Earth's specs

Somewhen in the 200s BC, a Greek named Eratosthenes measured the circumference of the Earth. He worked and lived in Alexandria, Egypt and knew of a place in Syrene, Egypt where, on the summer solstice, the image of the sun could be seen in a deep well, meaning that the sun was directly overhead. That placed Syrene on the equator. 

Eratosthenes assumed the Earth to be a sphere. If that were true, he reasoned that, if he stuck a rod in the ground vertically, it's line could be extended straight to the Earth's center to form an angle with a similar line from Syrene and that angle could easily be calculated. All he had to do was measure the angle formed of the line from the tip of the sun's shadow to the tip of the rod with the ground, subtract that from the 90° angle of the vertical rod with the ground, and he would have it...and "it" would also be Alexandria's latitude. It worked out to be about 7°.

By that time, everybody knew that the Earth was round and that the angular measure of any circle was 360°, and Eratosthenes knew that Alexandria was 5000 stadia from Syrene, he could figure out the circumference of a great circle on the Earth and, therefore, the Earth. His result was 250,000 stadia, or 39,385 kilometers, which is 1.4% off from the accurate circumference, 39,941 kilometers. Not too shabby!

So, on my recent hike to The Bluffs, I decided to do a modernized version of Eratosthenes' calculation.

The summer solstice was still a ways in the future so, not trusting nature to provide me with a good shot of the sun on demand, I measured the latitude and the distance between Arapahoe and Ridgegate Stations on the RTC southern light rail lines. I used Veiyra Software's Physics Toolbox Pro for the measurements. Here are the readouts.

Arapahoe Station

Ridgegate Station.

The distance, measured as the crow flies using Google Maps, is 5.6 miles or 9 kilometers.

"As the crow flies" is another way of saying "along a great circle on the globe," so I now have a way of converting degrees along the circumference of the Earth to kilometers and vice versa. By the way, I have it from a reputable source, namely, a crow, that crows do not always fly in straight lines.

The two stations are at almost the same longitude, so I can ignore that. The difference in latitude is .08 degrees.

But what about the stick in the ground? Well, that's another thing. It's called a gnomon and was a primary tool of ancient astronomers. It simply measured the angle of inclination of an astronomical object. Today we have astrolabes (basically a protractor with a plumb bob and a pointer) and the more advanced theodolite used by surveyors. I have a theodolite on my phone, the Dioptra app by Workshop512.

Since I really had all the information I needed, and I didn't know how far I was from the equator, I just wanted to do a modern version of Eratosthenes' trick to find my latitude by the sun. True to course, it was so cloudy on the summer equinox that I couldn't even tell which quadrant of the sky the sun was in, but I slapped a welder filter on my phone and took this shot from Dioptra the next day.

The angle of inclination was 51.1°, which was close to the actual measure on the equinox taken from the Time and Date website:

https://www.timeanddate.com/astronomy/usa/denver

Solar noon was at 1:07.

Angle of inclination was 50.5°, which placed my latitude at 90°-50.5°=39.5° . Looking at the Toolbox measurements above, I'm off by less than a tenth of a degree. The Dioptra measurement, which is also GPS is 39.58, so it's close.

But back to the real thing. The difference in measured latitude was 0.08° which is 4 minutes and 48 seconds (There are 60 minutes in a degree and 60 seconds in a minute). If 0.08 degrees is the same as 9 kilometers, 1 degree is 112.5 kilometers.

Okay, breath held, the moment of truth….112.5 kilometers times 360 degrees is 40,500 kilometers. The actual value is 39,941 kilometers. I was off by 1.01% Wow! I just impressed myself!

Of course, along with all the measurement error and such, the Earth is only approximately a sphere. The radius at the equator is larger than the radiuses at the poles.

We know the circumference of the Earth. The approximate volume is easy. The volume of a sphere is π\6 times the diameter cubed. The diameter is the circumference divided by π. Working backward, the diameter is 12,714 kilometers. So the volume is right at 10 to the 12th power cubic kilometers.

Okay, mass...mass is a bear. You measure mass with a balance and standard mass (remember the blog about mass and weight?) But Earth does have a mass. How in Sam Hill would you figure it out?

Well, obviously, you can't use a balance so any measurement has to be indirect. The first measurement to within 1% was made in 1798 by Henry Cavendish as a spin off of his accurate measurement of the gravitational constant. He used a torsion balance to do that and I can't even approach that kind of precision at home, so I'll just tell you how he did it. 

Isaac Newton figured out that the force of attraction (gravity) between any  two masses is directly proportional to the difference between their masses, and inversely proportional to the square of the distance between them. But to come up with an actual measurement, a proportionality constant was needed. He called it the Universal Gravitational Constant and never found it's value.

About 70 years later, Cavendish did it. Imagine a long, vertical, thin, flexible rod. At the bottom end is another rod forming an inverted T. At the end of that rod are two balanced heavy masses. His masses were  .73 kilograms each. He could set the bottom rod spinning back and forth and measure a slight force inhibiting the motion by comparing the frequency of oscillation with and without the force. The force, of course, would be another large mass close to one of the chunks of lead on the torsion balance. He knew the masses he was working with, the separation between them, and Newton's formula, so he was ready to calculate the Universal Gravitational Constant.

It was 6.67408 x 10^-11 m^3kg^-1s^-2 .

Believe it or not, that's what we need to calculate the mass of the Earth. Using Newton's formula we need the acceleration due to gravity (we found that approximately fooling around with the smartphone's accelerometer), multiplied by the radius of the Earth squared (we know that), divided by the Gravitational Constant.

So let's do it. Square the Earth first. The diameter is 12714 km so the radius is 6357 km. We need that in meters so 6357 x 10^3 meters. Square that to get 4.04 x 10^13 meters squared. The acceleration due to gravity is 9.18 meters per second square so the numerator is 3.71 x 10^14. Now we divide that by our Gravitational Constant, 6.67408 x 10^-11 m^3kg^-1s^-2  to get 5.56 x 10^24 kg (the accurate figure is 5.972 x 10^24 kg).

Actually, Cavendish didn't report the mass of the Earth. He stopped one step short by publishing the density of Earth which was 5.45 grams per cubic centimeter. He probably figured that, from there, it was easy to multiply that times the volume of Earth so, eh, let someone else do the easy part. 

If we look around and figure out what proportion of Earth is made of light rocks, heavy rocks, water, air... and come up with an average density we would say that it's around (and people before Cavendish had done just that) 2.7 grams per cubic centimeter, so where does all that mass come from?

Well, obviously, there's more underneath our feet than meets the eye. In fact, the deepest we've ever been is 12,262 meters and, although that's pretty deep, it barely scratches the surface. Still, the researchers expected temperatures around 212 degrees Fahrenheit and what they got was 356 degrees. It's hot down there.

But two things convince us that the core of the Earth is iron-rich molten metal. One is the surprising density of Earth. The other is something you don't see a lot of in the solar system...magnetism.

Earth is a magnet. The sun and gas giants like Jupiter and Neptune have strong magnetic fields. Mercury has a weak field. Some of the moons (but not ours) seem to be magnetic, but most of the smaller planets are magnetically inert.

We've used compasses that rely on the Earth's magnetic field for a long time. It wasn't until 1600 that William Gilbert proposed that Earth is a magnet. In fact, Earth is not a permanent magnet. It's an electromagnet.

Moving electrons (current) generates magnetic fields and our rotating molten metal outer core is one humongous magnetic field generator.

Our planet is special. We are just the right size. If we were too big, gravity would squash us. Too small and we wouldn't have enough gravity to hold onto our atmosphere. We get just enough sunlight for a healthy biosphere. We have plenty of that rare commodity - water. A nice balance of plants and animals conditions our air. And we have an effective magnetic shield that shunts dangerous solar radiations around the planet and out into space.

When I bought my current phone, I made sure it had a magnetometer in addition to the other regular sensors. Phones with GPS receivers will provide fairly accurate compass readings, but a magnetometer is more accurate and you can use it to measure both magnetic fields and electrical currents.

My Android has a AK09918 triaxial magnetometer. Since it's triaxial, it measures field strength in three directions (like the accelerometers). There are two common kinds of magnetometers in smartphones: magnetoresistive and Hall Effect. The AKM is a Hall Effect sensor that uses a flat conductive plate. A magnetic field causes electrons to deviate from their path and polarizes the plate. That can be sensed as a potential difference across the plate.

About a week ago, I hiked down a mile of  Little Dry Creek trail and used the Physics Toolbox Pro to record magnetic fields. I walked almost due west so I was cutting across the magnetic field lines.

The strength of a magnetic field is measured in teslas (in this case, in microteslas). A tesla is equal to a weber per square meter, and a weber is a kilogram per square second. If you understand induction (it makes transformers work), webers involve how much voltage you can crank out with a magnetic field. So with microteslas, don't expect geomagnetic electric generating stations any time soon.

I recorded the magnetic field in three directions at a rate of one measurement per second. Since I had the phone in my shirt pocket, the x direction was right-left, y was up-down, and z was forward-backward. I then saved the several thousands readings in a csv (comma separated values) file that I could pick up with DANSYS, my statistical spreadsheet.

Here's a graph of the tracings.

The tracings are pretty fuzzy, indicating a lot of noise. The inside of a smartphone has lots of electrical components crowded together and heat from those and the outside. Noise is to be expected and when you're measuring on the order of micro-anything, you can expect noise to blur the lines. 

All the lines have big spikes but the z component has the most. That is my forward and backward direction and I was walking in an urban environment, so power lines, underground cables…. yeah. So that's not the Earth's magnetic field, right? 

Many scientists call this the anthropocene epoch because the biggest influence on the Earth's environment, for the first time, is a single species - humanity. Every stray magnetic field alters Earth's magnetic field locally. Have you ever tried to get a compass to work in a house? You're likely to find it somewhat off the magnetic north.

But, we can sense some trends. There is a noticeable difference between the start of my recording and the latter part. That's because I started at my home and walked a ways more or less north before turning west on the trail.

The green line gives us the total field strength. It's measuring around 50 to 75 microteslas. The normal background magnetic field strength runs around 25 to 65 microteslas, so we're well within that range (once we get away from the houses.) The local residue from residences doesn't seem to spread out very far. The trail is generally about 200 to 300 feet (as measured by Google Maps) from the nearest houses.

Geophysics is the study of the physical attributes of our planet. After the barriers between East and West came down in 1957, scientists took the opportunity to focus on Earth and instituted the International Geophysical Year. You can learn a lot more with a team than you can alone. Perhaps you can join with some interested neighbors and have a Geophysical Year of your own!

Something about mass

I've had the delight of playing with a variety of balances from analytical balances that have to be protected from drafts and are precise to fractions of a milligram (that's a thousandth of a gram), to standard laboratory balances that will give you, oh, a few hundredths of a gram. And I've had several balances that came with various science kits. You won't find a statement if precision for those.

In one of my pharmacy labs, we had to synthesize aspirin and then purify it...because then we had to take it and, although the byproducts of aspirin are not horribly toxic (they didn't tell us that beforehand), they're not ideal snacks for happy-happy time. After taking our own aspirin, we measured the rate that it went through us. We were the people walking around campus with brown paper bags full of amber medicine bottles full of urine. Precision was important, but the laboratory scales gave us plenty for what we needed.

One of the most precise scales I've seen from a kit is the one from Penny Norman's Science Wiz Physics kit. Here, I use it to measure a gram of table salt.

[A gram of salt]

All this begs the question, "What is mass?"

I remember the stock answer from school, "mass is the amount of matter in a body," but I also remember the definition of matter, "Matter is that which has mass." That sounds a little too convenient...too circular. And what did they mean by "amount"?. Look at the picture at the top of this article. There's a gram of brass in the reference mass and a gram of table salt. It sure looks like there's more table salt (by volume) than there is brass.

I'm going to claim that the gram of table salt contains about 1x 10^22 molecules of sodium chloride and, to explain that, let me start close to the bottom.

Atoms are made of electrons, protons, and neutrons. Protons and neutrons are made of various other debris, notably quarks, but we don't need to go that far. A proton has a mass of 1.6726219 x 10^-24 grams. A neutron has a mass of 1.674927471 x 10^-24 grams. An electron has a mass of 9.10938 x 10^-28 grams. Electrons don't have enough mass to even consider, so let's forget them for the time being. The mass of the other two particles are so similar that we can just define an atomic mass unit as the mass of one proton or neutron. 

Table salt is impure sodium chloride and, to simplify things, let's ignore the impurities. What's the mass of a sodium atom? It has 11 protons and 12 neutrons so the mass of a sodium atom is 22.98976928 atomic mass units. Wait a second….but that's what my periodic table says. The fact is, the most common sodium atom has 11 protons and 12 neutrons, but there are other kinds of sodium atoms in nature that have more or less than 12 neutrons. It's the number of protons in an atom that makes it the element that it is. The number of neutrons can vary and you call the different kinds of sodium "isotopes" of sodium. If you take an average of the atomic masses of all the different isotopes of sodium according to their relative prominence in nature, you come up with 23.98976928 atomic mass units.

Avogadro's number is 6.0221409 x 10^23. That's the number of particles (atoms, molecules, etc.) in a mole of a substance and a mole is the number of grams that is the same as the number of atomic mass units of one particle. Since the atomic mass unit of sodium is about 24 and the atomic mass unit of chlorine is about 35.5, the atomic mass of sodium chloride (one sodium atom and one chlorine atom) is about 59.5. A mole of sodium chloride is 59.5 grams and a gram of sodium chloride is 1/59.5 mole. That means that you can divide Avogadro's number by 59.5 to find the (approximate) number of molecules of sodium chloride in a gram - 1x 10^22 molecules.

All of which gets us no closer to understanding what mass is. It has something to do with gravity. You find the mass of an object by comparing how hard gravity pulls on it to how hard gravity pulls on something else.

That "pull" is a problem, too. How does anything pull on anything? You might think you pull a wagon, but think again. Where do you apply pressure to the wagon...on the inside of the handle. You push against the inside of the wagon's handle. Can you really pull anything?

This bothered Isaac Newton all his life. He worked out all of how gravity works. He knew that mass is connected with gravity...somehow. By figuring out how planets have to interact to stay in their observable orbits, he knew that the force of attraction between bodies had to be the product of their masses divided by the square of the distance between them. A constant had to be thrown in to make the numbers work out but Newton never knew the value. Henry Cavendish came up with the value in 1798, 71 years after Newton's death.

What is gravity? The best Newton could do was "action at a distance". He was not amused.

Over the following years, there were all kinds of weird theories. One was that, as an object moved through some strange "ether" that filled the universe, it flowed around and would catch other objects up like things are pulled along in the wake of a fast moving boat. Imagine the dismay at the end of the 19th century and the beginning of the 20th when scientists had to accept that the universal ether does not exist.

At about the same time, Albert Einstein came along and figured out (part of) how it works. Here is mass and gravity according to Einstein.

[Gravity according to Einstein]

General and special relativity are weird...granted, but they are the most experimentally verified parts of physics, so there's little chance that that weirdness isn't a real part of our universe.

The gravity simulator above is a collection of Legos constructed to support an 11 inch embroidery hoop with a square of Lycra stretchy fabric clamped securely in it. The gooseneck assembly hanging over it holds my cell phone video camera.

Einstein's idea was that, instead of gravity being an attractive force between massive objects, any object with any mass distorts space around it and, then, other objects fall into the distortion just like the small ball bearing fell toward the large ball bearing.

The simulation isn't perfect. It suggests that mass distorts space into another spatial dimension. That isn't necessarily so.  It may just distort space into itself. The distortion is called a field, and there are other kinds of fields. If you set a magnet near one of these steel ball bearings, the magnet and the ball bearing will come together. 

This "action at a distance" of Newton can now be explained as a "falling together".

We're certainly going to be looking a lot in the future at these "falling togethers" and you'll see more of my gravity simulator very soon.

But the weirdness deepens. I've heard physicists express the conviction that, matter and energy are not the realities of our universe - the only things that are real are fields  So why do matter and energy seem so real to us and fields are so hard to wrap our brains around 

That has a lot to do with how our brains code the world around us. Brains are primarily interested in survival and the important things in our world related to survival are things like not being crushed in rock slides or falling off cliffs, building houses, finding food and water. To survive, we most need to be able to handle matter and energy. If fields are at the bottom of it all, that's interesting, but it's not what we need to pay attention to, so as humanity grew up, our brains learned to pay attention to survival things. 

We can measure fields, but we can't perceive them directly.

So, what is mass?

In 2012, physicists at the giant international particle accelerator at CERN confirmed the existence of a subatomic particle called the Higg's boson which is responsible for the attribute of matter called "mass". It creates a field in what we perceive as matter called the Higg's field, and that is what we recognize as mass.

This exercise in weirdness is as far as we are going to go in this blog. I try to crack open the world to show you how it works, but I'm a social psychologist that just happens to have a lot of other interests and I have my limits. Do physicists actually understand the weirdness they're dredging up? Maybe, maybe not, but all they do know makes the weirdness a necessary corollary. I've also heard physicists say that, if the universe was the way it seems to be, it wouldn't exist, so we have a ways to go.

It's only fair that I don't leave you thinking that there's no mystery in the universe...that everything is straight forward and that I can just open anything up and let you see inside.

But now, I'm going to back up to the part I can poke around in and start at the beginning...pretty much the world of Newton. Things will get quite weird enough.

As Jason Nesmith says, "Never give-up, never surrender!" (movie reference, there). Physics can be weird, but don't let that stop you. Amateur scientists make important discoveries and any swimmer will tell you - going out into the deep end is fun. The Teaching Company, MIT Opensourceware, local colleges and universities, many other resources are out there waiting for you...waiting to show you how deep you can go…..oooh, scary!

I once worked on a pipeline barge as a welder helper. I was in pretty good shape and I stayed that way by exercising regularly. I would do push-ups - standard and inverted (with my feet up on the wall). On land, push-ups are one thing - on the Gulf of Mexico, they're something else. Pushing up as the barge rode down the slope of a wave, I would almost lift off the floor, but if I wasn't ready when the barge was lifted by the next wave, I could find my face in a collision with the floor.

Something was changing as the barge bobbed around in the water. Was it my weight? Was it my mass? Does it make a difference?

I have an assortment of tools for measuring weight and mass. Here's a picture of some of it.

[Tools]

There are some weight (or mass) sets in the center. (I apologize but the vocabulary of weight and mass is incurably tangled.) To the left are some tools to measure weight - I'll call those "scales", although the word is also used for things that measure mass. To the right are tools to measure mass - I'll call those "balances" although that word is also used to refer to weight measuring devices.

You might say I have a problem, here. Let's look at the three groups separately.

[Weights]

There are standards of both weight and mass and I have lots of little pieces of metal and other materials that have been created to conform to those standards. For instance, the open black box in the center contains very precise (I had to buy it for lab work when I was in pharmacy school) pieces of metal with gram masses and ounce weights.

The cubes below it are called "density samples" because, despite the fact that they're the same size, they have very different masses. Density is defined as mass per unit volume.

There's also a stack of brass masses on a hook that are just right for hanging.

[Scales]

You've likely seen many scales. The things you use to measure the weight of produce at a grocery store are scales. You also weigh yourself on scales - bathroom scales.

Usually, a scale measures how hard an object pulls on a spring (like the set of spring scales on the lower right, or the Jolly Balance (which is actually a scale), the yellow plastic thing in the upper right corner - it also measures density. Alternately, a scale might measure how hard an object pushes down on an electronic component, like the digital bathroom scales in the picture.

[Balances]

On the other hand, a balance literally balances two objects. If they balance evenly, they are pulling down with equal force. Many science kits include inexpensive balances.

The blue velvet lined box in the picture contains a brass assayer's balance like the ones used long ago to "weigh" gold nuggets. There is also a pocket postal scale (which is actually a balance) and a tiny, three beam balance. It works like the "scales" your doctor uses to weigh you. In that case, the doctor balances you against the slider weights on the bars that are about eye height in front of you. A system of levers magnify the weights of the sliders and the machine calculates your weight when you're balanced. 

Scales measure weight and balances measure mass.

I carried some equipment to the Ross-University Hills branch of the Denver Public Library and rode their elevator to see what would happen when I measured the weight and mass of objects.

[Riding an elevator with a scale]

First, I used a portable electronic scale to weigh a mass. Yeah, I know it's a 20 gram mass that the scale says is 30 grams - I didn't zero the scale, but you can tell that the indicated weight (actually weight translated into grams - more about that below) changes as the elevator goes up and then returns to the first floor.

[Riding an elevator with a balance]

On the other hand, the balance stays balanced. You can tell because the vertical point stays vertical. Why did the weight change but the mass did not?

Mass is simply the amount of matter in an object and that doesn't change as long as the object is intact. The mass of an object is measured by comparing it to another object of known mass.

Weight is actually the force that an object directs straight down vertically. Newton's second law of mechanics, and the one most central to everything, defined force as mass times acceleration, so I need to go over a few technicalities.

When a thing changes position, it moves at a particular speed. In a car, speed is usually measured by a speedometer in miles per hour (at least in America. Everywhere else, it's kilometers per hour.) Speed is measured in distance per time, or distance divided by time.

Physicist do not usually work with speed. They prefer to work with velocity. Velocity is speed in a specified direction. It's called a vector quantity because you have to give more than one measure to fully specify it.

When you're driving a car, you don't maintain a constant speed. Acceleration is how fast you're speeding up or slowing down. Acceleration is measured as speed per time. That means it is measured as the distance traveled per unit time per unit time, or distance per unit time squared. In physics, the most common measure is meters per second squared.

Now we come to force. When I say that force is mass times acceleration, think in terms of pushing an object so that it speeds up faster and faster at a constant rate. Force makes things go faster or slows them down. A common measure of force is the newton which is the amount of push required to accelerate a one kilogram object one meter per second squared.

And weight, being a force, is often measured in newtons (notice that, when "Newton" is a name, it's capitalized, but when it's a unit of force, it's written in lower case.). Weight is mass times acceleration. What acceleration? The acceleration of gravity. That's why the weight of a body can change. The acceleration that gravity imposed on a body in freefall is 9.764 meters per seconds squared...at sea level on the Earth and, although it is different at different places on the Earth's surface, the variance is usually too small to worry about. (Geologists actually do worry about it because large deposits of metal ore will present a slightly different gravitational pull than other rocks and they use of a very sensitive instrument called. "gravitometer" to measure the pull.) As you move out away from the Earth, though, it's pull becomes weaker and your weight also decreases.

The moon is smaller than the Earth and, therefore, has less gravitational pull. Acceleration due to it's gravity is only 1.625 meters per second squared on the moon. I weigh 185 pounds on Earth. On the moon, I would only weigh 185 times 1.625/9.774, or 30.8 pounds.

I recorded my phone's accelerometer on the elevator using Google's Science Journal. It looked like this.

[Elevator ride]

Another digression...it can be confusing which direction is which on a phone. Just remember the graphs you drew in algebra. The x axis went left to right, the y axis went up and down and if you were working with three dimensions, the z axis was into and out of the page. It's the same for the phone. Holding the phone flat in front of me, the direction of the elevator's motion was along the z axis. All of the accelerometers produced jagged lines, but look at the scales. The x and y accelerometers showed accelerations around zero and one m/s2. The z accelerometer measured around 9.5 m/s2. That should look familiar - it's the acceleration due to gravity.

When the elevator starts up, weights in the elevator opposes it's motion with an equal but opposite force, (that's Newton's third law). So, add the elevator's acceleration to that due to gravity. Since the accelerometer measures up to twelve m/s2, the elevator must be accelerating at about two and a half meters per second squared until it reached a constant speed, and the tracing smoothed out. At the top, the elevator slows down at about 1.5 meters per second squared and objects lighten up. As the elevator starts back down, objects in it lose weight again, to regain it at the bottom.

That's actually how the phone's accelerometers work. They are tiny (You might have heard of nanotechnology. Cell phone accelerometers are nanotech.) combs that have tiny weights at the end of their times. As the weights accelerate, they move with the acceleration and sensors pick up the motion.

It's not entirely bogus that my electronic scales claim to measure grams (mass). It actually measures weight but, on Earth, weight is mass times a constant 9.764 acceleration due to gravity so the electronics just have to divide the weight by 9.774 to get the mass….but not on the moon.

Riding in a car, notice how you lean as it slows down, speeds up, or turns a corner. That's forces at work. If you have an elevator handy, you might try riding it with a bathroom scales and see how your weight changes and remember...your mass stays the same.