moon, while Mercury and Venus have none and all the other planets in our solar system have two or more.

The leading explanation for how Earth's moon formed is that a giant impact knocked the raw ingredients for the moon off the primitive, molten Earth and into orbit. Scientists have suggested that the object that hit the planet had roughly 10 percent the mass of Earth, about the size of Mars.

Life on Earth

Earth is the only planet in the universe known to possess life. The planet boasts several million species of life, living in habitats ranging from the bottom of the deepest ocean to a few miles into the atmosphere. And scientists think far more species remain to be discovered.

Researchers suspect that other candidates for hosting life in our solar system — such as Saturn's moon Titan or Jupiter's moon Europa — could house primitive living creatures. Scientists have yet to precisely nail down exactly how our primitive ancestors first showed up on Earth. One solution suggests that life first evolved on the nearby planet Mars, once a habitable planet, then traveled to Earth on meteorites hurled from the Red Planet by impacts from other space rocks.

"It's lucky that we ended up here, nevertheless, as certainly Earth has been the better of the two planets for sustaining life," biochemist Steven Benner, of the Westheimer Institute for Science and Technology in Florida, told Space.com. "If our hypothetical Martian ancestors had remained on Mars, there might not have been a story to tell."

https://www.space.com/54-earth-history-composition-and-atmosphere.html

Foundations of Algebra

Function Notation

The typical notation for a function is f(x). This is read as "f of x" This does NOT mean f times x. This is a special notation used only for functions.

However, f(x) is not the only variable used in function notation. You may see g(x), or h(x), or even b(a). You can use any letters, but they must be in the same format - a variable followed by another variable in parentheses.

FYI

The f(x) function notation was first used by a mathematician named Leonhard Euler in the 1700's.

Often times functions are written as an abbreviation. For example, if you are writing an equation to calculate the square of x. You may write this as a function and name it s(x). This is read as "s of x" for the "square of x".

Another example would be if I were writing an equation to determine the distance a car travels based on a certain time driving. I may write the function as d(t) for "the distance based on the time". This way, I know that t, which represents "time" is my independent variable and d(t) is the outcome.

Ok.. what does this really mean?

Remember when we graphed linear equations? Every equation was written as y = ..... Well, now instead of y = , you are going to see f(x) .....

f(x) is another way of representing the "y" variable in an equation.

Let's take a look at an example.

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Notice y is replaced with f(x), g(x), even h(a).

This is what is known as function notation. They all mean exactly the same thing. You graph all of these exactly as you would y = 2x +3. We are just using a different notation.

Evaluating Functions

In our introduction to functions lesson, we related functions to a vending machine. You "input" money and your "output" is candy or chips!

We're going to go back to that visual as we begin evaluating functions. We are going to "input" a number and our "output" is the answer.

If you can substitute and evaluate a simple equation, then you can evaluate functions. Remember, a function is basically the same as an equation. The only difference is that we use that fancy function notation (such as "f(x)") instead of using the variable y.

Pay close attention in each example to where a number is substituted into the function. I promise you will have no trouble evaluating function if you follow along. Take a look....

Example 1 - Evaluating Functions

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9Save

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Example 2 - Evaluating Functions

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http://www.algebra-class.com/function-notation.html

Week of November 5, 2018

Astronomy

THE DIFFERENT TYPES OF GALAXIES

OUR GALAXY CLASSIFICATIONS

As discussed in the section on galaxy classifications, Hubble found four distinct types of galaxies: elliptical, spiral, spiral barred and irregular. Although there are different types, we also learned that each galaxy contains the same elements, but these are arranged differently for each type. Just as every human is created with the same proteins that are configured uniquely, so are the galaxies uniquely configured with gasses, dust, stars and other elements.

SPIRAL GALAXY

Spiral galaxies are easily identified by observing three components common to all spiral galaxies. A spiral galaxy has a disk, a bulge, and a halo. The center of the galaxy is like a nucleus, containing a sphere shaped bulge that houses old stars and is devoid of dust and gas. The circular shape of the galaxy composes the disk. The arms of the spiral galaxy originate in the disk and are where new stars will form in a galaxy.

The sun in our galaxy is located in one arm and its stars are created in this portion of the galaxy, which contains the most gas in the galaxy. This area is rich in blue stars. The Halo is a spherical shaped collection of old stars and clusters known as globular clusters that is found in the outer edge of the galaxy. This stunning view of Spiral Galaxy Messier 74 from NASA taken with the Hubble telescope shows a bright bulge in the center with the arms spiraling outward.

When a spiral galaxy has no arms, S0, it is termed lenticular. These galaxies are so similar to E7 that identifying them can be tricky. Lenticular galaxies also occur with barred spiral and are classified as SB0. Spiral galaxies are the most common galaxy of the known universe, comprising about 77% of all known galaxies.

BARRED SPIRAL GALAXY

Barred spiral galaxies share the same features and functions as regular spiral galaxies, but they also have a bar of bright stars that lie along the center of the bulge, and extend into the disk. The bright bulge has very little activity here and contains mostly older, red stars. The bar and arms have lots of activity.including star formation.

While the classification for barred spirals is the same as it is for regular spiral galaxies, the bar must be considered as well. Short bars correlate to tighter galaxies and will be included in the designation SBa. SBb have longer bars and SBc are the longest. Most astronomers now agree that the Milky Way is a barred spiral galaxy.

ELLIPTICAL GALAXY

Elliptical galaxies can be recognized by their elongated spherical shape and their lack of nucleus or bulge at the center. Although there is no nucleus, the galaxy is still brighter in the center and becomes less bright toward the outer edges of the galaxy. Stars, gases and other materials are spread throughout the elliptical galaxy. An elliptical galaxy can be nearly round or long and cigar shaped.

It is believed that a great deal of the mass in an elliptical galaxy is due to the presence of a central black hole. These galaxies have very little activity and contain mostly old stars of low mass, because there aren’t the gasses and dust needed to form new stars.

IRREGULAR GALAXY

Irregular galaxies are composed of gasses, dust, stars, nebulous formations, neutron stars, black holes and other elements common to all galaxies. Irregular galaxies are named so because they have no definite shape, but like all galaxies, they are in constant motion, moving outward and away from the center of our universe. Irregular galaxies are divided into two classifications: Im and IO.

Im galaxies occur most often among irregular galaxies and may show a trace of the spiral galaxy arms. IO galaxies are completely random and can be called chaotic in nature. The Magellanic clouds that border our own Milky Way Galaxy are examples of Im galaxies. Approximately 20% of our galaxies are classified as irregular.

https://theplanets.org/types-of-galaxies/

Rate of Change

A rate of change is a rate that describes how one quantity changes in relation to another quantity. If x$x$ is the independent variable and y$y$is the dependent variable, then

rate of change=change in ychange in x$\text{rate}\text{of}\text{change}=\frac{\text{change}\text{in}y}{\text{change}\text{in}x}$

Rates of change can be positive or negative. This corresponds to an increase or decrease in the y$y$ -value between the two data points. When a quantity does not change over time, it is called zero rate of change.

Positive rate of change

When the value of x$x$ increases, the value of y$y$ increases and the graph slants upward.

Negative rate of change

When the value of x$x$ increases, the value of y$y$ decreases and the graph slants downward.

Zero rate of change

When the value of x$x$ increases, the value of y$y$ remains constant. That is, there is no change in y$y$ value and the graph is a horizontal line .

Example:

Use the table to find the rate of change. Then graph it.

Time Driving (h) x246Distance Travelled (mi) y80160240$\begin{array}{|cc|}\hline \text{Time}\text{Driving}(\text{h})x& \text{Distance}\text{Travelled}(\text{mi})y\\ 2& 80\\ 4& 160\\ 6& 240\\ \hline\end{array}$

A rate of change is a rate that describes how one quantity changes in relation to another quantity.

rate of change=change in ychange in x=change in distancechange in time=160−804−2=802=401$\begin{array}{l}\text{rate}\text{of}\text{change}=\frac{\text{change}\text{in}y}{\text{change}\text{in}x}\\ =\frac{\text{change}\text{in}\text{distance}}{\text{change}\text{in}\text{time}}\\ =\frac{160-80}{4-2}\\ =\frac{80}{2}\\ =\frac{40}{1}\end{array}$

The rate of change is 401$\frac{40}{1}$ or 40$40$ . This means a vehicle is traveling at a rate of 40$40$ miles per hour.

 https://www.varsitytutors.com/hotmath/hotmath_help/topics/rate-of-change