Gravitational Potential Energy
Discussion
introduction
Short bit of calculus.
| Ug = − | ⌠ ⌡ |
F · ds |
| r | ||||
| Ug = − | ⌠ ⎮ ⌡ |
− | Gm1m2 | dr |
| r2 | ||||
| ∞ |
| Ug = − | Gm1m2 | ⎛ ⎜ ⎝ |
1 | − | 1 | ⎞ ⎟ ⎠ |
| r | ∞ |
and here it is…
| Ug = − | Gm1m2 |
| r |
where…
| Ug = | gravitational potential energy |
| m1m2 = | masses of any two objects |
| r = | separation between their centers |
| G = | universal gravitational constant (6.67 × 10−11 N m2/kg2) |
Note that there is no ∆ in this expression. Discuss here.
Discuss potential vs. potential energy somewhere.
small distance approximation
What about the old equation?
∆Ug = mg∆h
It's hidden in the new equation.
| Ug = − | Gm1m2 |
| r |
Let me show you.
| ∆Ug = | Uf | − | Ui | |
| ∆Ug = | Ug(r + ∆h) | − | Ug(r) | |
| ∆Ug = | − | Gm1m2 | + | Gm1m2 |
| r + ∆h | r |
Combine terms over a common denominator.
| ∆Ug = |
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| ∆Ug = |
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Multiply by "one".
| ∆Ug = | Gm1m2∆h | r | |
| r(r + ∆h) | r |
Swap terms in the denominators.
| ∆Ug = | Gm1m2∆h | r | |
| r2 | r + ∆h |
Factor some stuff out of the numerator.
| ∆Ug = m2Δh | Gm1 | r | |
| r2 | r + ∆h |
Do you see it? If r is the radius of the Earth, m1 is the mass of the Earth, and m2 is the mass of something being lifted, then…
| g = | Gm1 |
| r2 |
is the acceleration due to gravity on the Earth's surface. Making this substitution (and dropping the subscript, since we only have one mass left), we get…
| ∆Ug = mg∆h | r |
| r + ∆h |
The first part of this expression is our old friend, the original equation for gravitational potential energy. The second term is a correction factor. For ordinary heights, this term is essentially one. Let's confirm this using a really high height — the top of the spire on the Burj Khalifa in the United Arab Emirates (818 m).
| r | = | 6,371,000 m |
| r + ∆h | 6,371,000 m + 818 m | |
| r | = | 0.999872 |
| r + ∆h |
The engineers who designed the Burj would have an error in the fourth decimal place of their calculations if they used ∆Ug = mg∆h instead of the more general equation. This deviation is probably smaller than the uncertainty in the mass of the girders used to construct the building, which is why ∆Ug = mg∆h is totally acceptable for most down-to-earth applications
Now let's try something astronomical. Can ∆Ug = mg∆h be used to measure the gravitational potential energy of the moon? The Earth-moon distance (384,400,000 m) is measured from the center of the Earth, not it's surface. In this case, r + ∆h will actually be a difference in two numbers.
| r | = | 6,371,000 m |
| r + ∆h | 384,400,000 m − 6,371,000 m | |
| r | = | 0.016853 |
| r + ∆h |
This number is obviously closer to zero than to one, which is why…
| Ug = − | Gm1m2 |
| r |
is used for astronomical applications.
Escape velocity
What goes up, must come down. Right?
Umm. No. Not necessarily.
If an object is thrown upward fast enough it will go up and never come down. The minimum speed needed to do this is called the escape velocity.
No human has ever traveled faster than the escape velocity of the Earth. The Apollo astronauts got very close, but they were headed to the moon, which is trapped by Earth's gravity into a closed orbit. In some sense they didn't really want to escape the Earth. They did manage to travel faster than the escape velocity of the moon, however, which is why they were able to return to the Earth.
Any spacecraft that has ever traveled to another planet or asteroid has managed to exceed the escape velocity of the Earth. Counting them all is too much work. It's somewhere in the low hundreds.
Five space probes are currently on trajectories that will take them out of the solar system, which means they have exceeded the escape velocity of the Sun. They are the Pioneer 10 and 11 missions to Jupiter and Saturn, the Voyager 1 and 2 missions to all four Jovian planets (Jupiter, Saturn, Uranus, Neptune), and the New Horizons mission to Pluto. In 2012 Voyager 1 became the first human made object to cross into interstellar space at a distance from the Sun of 120 astronomical units (120 times the Earth-sun distance or 4 times the Neptune-sun distance).
Let's consider the factors that affect escape velocity.
- All objects with mass have an escape velocity, but the term is normally only used for astronomical objects — things with a lot of mass. The escape velocity of the Earth is quite fast (11 km/s), which is why most people have never directly seen an object escape the Earth. The escape velocity of person is imperceptibly small (0.3 mm/s), which is why everything any person has ever let go of has never returned to them due to their gravitational pull. Escape velocity increases with mass (but the two quantities are not directly proportional).
- Escape velocity is sometimes described as the speed needed to escape the gravity of a object, but that's not quite right. You can never escape the pull of gravity. Never! Gravity decreases with increasing separation between objects, but it never equals zero. Gravity is infinite in its reach. Since gravity decreases with separation, the escape velocity does as well (but the two quantities are not inversely proportional).
Time for some math. Start with the law of conservation of energy — total energy at the start is the same as total energy at the end.
E0 = E
For an object tossed upwards, the relevant energies are kinetic and gravitational potential.
K0 + U0 = K + U
Replace the energy symbols with their equations. (Remember that kinetic energy is always positive and that gravitational energy is always negative.)
| ½m1v02 − | Gm1m2 | = ½m1v2 − | Gm1m2 |
| r0 | r |
Let…
| m1 = | the mass of object that's doing the escaping (a spacecraft, explosion debris, meteorite impact debris, etc.) |
| m2 = | the mass of the astronomical object it's escaping from (the Earth, sun, Milky Way, etc.) |
| v0 = | the initial speed of the escaping object |
| v = | the speed of the escaping object at some later time |
| r0 = | the initial separation between the two objects |
| r = | the separation between the two objects at some later time |
| G = | the universal gravitational constant |
Throw an object upwards and what happens? Gravity pulls on it, slowing it down as it rises. Eventually the object will stop, turn around, and fall back to Earth. Throw it faster and that turnaround point will be higher. Throw it even faster and that point will be even higher. Take this situation and push it to the extreme. Make that turnaround point be infinitely far away. In the language of limits, let r → ∞ and v → 0. When we do this, both terms on the right side of the equals sign disappear. If you want to escape the Earth, or the Sun, or the Milky Way, just have zero total energy.
| ½m1v02 − | Gm1m2 | + = 0 − 0 |
| r0 |
Algebra says we can eliminate m1 since it appears in every term. Since there's only one mass left I see no reason to write a subscript anymore. We can also do the same for the subscripts on v0 and r0. With no other values of speed or separation to worry about, why bother with subscripts? Let's also do a little bit of rearrangement to reduce the number of terms and eliminate the minus sign.
| ½v2 = | Gm |
| r |
Solving for v gives us the following equation for escape velocity…
| v = √ | 2Gm |
| r |
where…
| v = | escape velocity (the minimum speed an object needs to travel infinitely far from an astronomical object at a specific location) |
| G = | universal gravitational constant |
| m = | mass of the astronomical object |
| r = | initial separation between the two objects |
Note that although it's refereed to as an escape velocity, it's really an escape speed. It's not a vector it's a scalar. As long as the direction you're headed is not into the astronomical object, the escaping object will escape. We approached this as a conservation of energy problem and energy is a scalar quantity. It doesn't care about direction. Recall too that gravity is described as a conservative force, which means it doesn't care about the path you take.
event horizon
A black hole is a star that has collapsed down to a point. Within a certain radius, known as the event horizon, the escape velocity is greater than the speed of light. Since nothing can exceed the speed of light, anything crossing over the event horizon becomes trapped forever within a black hole.
Black holes destroy volume, but not mass, energy, angular momentum, charge, entropy, etc.
Event horizon a.k.a. Schwarzschild radius. The point of no return.
First theorized by John Michel in 1784
Starting from the escape velocity formula, derive an equation for the radius of the event horizon in terms of m (the mass of the black hole), G (the gravitational constant), and c (the speed of light).
| v = c = √ | 2Gm |
| rs |
| c2 = | 2Gm |
| rs |
| rs = | 2Gm |
| c2 |
Paraphrase this, "it might be possible to make a black hole in a laboratory, nothing lighter than 10 μg can make a black hole, this energy is still out of the range of current particle colliders, but not very energetic cosmic rays, since the Earth hasn't been gobbled up by baby black holes, we can assume they are unstable."
cosmic expansion
Hubble's law
When we look at galaxies and other objects outside our own Milky Way we see that they are generally moving away from us and that their recessional velocities us are nearly directly proportional to their distance. That is, the farther away a particular galaxy is from us, the faster it's running away from us. If one galaxy is twice as far away from the Milky Way as another, it's speed will also be twice a great. Three times farther way means three times faster, and so on. This observation was first made in 1929 by the American astronomer Edwin Hubble (1889–1953) and it has since become known as Hubble's law. Mathematically, Hubble's law is written as…
v = Hr
where…
| v = | recessional velocity (the component of the object's velocity away from the Milky Way) |
| r = | distance from the Milky Way |
| H = | constant of proportionality known as the Hubble constant. This constant is assumed to vary with time. When referring to its current value we use H0 |
Hubble's law is important because it tells us that the universe is expanding and, if we extrapolate backward in time, that the universe was initially infinitely small and infinitely dense. It is one of many pieces of evidence for the big bang theory. Space-time came into being some 13.8 billion years ago, was filled with all the mass-energy that exists now and ever will exist, and then inflated rapidly from a region far smaller than a proton to one about the size of a grapefruit in an unbelievably small time of 10−32 seconds. This initial out rush of space-time filled with mass-energy was carried by its own "momentum" (for lack of a better word) until the observable universe grew to the dimensions we currently see — roughly 13.8 billion light years in all directions.
Hubble constant
Distances in astronomy are so vast that the meter is so hilariously small as to be useless. (Prefixes indicating muliples of the meter big enough to be used for the Hubble constant weren't invented until decades later.) To get around this, astronomers invented two units: the light year and the parsec. Of the two, the more intuitive to me is the light year, which is the distance a beam of light would travel in one year (9.46 × 1015 m).
The other unit, the parsec, is a geometric rather than physical unit. Astronomical bodies near to us (well, near in astronomical terms) will appear to shift their position in the sky as the Earth moves around in its orbit. The word parsec stands for "parallax of one arc second". Thus, an object that appears to shift by one arc second (13600°) as the Earth moves from one side of the Sun to the other six months later is said to be one parsec away. For observational astronomers of the Nineteenth and early Twentieth Centuries, the parsec was a more convenient unit for professional use than the light year. One parsec [pc] is approximately 3.26 light years or 3.09 × 1016 m. The nearest stars to the Earth other than the Sun are a bit more than one parsec away. The edge of the Milky Way is several thousand parsecs away — several kiloparsecs [kpc]. Cosmic distances, like those between galaxies, are measured in millions of parsecs or megaparsecs [Mpc]. This is the unit that Hubble used in his work.
In Hubble's original 1929 paper he reported a value for his constant of approximately 500 km/s/Mpc. I have included his original data in the section of this book entitled Curve Fitting. You can analyze the data yourself if you wish. Using a standard linear regression analysis, I came up with a value of H = 463 km/s/Mpc. The interesting thing about this value is that it is now universally recognized as terribly wrong. Distance measurements of extra galactic bodies in Hubble's era were later found to be seriously flawed. Still, the theory turned out to be right even if the data used to derive it were all undervalued.
Determination of H has progressed slowly but surely since the early 20th century and the constant has gone through several revisions. The current most accurate value comes from NASA's Wilkinson Microwave Anisotropy Probe (WMAP). As of 2014, the WMAP scientists have come up with a value of H0 = 69.3 ± 0.8 km/s/Mpc. Let's convert this value to SI units for some perspective…
| H0 = | 69.3 ± 0.8 km/s/Mpc |
| 3.08568 × 1019 km/Mpc |
| H0 = | 2.25 × 10−18 | 1 |
| s |
Seeing the Hubble constant in inverse second form makes it a bit more accessible. The space around us is expanding at a rate of roughly one part in 1018 every second. Given that the diameter of a proton or neutron is roughly 10−15 m, and that 18 orders of magnitude greater than this 1000 meters, a good phrase to tell your family, friends, and neighbors is that one kilometer of space expands at a rate equivalent to the diameter of one proton every second. If you'd like, we could scale this up a bit timewise.
| H0 = | 2.25 × 10−18 | 10 × 365.25 × 24 × 60 × 60 s | |
| s | decade |
| H0 = | 7.09 × 10−10 | 1 |
| decade |
This is a bit bigger than the diameter of a typical atom. Thus, one meter of space expands at a rate equivalent to the diameter of an atom every decade. An expansion this slow is imperceptible on the scales we humans are used to. Our lifetimes are too short and the sizes we have to deal with on a daily basis are too small.
To begin to appreciate the Hubble constant we need to scale things up a bit sizewise — a vast bit — 23 orders of magnitude and beyond. We need to look at the universe as a collection of galaxies; the closest of which are a few million light years away (1023 m) and the farthest of which are ten billion light years away (1026 m). The current most distant known objects are quasars. A quasar is a galaxy with a super massive black hole at its core that is actively gobbling up stars. The most distant quasar is moving away from us at 90% of the speed of light. Using Hubble's law, this gives us a distance of 12.7 billion light years.
| r = | v | = | 0.90 c | |
| H0 | 2.25 × 10−18 s−1 | |||
| r = | 4.01 × 1017 light seconds | |||
| r = | 4.01 × 1017 light seconds | |||
| 365.25 × 24 × 60 × 60 s | ||||
| r = | 12.7 × 109 light years | |||
The ultimate fate of the universe
The universe began with a big bang and is still growing, albeit at a much slower rate than in its early life. The inflationary period when space grew exponentially was quite short. The vast majority of the universe's history has been one of nearly constant growth overall, sprinkled with pockets of local contraction. Stars contract, galaxies collide and coalesce, but the distance between clusters of galaxies is getting larger and larger. The increase in the overall size of the universe has meant that the pull of gravity on the cosmic scale is getting weaker and weaker.
We have ample evidence to support the birth of the universe in a big bang, but the ultimate fate of the universe is still an open question. Will the expansion carry the galaxies so far apart that they no longer exert any significant gravitational pull on one another? This possible outcome is known as the heat death — a somewhat inappropriate name since the subsequently isolated galaxies would contain nothing but cold, dead stars. Or will gravity prevail so that everything comes crashing back in on itself? (Expansion be damned.) This cosmic reversal of the big bang is known as the big crunch.
The answer to this question can be found by combining Hubble's law with the formula for escape velocity. A universe that expands forever will have a density that produces an escape velocity less than that of its observed expansion. A universe that comes crashing back in upon itself will have an escape velocity greater than this.
Start by setting the escape velocity formula equal to the velocity from Hubble's law.
| v = H0r = √ | 2Gm |
| r |
Square both sides to eliminate the square root.
| H02r2 = | 2Gm |
| r |
Here comes the tricky part. What's the mass of the universe? Determining this number is quite difficult, but by looking at a large portion of the universe we can determine its density. Density times volumes gives us mass. Looking out from our vantage point in the Milky Way, we can see equally well in nearly all directions. (The Milky Way is rather congested, which prevents us from seeing those portions of the universe lying along its plane, but this doesn't hurt things much.) The locus of all points equidistant from a fixed point generates a sphere in space. Multiplying the density of the observable universe times the volume of the sphere that contains it gives us its mass.
m = ρV = ρ(43πr3) = 43πρr3
Substitute this expression into the one derived above it…
| H02r2 = | 2G | 4πρr3 | |
| r | 3 |
and solve for density.
| ρ0 = | 3H02 |
| 8πG |
This is the critical density separating eternal expansion from eventual collapse. Let's compute its value using the current best estimate of the Hubble constant in SI units.
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Anything greater than this density will resist the expansion of space. You and I and everything we see in our ordinary lives has a density much greater than this and will retain its shape and size for as long as all other things remain constant. People and other animals have a density about the same as water — 103 kg/m3 or 30 orders of magnitude larger than the critical density.
Looking at the universe as a whole, however, the situation is somewhat different. Given that most of the universe is hydrogen and that the mass of one hydrogen atom is 1.67 × 10−27 kg, this corresponds to a density of a half dozen hydrogen atoms per cubic meter.
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Our galaxy, and probably every other galaxy in sight, has a density of roughly one hydrogen atom per cubic centimeter. Since there are a million cubic centimeters in a cubic meter, galaxies are more than dense enough. About the only thing that can't resist the expansion of space is the space between galaxies, where the density is apparently on the order of one hydrogen atom for every four cubic meters.
A third possible outcome
The Hubble constant appears not to be constant. Observations first made in 1998 show that the rate of expansion of the universe is increasing. If the universe began with a big bang 13.8 billion years ago it just might end with a big rip, 20 billion years in the future. The reason is that there's more to the universe than meets the eye. Much more.
What you see isn't what you get.
Only 4% of the universe is thought to be made of ordinary stuff — atoms, nuclei, electrons, photons, and neutrinos. Another 23% is dark matter — "dark" because it interacts gravitationally like ordinary matter, but not electromagnetically. Thus it cannot be "seen" with electromagnetic radiation like light, x-rays, or radio waves. (Dark matter is discussed in more detail in a previous section of this book.) The remaining 73% of the universe — the overwhelming majority — is in the form of dark energy. Dark energy is really odd, not only because it can't be seen, but also because it acts a lot like negative mass. That is, it strives to expand space rather than compress it. Despite its preeminence, dark energy has only the feeblest effect on the local space around us. (In this context "local" refers to everything within out galaxy and nearly everything from here to the nearest thousand galaxies.) Small though it is, this tiny effect adds up until it dominates the behavior of the space in the universe as a whole. Dark energy may be tiny but it is everywhere. Like a swarm of mosquitoes at a summer picnic, dark energy has enough of an effect to ruin the whole party.
With the way things are going, in another ten billions years or so the distance between galaxies will have expanded to the point where light won't be able to traverse the space between them fast enough. Space will be expanding so rapidly that every galaxy will be independent of and invisible from every other galaxy. Our visible universe would then be reduced from its current 100,000 galaxies to just one — our own Milky Way.
Now it gets really weird. If there's nothing to stop the acceleration of the expansion, eventually the galaxies themselves will start to come undone. (This is the so called phantom energy scenario.) First, the 100 billion stars of the Milky Way will split off to form isolated solar system universes. Then the space within these solar systems will begin to increase noticeably. The Earth and other planets will recede away from each other in our own solar system. The sun would grow ever smaller in the sky until it disappeared. At this point, life on Earth would get really bad. (Assuming the Earth was still around.) Worse than the thought of isolation from the rest of the universe is the thought of losing touch with the Sun. This would be our effective end. Were anyone around to witness what happened next, they would eventually see the rate of expansion increase to the point where the Earth exploded, then everything on the Earth would be reduced to vapor, then molecules would lose coherence, followed by atoms and nuclei. The final end of it all would occur when the fundamental particles themselves were torn to shreds and there was nothing left of the universe to say that it ever existed.
I'll tell you what I like about this stuff. It forces you to think beyond yourself. What will the world look like when you die? The answer is culturally quite different (it's the nature of times we live in) but physically pretty much the same. Multiply this by ten, by a hundred, by a thousand, by a million lifetimes! Care to make a prediction? Keep going. A billion. A trillion lifetimes. Now the time scales have gotten so vast that only a few would care to speculate. This is the realm of theoretical physics and, amazingly, it is not beyond our comprehension.