Posted on by Jean Tate

Gravity Equation

There is not one, not two, not even three gravity equations, but many!

The one most people know describes Newton’s universal law of gravitation:

F = Gm1m2/r2,
where F is the force due to gravity, between two masses (m1 and m2), which are a distance r apart; G is the gravitational constant.

From this is it straightforward to derive another, common, gravity equation, that which gives the acceleration due to gravity, g, here on the surface of the Earth:

g = GM/r2,
Where M is the mass of the Earth, r the radius of the Earth (or distance between the center of the Earth and you, standing on its surface), and G is the gravitational constant.

With its publication in the early years of the last century, Einstein’s theory of general relativity (GR) became a much more accurate theory of gravity (the theory has been tested extensively, and has passed all tests, with flying colors, to date). In GR, the gravity equation usually refers to Einstein’s field equations (EFE), which are not at all straight-forward to write, let alone explain (so I’m going to write them … but not explain them!):

G?? = 8?G/c4 T??

G (without the subscripts) is the gravitational constant, and c is the speed of light.

Finally, here’s a acceleration of gravity equation you’ve probably never heard of before:

a = ?(GMa0/r),

where a is the acceleration a star feels, due to gravity under MOND (MOdified Newtonian Dynamics), an alternative theory of gravity, M is the mass of a galaxy, r the distance between the star in the outskirts of that galaxy and its center, G the gravitational constant, and a0 a new constant.

Some websites which contain more on gravity equations, for your interest and enjoyment: Newton’s Theory of “Universal Gravitation” (NASA), Einstein’s equation of gravity (University of Wisconsin Madison – heavy), and Gravity Formula (University of Nebraska-Lincoln).

Universe Today, as you would expect, has several stories relevant to gravity equations; here are a few: See the Universe with Gravity Eyes, A Case of MOND Over Dark Matter, and Flyby Anomalies Explained?. Here’s an article about 0 gravity.

Gravity, an Astronomy Cast episode, has more on gravity equations, as do several Astronomy Cast Question Shows, such as September 26th, 2008, and March 31st, 2009.

Sources:
University of Nebraska-Lincoln
NASA
UT-Knoxville

Posted on by Jean Tate

Pioneer Anomaly

Artist impression of the Pioneer 10 probe (NASA)

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Named after the Pioneer 10 and 11 space probes, the Pioneer anomaly refers to the fact that they seem to be moving a teensy bit different from how we think they should be moving (or, more technically, the spacecraft seem to be subject to an unmodeled acceleration whose direction is towards the Sun).

The anomaly was first noticed, by John Anderson, in 1980, when analysis of tracking data from the spacecraft showed a small, unexplained acceleration towards the Sun (this was first published in 1995, with the main paper appearing in 1998). Since then it has been studied continuously, by quite a few scientists.

The Pioneer anomaly is one of the (very few!) true mysteries in contemporary physics, and is a great example of how science is done.

The first step – which Anderson and colleagues took – was to work out where the spacecraft were, and how fast they were traveling (and in what direction), at as many times as they could. Then they estimated the effects of gravity, from all known solar system objects (from the Sun to tiny asteroids and comets). Then they estimated the effects of things like radiation pressure, and possible outgassing. Then … They also checked whether other spacecraft seemed to have experienced a similar anomalous acceleration (the net: not possible to get an unambiguous answer, because all others have known – but unmodelable – effects much bigger than the Pioneer anomaly). Several independent investigations have been conducted, using different approaches, etc.

In the last few years, much effort has gone into trying to find all the raw tracking data (this has been tough, many tapes have been misplaced, for example), and into extracting clean signals from this (also tough … the data were never intended to be analyzed this way, meta-data is sorely lacking, and so on).

And yet, the anomaly remains …

… there’s an unmodeled acceleration of approximately 9 x 10-10 m/s2, towards the Sun.

The Planetary Society has been funding research into the Pioneer anomaly, and has a great summary here! And you can be a fly on the wall at a meeting of a team of scientists investigating the Pioneer anomaly, by checking out this Pioneer Explorer Collaboration webpage.

Universe Today has several stories on the Pioneer anomaly, for example The Pioneer Anomaly: A Deviation from Einstein Gravity?, Is the Kuiper Belt Slowing the Pioneer Spacecraft?, and Ten Mysteries of the Solar System.

Astronomy Cast has two episodes covering the Pioneer anomaly, The End of Our Tour Through the Solar System, and the November 18th, 2008 Questions Show.

Source:
The Planetary Society

Posted on by Jean Tate

What is Loop Quantum Gravity?

In this illustration, one photon (purple) carries a million times the energy of another (yellow). Some theorists predict travel delays for higher-energy photons, which interact more strongly with the proposed frothy nature of space-time. Yet Fermi data on two photons from a gamma-ray burst fail to show this effect. The animation below shows the delay scientists had expected to observe. Credit: NASA/Sonoma State University/Aurore Simonnet

The two best theories we have, today, in physics – the Standard Model and General Relativity – are mutually incompatible; loop quantum gravity (LQG) is one of the best proposals for combining them in a consistent way.

General Relativity is a theory of spacetime, but it is not a quantum theory. Since the universe seems to be quantized in so many ways, one approach to extending GR is to quantize spacetime … somehow. In LQG, space is made up of a network of quantized loops of gravitational fields (see where the name comes from?), which are called spin networks (and which become spin foam when viewed over time). The quantization is at the Planck scale (as you would expect). LQG and string theory – perhaps the best known of theories which aim to both go deeper and encompass the Standard Model and General Relativity – differ in many ways; one of the most obvious is that LQG does not introduce extra dimensions. Another big difference: string theory aims to unify all forces, LQG does not (though it does include matter).

Starting with the Einstein field equations of GR, Abhay Ashtekar kicked of LQG in 1986, and in 1988 Carlo Rovelli and Lee Smolin built on Ashtekar’s work to introduce the loop representation of quantum general relativity. Since then lots of progress has been made, and so far no fatal flaws have been discovered. However, LQG suffers from a number of problems; perhaps the most frustrating is that we don’t know if LQG becomes GR as we move from the (quantized) Planck scale to the (continuum) scale at which our experiments and observations are done.

OK, so what about actual tests of LQG, you know, like in the lab or with telescopes?

Well, there are some, potential tests … such as whether the speed of light is indeed constant, and recently the Fermi telescope team reported the results of just such a test (result? No clear sign of LQG).

Interested in learning more? There is a lot of material freely available on the web, from easy reads like Quantum Foam and Loop Quantum Gravity and Lee Smolin’s Loop Quantum Gravity, to introductions for non-experts like Abhay Ashtekar’s Gravity and the Quantum, to reviews like Carlo Rovelli’s Loop Quantum Gravity, to this paper on an attempt to explain some observational results using loop quantum gravity (Loop Quantum Gravity and Ultra High Energy Cosmic Rays).

As you’d expect, Universe Today has several articles on, or which feature, loop quantum gravity; here is a selection What was Before the Big Bang? An Identical, Reversed Universe, Before the Big Bang?, and Before the Big Bang.

Source: Wikipedia

Posted on by Jean Tate

Gravity Constant

Anaglyph images created from an ESA video animation of global gravity gradients. A more accurate global map will be generated by ESA's GOCE craft. Credit: ESA and Nathaniel Burton Bradford.

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The constant of gravity, or gravity constant, has two meanings: the constant in Newton’s universal law of gravitation (so is commonly called the gravitational constant, it also occurs in Einstein’s general theory of relativity); and the acceleration due to gravity at the Earth’s surface. The symbol for the first is G (big G), and the second g (little g).

Newton’s universal law of gravitation in words is something like “the gravitational force between two objects is proportional to the mass of each and inversely proportional to the square of the distance between them“. Or something like F (the gravitational force between two objects) is m1 (the mass of one of the objects) times m2 (the mass of one of the other object) divided by r2 (the square of the distance between them). The “is proportional to” means all you need to make an equation is a constant … which is G.

In other words: F = Gm1m2/r2

The equation for little g is simpler; from Newton we have F = ma (a force F acting on a mass m produces an acceleration a), so the force F on a mass m at the surface of the Earth, due to the gravitational attraction between the m and the Earth is F = mg.

Little g has been known from at least the time of Galileo, and is approximately 9.8 m/s2 – meters per second squared – it varies somewhat, depending on how high you are (altitude) and where on Earth you are (principally latitude).

Obviously, big G and little g are closely related; the force on a mass m at the surface of the Earth is both mg and GmM/r2, where M is the mass of the Earth and r is its radius (in Newton’s law of universal gravitation, the distance is measured between the centers of mass of each object) … so g is just GM/r2.

The radius of the Earth has been known for a very long time – the ancient Greeks had worked it out (albeit not very accurately!) – but the mass of the Earth was essentially unknown until Newton described gravity … and even afterwards too, because neither G nor M could be estimated independently! And that didn’t change until well after Newton’s death (in 1727), when Cavendish ‘weighed the Earth’ using a torsion balance and two pairs of lead spheres, in 1798.

Big G is extremely hard to measure accurately (to 1 part in a thousand, say); today’s best estimate is 6.674 28 (+/- 0.000 67) x 10-11 m3 kg-1 s -2.

The Constant Pull of Gravity: How Does It Work? is a good NASA webpage for students, on gravity; and the ESA’s GOCE mission webpage describes how satellites are being used to measure variations in little g (GOCE stands for Gravity field and steady-state Ocean Circulation Explorer).

The Pioneer Anomaly: A Deviation from Einstein’s Gravity? is a Universe Today story related to big G, as is Is the Kuiper Belt Slowing the Pioneer Spacecraft?; GOCE Satellite Begins Mapping Earth’s Gravity in Lower Orbit Than Expected is one about little g.

No surprise that the Astronomy Cast episode Gravity covers both big G and little g!

Posted on by Abby Cessna

Gravity for Kids

Gravity of the Sun and Earth. Image credit: NASA

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What keeps us from floating off into space? Why does something I drop fall to the ground? Kids are famous for asking questions like this, which usually cause parents to mumble something about gravity or tell them they will learn it when they get older. Here are a number of resources that can answer some of those questions.

Kidipede explains what gravity is on the Earth and in the universe.

How Stuff Works has a number of experiments for kids regarding the laws of gravity.

Science Experiments  offers a simple science experiment about gravity for kids, including a video clip showing how to do the experiment.

The USGS has a simple definition of what gravity is.

This site explains that the reason things do not fall off the Earth is because of gravity.

Physics 4 Kids has information on gravity for children. It also covers the topics of planetary gravity and the Moon. Additionally, the site has other links to different resources.

Spaghetti Box Kids has an experiment that teaches kids about density and gravity. The project involves making miniature hot air balloons.

About.com offers information on Sir Isaac Newton and tells about his work regarding gravity and his three laws.

Teacher Tech has an entire lesson plan mapped out around Sir Isaac Newton. It teaches about Newton and his three laws of motion. Additionally, it has a quiz for students and two science experiments involving gravity and motion.

Science Monster makes learning about gravity fun and easy. In addition to providing easy to understand definitions of gravity and intertia, the website has a game you can play that further reinforces the concepts.

This is a video clip from NASA showing how important gravity is in our everyday lives. It also has links to other video clips from NASA. This material is rated for grades 5 through 12 according to NASA.

Kids Konnect  has links to a variety of sources related to gravity including NASA. The site also has a number of links to information about Sir Isaac Newton who is famous for his work regarding gravity.

Universe Today has articles on planets for kids and Solar System projects for kids.

If you are looking for more information, check out Kids Astronomy and Primary Games.

Astronomy Cast has an episode on gravitational waves.

Posted on by Fraser Cain

Artificial Gravity

An artist's representation of a rotating space station.

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Have you ever noticed that astronauts float around in the space shuttle and in the International Space Station, while space travelers on television and in the movies keep their feet firmly on the ground. That’s because it would be very difficult (and expensive) to have your actors floating around in every scene. So science fiction writers invent some kind of artificial gravity technology, to keep everyone standing on the ground.

Of course, there’s no technology that will actually generate gravity in a spaceship. Gravity only comes from massive object, and there’s no way to cancel the acceleration of gravity. And so if you wanted to have a spacecraft that could generate enough artificial gravity to keep someone’s feet on the ground, the spaceship would need to have the mass of the Earth.

Floating in space is actually very hard on astronauts’ bodies. The lack of gravity softens their bones and causes their muscles to weaken. After any long trip into space, astronauts need several days and even weeks to recover from traveling in microgravity.

But there a couple of ways you could create artificial gravity in a spaceship. The force we feel from gravity is actually our acceleration towards a massive body. We’d keep falling, but the ground is pushing against us, so we stand on the ground. If you can provide an alternative form of acceleration, it would feel like gravity, and provide the same benefits of standing on the surface of a planet.

The first way would be through accelerating your spaceship. Imagine you wanted to fly your spaceship from Earth to Alpha Centauri. You could fire your rockets behind the spacecraft, accelerating at a smooth rate of 9.8 meters/second2. As long as the rocket continued accelerating, it would feel like you were standing on Earth. Once the rocket reached the halfway point of its journey, it would turn around and decelerate at the same rate, and once again, you would feel the force of gravity. Of course, it takes an enormous amount of fuel to accelerate and decelerate like this, so we can consider that pretty much impossible.

A second way to create acceleration is to fake it through with some kind of rotation. Imagine if your spaceship was built like a big donut, and you set it spinning. People standing on the inside hull would feel the force of gravity. That’s because the spinning causes a centrifugal force that wants to throw the astronauts out into space. But the spaceship’s hull is keeping them from flying away. This is another way to create artificial gravity.

There are no spacecraft that use any form of artificial gravity today, but if humans do more space exploration, we will likely see the rotational method used in the future.

We have written several articles about artificial gravity for Universe Today. Here’s an article about how mice might be used to test out artificial gravity, and here’s more information about future technologies that might use artificial gravity.

Here’s a podcast from Scientific American that talks about the effect of artificial gravity.

We have recorded an episode of Astronomy Cast that talks about science fiction technologies. Listen to it here: Episode 104 – Science Fiction at Dragon*Con

Sources:
Wikipedia
NEWTON, Ask A Scientist!
Wise Geek

Posted on by John Carl Villanueva

What Causes Tides?

The Earth is a water-dominated planet. (Image credit: Ian O'Neill)

Tides refer to the rise and fall of our oceans’ surfaces. It is caused by the attractive forces of the Moon and Sun’s gravitational fields as well as the centrifugal force due to the Earth’s spin. As the positions of these celestial bodies change, so do the surfaces’ heights. For example, when the Sun and Moon are aligned with the Earth, water levels in ocean surfaces fronting them are pulled and subsequently rise.

The Moon, although much smaller than the Sun, is much closer. Now, gravitational forces decrease rapidly as the distance between two masses widen. Thus, the Moon’s gravity has a larger effect on tides than the Sun. In fact, the Sun’s effect is only about half that of the Moon’s.

Since the total mass of the oceans does not change when this happens, part of it that was added to the high water regions must have come from somewhere. These mass-depleted regions then experience low water levels. Hence, if water on a beach near you is advancing, you can be sure that in other parts of the world, it is receding.

Most illustrations containing the Sun, Moon, Earth and tides depict tides to be most pronounced in regions near or at the equator. On the contrary, it is actually in these regions where the difference in high tide and low tide are not as great as those in other places in the world.

This is because the bulging of the oceans’ surface follows the Moon’s orbital plane. Now, this plane is not in line with the Earth’s equatorial plane. Instead, it actually makes a 23-degree angle relative to it. This essentially allows the water levels at the equator to seesaw within a relatively smaller range (compared to the ranges in other places) as the orbiting moon pulls the oceans’ water.

Not all tides are caused by the relative positions of these celestial bodies. Some bodies of water, like those that are relatively shallow compared to oceans, experience changing water levels because of variations in the surrounding atmospheric pressure. There are also other extreme situations wherein tides are manifested but have nothing to do with astronomical positioning.

A tidal wave or tsunami, for example, makes use of the word ‘tide’ and actually exhibits rise and fall of water levels (in fact, it is very noticeable). However, this phenomena is caused entirely by a displacement of a huge amount of water due to earthquakes, volcanic eruptions, underwater explosions, and others. All these causes take place on the Earth’s surface and have nothing to do with the Moon or Sun.

A thorough study of tides was conducted by Isaac Newton and included in his published work entitled Philosophiæ Naturalis Principia Mathematica.

We have some related articles here that may interest you:

There’s more about it at NASA. Here are a couple of sources there:

Here are two episodes at Astronomy Cast that you might want to check out as well:

Sources:
Princeton University
NASA
NOAA

Posted on by John Carl Villanueva

What is the Gravitational Constant?

Visualization of a massive body generating gravitational waves (UWM)

The gravitational constant is the proportionality constant used in Newton’s Law of Universal Gravitation, and is commonly denoted by G. This is different from g, which denotes the acceleration due to gravity. In most texts, we see it expressed as:

G = 6.673×10-11 N m2 kg-2

It is typically used in the equation:

F = (G x m1 x m2) / r2 , wherein

F = force of gravity

G = gravitational constant

m1 = mass of the first object (lets assume it’s of the massive one)

m2 = mass of the second object (lets assume it’s of the smaller one)

r = the separation between the two masses

As with all constants in Physics, the gravitational constant is an empirical value. That is to say, it is proven through a series of experiments and subsequent observations.

Although the gravitational constant was first introduced by Isaac Newton as part of his popular publication in 1687, the Philosophiae Naturalis Principia Mathematica, it was not until 1798 that the constant was observed in an actual experiment. Don’t be surprised. It’s mostly like this in physics. The mathematical predictions normally precede the experimental proofs.

Anyway, the first person who successfully measured it was the English physicist, Henry Cavendish, who measured the very tiny force between two lead masses by using a very sensitive torsion balance. It should be noted that, after Cavendish, although there have been more accurate measurements, the improvements on the values (i.e., being able to obtain values closer to Newton’s G) have not been really substantial.

Looking at the value of G, we see that when we multiply it with the other quantities, it results in a rather small force. Let’s expand that value to give you a better idea on how small it really is: 0.00000000006673 N m2 kg-2

Alright, let’s now see what force would two 1-kg objects exert on one another when their geometrical centers are spaced 1 meter apart. So, how much do we get?

F = 0.00000000006673 N. It really doesn’t matter much if we increase both masses substantially.

For example, let’s try the heaviest recorded mass of an elephant, 12,000 kg. Assuming we have two of these, spaced 1 meter apart from their centers. I know it’s difficult to imagine that since elephants are rather stout, but let’s just proceed this way because I want to put emphasis on the significance of G.

So, how much did we get? Even if we rounded that off, we’d still obtain only 0.01 N. For comparison, the force exerted by the earth on an apple is roughly 1 N. No wonder we don’t feel any force of attraction when we sit beside someone… unless of course you’re a male and that person is Megan Fox (still, it’d be safe to assume that the attraction would only be one way).

Therefore, the force of gravity is only noticeable when we consider at least one mass to be very massive, e.g. a planet’s.

Allow me to end this discussion with one more mathematical exercise. Assuming you know both your mass and your weight, and you know the radius of the earth. Plug those into the equation above and solve for the other mass. Voila! Wonder of wonders, you’ve just obtained the mass of the Earth.

You can read more about the gravitational constant here in Universe Today. Want to learn more about a new study that finds fundamental force hasn’t changed over time? There’s also some insights you can find among the comments in this article: Record Breaking “Dark Matter Web” Structures Observed Spanning 270 Million Light Years Across

There’s more about it at NASA. Here are a couple of sources there:

Here are two episodes at Astronomy Cast that you might want to check out as well:

Sources:

Posted on by Jerry Coffey

How Strong Is Jupiter’s Gravity?

Clouds on Jupiter. Image credit: NASA/JPL

Jupiter is the most massive planet in our Solar System and; therefore, the gravity of Jupiter is the most intense in the Solar System. The gravity of Jupiter is 2.5 times what it is here on Earth.

In the 1990s Jupiter’s gravity tore apart Comet P/Shoemaker-Levy 9 and pulled the broken pieces into the to planet. This marked the first time that humans had direct observation of two extraterrestrial Solar System bodies colliding. Jupiter had actually captured the asteroid between 20 and 30 years prior to impact and it had been orbiting the planet since. In 1992, the asteroid entered Jupiter’s Roche limit and was broken apart by the planet’s tidal forces. The asteroid resembled a string of pearls until its fragments impacted the surface July 16-22 of 1994. The fragments were as large as 2 km each and hit the surface at 60 km/s. The impacts allowed astronomers to make several new discoveries about Jupiter.

Astronomers found several chemicals within the Jovian atmosphere that had not been seen prior to the impacts. Diatomic sulfur and carbon disulfide were the most interesting. This was only the second time that diatomic sulfur had been detected in any astronomical object. Ammonia and hydrogen sulfide were detected for the first time on Jupiter. You can read up on other discoveries made during and shortly after these impacts by reading this article and this pdf from C.A. Olano.

Some scientists, including Jacques Laskar of the Paris Observatory, as well as Konstantin Batygin and Gregory Laughlin of the University of California, Santa Cruz believe that Jupiter’s gravity may lead to the destruction of Mercury. After running some simulations the group found that Jupiter is perturbing Mercury’s already eccentric orbit. They arrived at four possible end results: Mercury will crash into the Sun, Mercury will be ejected from the solar system altogether, Mercury will crash into Venus, or Mercury will crash into Earth. None is pleasant for Mercury and the last would be even less pleasant for humans. Not to fear though, none of these possible outcomes will happen in the next 5-7 billion years anyway.

The gravity of Jupiter affects every planet to one degree or another. It is strong enough to tear asteroids apart and capture 64 moons at least. Some scientist think that Jupiter destroyed many celestial objects in the ancient past as well as prevented other planets from forming. How’s that for a powerful neighbor?

Here’s an article from Universe Today about how Jupiter’s gravity might actually wreck the Solar System, and here’s an article about how big planets like Jupiter could get.

Use this site to calculate your weight on other worlds, and here’s more information about Comet P/Shoemaker Levy 9.

We’ve also recorded an entire show just on Jupiter for Astronomy Cast. Listen to it here, Episode 56: Jupiter, and Episode 57: Jupiter’s Moons.

Sources:
http://www2.jpl.nasa.gov/sl9/
http://adsabs.harvard.edu/full/1996EM%26P…73..147H