Tag: math

BAFact Math: Jupiter is big enough to swallow all the rest of the planets whole

By Phil Plait | August 22, 2012 9:46 am

[BAFacts are short, tweetable astronomy/space facts that I post every day. On some occasions, they wind up needing a bit of a mathematical explanation. The math is pretty easy, and it adds a lot of coolness, which I’m passing on to you! You’re welcome.]

Today’s BAFact: Jupiter is so big you could fit every other planet in the solar system inside it with room to spare.

Volume is a tricky thing. Our brains are pretty good at judging relative linear sizes of things: this thing is twice as long as that thing, for example. But volume increases far more rapidly than linear size. Take a cube where each side is one centimeter. It has a volume of one cubic centimeter (cc). Now double the length of each side to 2 cm. It looks twice as big, but its volume goes up to 8 cc! The volume of a cube is a the length x width x height, so there you go.

Spheres are the same way: the volume increases with the cube of the radius. Specifically, volume = 4/3 x π x (radius)3. So one sphere might look slightly larger than another, but in fact have a lot more volume.

Such is the way of Jupiter. I see pictures of it compared to the other planets, and honestly Saturn looks only slightly smaller – Saturn’s radius is about 60,000 km compared to Jupiter’s 71,000. But that turns out to make a huge difference in volume!

Here’s a table I created to compare the planets. The first number column is the planet’s equatorial radius in kilometers (the biggest planets aren’t perfect spheres, but as you’ll see this doesn’t matter). The second number column is the volume in cubic km based on that radius. The third is the volume of the planet divided by the volume of Jupiter (so that ratio = 1 for Jupiter itself). The last column is the same, but rounded to two decimal places to make it easier to read.

The big conclusion here is pretty obvious when you look at that last column. Even though Saturn is only a little smaller than Jupiter, it only has 60% of the big guy’s volume! Uranus and Neptune together are only another 9%. If you combine all the planets in our solar system, they add up to only about 70% of Jupiter’s volume. That leaves a lot of room left over for all the moons and asteroids in the solar system, too!

So Jupiter really is a monster. There’s a half-joke astronomers say: The solar system consists of the Sun, Jupiter, and assorted rubble. As you can see, that’s really not that far off from the truth!

Image credit: NASA

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Announcing BAFacts: a daily dose of sciencey fun

MORE ABOUT: BAFacts, Jupiter, math, Saturn, volume

Dropping a dime on the Moon

By Phil Plait | May 5, 2012 10:59 am

So, tonight is the so-called Supermoon, when the Moon happens to be full at the same time it’s at perigee, the point in its orbit closest to the Earth. This makes it somewhat larger and brighter than normal, and that’s getting a lot of attention in the press. I pointed out a few days ago that in reality, you almost certainly won’t notice the difference between this full Moon and any other, mostly because the difference is small, and our eyes and brain are terrible at judging things like that without something to directly compare it to.

I was thinking about this last night as I watched the almost-full Moon rise in the east (which, I’ll add, ironically looked huge due to the Moon Illusion!), and thought of something that might help illustrate this last point.

Monetary eclipse

Imagine you go outside tonight to look at the full Supermoon rising in the east. Imagine also you’re holding a US dime in your hand (if you live in another country, feel free to substitute your local currency, but beware of the math; hang on a minute to see).

Let me ask you this: How far away would you have to hold the dime so that it appears as big as the Moon to you?

A few inches? A foot? (Convert to metric if you wish). Go ahead, guess!

… OK, ready? [Answer is below the fold so as not to spoil it.]

Read More

Mine goes to 11 11 11

By Phil Plait | November 11, 2011 1:11 am

Geeks across teh intertubez are giddy with delight today. Why? Because it’s November 11, 2011, of course! And if you’re in the US, you’d write this as 11/11/11, so of course this tickles the heart of any true mathematically-inclined nerd such as me. now, in the US we put the month first — MM/DD/YY — which is somewhat silly; in England and other realms they write it more logically with the units getting bigger left-to-right, so for them it’s DD/MM/YY, or, contrary to us yanks, 11/11/11.

OK, so anyway, it’s all ones. Why is that cool?

Well, it just is. Duh. But deep down, this goes to the root ("root"! HAHA! Oh man, I’m a math riot) of how we count. And I am never one to miss a chance to lecture on nerdalicious topics, so stick with me for a bit.

The power of ten compels you!

On the web, which consists of 87% dorks (look it up!), this date is special because it looks binary. For those of you unfamiliar with this, we humans tend to use the number ten as the basis of our counting. Our numbers reflect this: we break things down into powers of ten when we write out a number.

For example, the number 1234 — one thousand, two hundred, thirty-four — has four digits, each representing a power of ten. On the right, we have the "ones" place, where 1 = 100. For our example, there are four of them.

Next, moving left, is the "tens" place, and 10 = 101. For 1234, we have three tens, or thirty.

See how this works? Next is the "hundreds" place, and 100 = 102. Two of those is two hundred.

Last, all the way to the left is the " thousands" place, or 103. We have one of those, for one thousand.

Add ’em together, and you get 1 thousand, 2 hundred, thirty four. This is actually a very clever way to write down a number. Compact, efficient, and makes simple arithmetic possible. It’s not that hard to learn how to add fairly large numbers in your head due to this notation. Try that in Roman numerals!

An important note: we use single digits to represent numbers from 0 – 9, then two digits for 10 – 99. Obvious, right? But if you think about it, you’ll see there’s a reason: you don’t need a one digit numeral for "ten", because it has its own column. Counting up from 0, once you reach the base number of ten you just put a 1 in the next column to the left and a 0 in the column on the right. Simple, neat, and efficient.

I’d even say it’s a brilliant innovation in notation, and is what allows us to represent huge numbers simply. Roman numerals use symbols for certain numbers, and you just mash them together to represent a bigger number (sometimes subtracting them, too, which is truly awful). Our number 1234 would be MCCXXXIV, which is unwieldy. And adding a number to that is completely nonintuitive. It’s more like a code than a system of notation for numbers. Our current method is way, way better. In fact, I’m not really sure why Roman notation is even taught anymore. Seriously, who needs it? Movie copyrighters and SuperBowl fans. That’s about it.

All your base are belong 2 us

But it turns out, you don’t have to use base 10. We have ten fingers, so it’s somewhat natural for us. But in fact other bases are possible, and sometimes even preferred. Like binary.

Binary is the simplest system. It’s base 2. So when you write a number, you use powers of two in the places, not ten. So the columns go from right to left like this:

20 = 1
21 = 2
22 = 4
23 = 8
24 = 16
25 = 32

and so on. You can only use a 0 and 1 in this case, and that makes sense. Why? Because, like base 10, you use a two-digit numeral to represent your base. What we think of as "2" in base ten becomes 10 in binary. It’s the base to the power of 1, just like it is in base 10 (which is called decimal, by the way). I’ll add that when you use the number 10 in decimal you call it "ten", but when you use 10 in binary you call it " one zero" to avoid confusion. If you call it "ten" then all the math people will laugh and make fun of you, and not invite you to their Star Trek marathon*.

In binary, just like in base 10, we add the columns together to make a number. So let’s pick an arbitrary number, like 42. If we look to the powers of 2, we see it’s 25 (32) + 23 (8) + 21. So we’d write it in base 2 as 101010. You have to put in the zeroes as place holders, or else you can’t see what power is what. But that makes sense: it’s 1 x 25 + 0 x 24 + 1 x 23 + 0 x 22 + 1 x 21 + 0 x 20: 101010.

It may seem more cumbersome than base 10, since 42 is only two digits in decimal but 6 in binary. True, but it’s really easy to represent numbers in base 2, since a 0 and 1 can be represented in lots of ways, like an arrow pointing up or down, or a section of a DVD with a tiny laser-burned microscopic pit or no tiny laser-burned microscopic pit, and so on. Anything that exists in two states (on/off, filled/empty) can be used to count in binary. Electric circuits do that, they can be made small and fast, and hey, don’t computers run on electricity?

So yeah. That’s why binary is used in computers.

Will you still need me, will you still feed me, when I’m sixty-three?

And finally, that brings us back to the date! November 11, 2011 is 11/11/11 or just 111111. And that looks like a binary number!

So what is binary 111111 in decimal form?

It’s 1 x 20 + 1 x 21 + 1 x 22 + 1 x 23 + 1 x 24 + 1 x 25 = 1 + 2 + 4 + 8 + 16 + 32 = 63.

That’s just one less than the next higher power of two, 64 (26). In binary, a number full of 1s is like a decimal number full of 9s. Add one to it, and you bump up to the next power of your base.

And that’s why some dorks people think today is cool. It’s the last binary number this year, and in fact we can’t get another date that looks like a binary number until 01/01/00, or January 1, 2100! That’s the first day of the last year of the 21st century. It’s a long wait.

And? That date will be 010100, or 20 in decimal.

I don’t know if people will hold binary parties tonight (you can either go or not go) or how they will celebrate — one person on Twitter said he’s getting married today, and I have to admire that — but for me, it’s just fun to think about the numbers.

But then, I’m a dork too.

The Ramans do everything in threes…

So I’ll leave you with a quiz. Base 3 (called ternary) is fun as well, and I don’t want to leave it out!

In base 3, you can use the numerals 0, 1 and 2. As it happens, today’s date consists of those numerals! So we can write out our entire date in ternary, including the full year: 11/11/2011. I ask you: what’s that in decimal? (Those of you who are in other parts of the world, where you’d say it’s 2011/11/11, you’re invited too). You could cheat and look it up online, but that’s no fun. Being a dork means doing it long hand sometimes!

And fun fun fun, as it happens, that ability to use a 2 means more dates in ternary are coming soon. So here’s a semi-trick question: when’s the next all-ternary date?

And, of course: Happy binary 63 day!

Image credits: The Flickr streams of Kichigai Mentat, Joe Shlabotnik, goldberg, and sgilliesm all licensed under Creative Commons.

* Spare me the nerd rage, please. I’ll be watching Stargate.

OK, so it’s not arbitrary. It’s a pronic number!

CATEGORIZED UNDER: Cool stuff, Geekery

The topographic Earth

By Phil Plait | October 18, 2011 1:30 pm

This is pretty nifty: a new elevation map of the Earth has just been released by NASA and Japan. It’s a "significantly improved" version of one that came out in 2009.

It uses Japan’s ASTER, the Advanced Spaceborne Thermal Emission and Reflection Radiometer, an instrument on board NASA’s Terra satellite. Terra is an Earth-observer, with detectors on board used to study various properties of our planet. ASTER looks both straight down and slightly behind the satellite’s track on the Earth is it passes. Over time stereo image pairs are created, and these can be used to create very high-resolution elevation maps (called topographic maps) of the surface of the Earth.

The new images are higher-res than before, and cover the Earth better to the tune of 260,000 more images. As an example of what can be done, they used it to make this map of the Grand Canyon:

[Click to enmesanate.]

One thing that struck me as funny when I read it: the coverage of ASTER’s observations goes from the Equator to as far north and south as 83° latitude… and they say that this is 99% of the Earth! That sounds odd, doesn’t it? You’d think the north and south poles of the Earth from 90° to 83° would be more than that, but in fact it’s true.

The portion of a sphere above a certain latitude line is called a cap, and the area of that cap depends on the latitude in question, and the radius of the sphere. I drew myself a diagram, fiddled with the numbers a bit, and found that the area of the Earth north of 83° compared to the surface area of the northern hemisphere is about 0.75%! So in fact, ASTER covered a bit more than 99% of the Earth’s surface, even if it never got past that 83°latitude.

Math! Surprising people since the time of Pythagoras.

Anyway, if you want to download the ASTER data yourself, you can: it’s public. Japan has a copy, and so does the USGS. I imagine it won’t be long before it’s integrated into Google Earth and all that too. Living in the future is pretty cool.

Image credit: NASA/GSFC/METI/ERSDAC/JAROS, and U.S./Japan ASTER Science Team

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Staring down an active volcano’s throat
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Satellite view of a volcanic pressure valve
Volcano study in red

CATEGORIZED UNDER: Cool stuff, NASA, Pretty pictures

Neptune is *really* far away

By Phil Plait | September 19, 2011 10:30 am

Mike Brown is an astronomer, specifically one who studies Kuiper Belt Objects, those giant frozen iceballs that haunt the solar system out past Neptune.

In fact, Neptune’s biggest moon Triton has a lot of characteristics similar KBOs — it may be one captured by Neptune — so observing it gives an interesting opportunity for a compare-and-contrast study. So this past weekend Mike was using the Keck telescope in Hawaii to observe Triton along with its (adoptive?) parent planet, and took this fantastic image of the pair:

[Click to poseidenate.]

This false-color image shows the two worlds in the infrared, specifically at a wavelength of about 1.5 microns, twice what the human eye can see. Methane strongly absorbs this color of light, so where Neptune (in the upper left) looks dark you’re seeing lots of methane clouds, and where it’s bright there are clouds higher up, above the methane. Triton is in the lower right, and is bright because it’s covered in ice which is highly reflective.

Now this is all very pretty and interesting and sciencey, but if you know me at all you know there’s more to this story.

Mike tweeted about the image, and I oohed and ahhhed at it, of course. But then he tweeted again, saying he was also observing Jupiter’s moon Europa, but it was too bright to get good images using the monster 10-meter Keck telescope. It "saturated the detector" which is astronomer-speak for "overexposed".

That’s funny, I thought. Neptune looks fine in the image, and the random noisy grain in it makes it clear Mike wasn’t anywhere near saturating the image. Now I know Europa is closer to the Earth, so it should look brighter, but geez, it’s a moon, and a lot smaller than Neptune. How could it be too bright to image?

It turns out my all–too–human and all–too–miserable sense of scale has failed me again. Math to the rescue!

Read More

CATEGORIZED UNDER: Astronomy, Cool stuff, Pretty pictures

Nature by Numbers

By Phil Plait | January 13, 2011 7:00 am

Don’t think math is beautiful? Watch this:

Make sure you set the resolution to 720p!

Tip o’ the Fibonacci series to Dan Durda, who sat next to me in countless math classes in college.

CATEGORIZED UNDER: Cool stuff, Science

30 years, a half million asteroids

By Phil Plait | August 27, 2010 7:00 am

By now you may have heard about this interesting video showing how many asteroids we’ve discovered since 1980. It’s pretty cool!

I have no idea how accurate it is, but the numbers seem about right; I know there are several hundred thousand known asteroids in the main belt between Mars and Jupiter [Note: the creator of the video talks about this in the comments below]. Note that a lot of the ones you see toward the end get close to Earth; according to the JPL Near-Earth Object site, almost 7200 near-Earth asteroids have been cataloged as of August 20, 2010! Of these, 815 are larger than about 1 kilometer in diameter, and 1137 are considered to be potentially hazardous; that is, have a chance (however small) of hitting the Earth.

It’s interesting to see where the asteroids are when discovered; usually in the opposite direction of the Sun, because that’s where surveys tend to look. Right at the end you’ll see two white patches at 90° from the direction of the Sun on either side. If I were a betting man — and I am — I’d wager those were from WISE, an infrared survey satellite. It scans the sky constantly, looking at right angles to the Sun, and I know it’s designed to find asteroids.

More interesting, to me, is how crowded the asteroid belt looks! But don’t be deceived. Read More

CATEGORIZED UNDER: Astronomy, DeathfromtheSkies!

SMBC gets the finger

By Phil Plait | June 21, 2010 10:24 am

smbc_20100620I’ve not seen this trick before, but Zach Weiner at Saturday Morning Breakfast Cereal is correct. Not only is he correct, but his math is correct, and his philosophical punch line is funny and correct. And I’m not saying that just because I’ll see him at Comic Con soon and I’m trying to get him to buy the first round. I swear, sometimes when I don’t get a math trick, that guy in the panel really is me.

Also. Don’t forget to hold your mouse over the red button at the bottom for extra bonus Zach-ish goodness*.

* He really does look like that.

CATEGORIZED UNDER: Cool stuff, Geekery, Humor
MORE ABOUT: math, SMBC, Zach Weiner

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