Newton’s law of motion is that the force acting on a body equals the mass of that body times its acceleration, *F* equals *ma*, or *m* *dv/dt*. And Newton’s law of gravity is *F* equals minus-*G *times big-*M *times little-*m* over *r*-squared times *r-*hat, where in this case the big-*M* is the Sun’s mass and little-*m* is the planet’s mass. So, now, let’s roll up our sleeves and show how these laws relate to Kepler’s laws.

One might expect that we’d go in order, starting with Kepler’s first law. But it turns out Kepler’s second law is the most fundamental. So, let’s start there. The key concept is the conservation of angular momentum. Before getting to angular momentum, though, let’s think a moment about momentum and velocity.

Momentum is mass times velocity, *p *equals* mv*. And the velocity has 2 components. In a time *dt*, the velocity takes the planet from one position to another, changing both *r *and *theta*. That means the velocity has a radial component, toward or away from the origin, and an angular component, in the perpendicular direction.

The radial component, *v*-sub-*r *is equal to *dr/dt*. And the angular component is *rd-theta*, the distance moved in the direction of increasing *theta*, divided by *dt*. So, *v_theta* equals *rd-theta*/*dt*.

Angular momentum, which we’ll consider as *L*, is defined as *r* times *mv_theta*. Only the angular component, the ‘sideways component’, of the velocity matters. And since *v_theta* is *rd-theta*/*dt*, we can write *L* as *mr*–squared *dtheta/dt*.

In vector language, *L* equals *r-*cross *mv*, the cross product being the way to pick out only the perpendicular component of a vector. It has a magnitude of *r* times the component of *mv* that’s perpendicular to *r*.

This article comes directly from content in the video seriesIntroduction to Astrophysics.Watch it now, on Wondrium.

Angular momentum, in some circumstances, is conserved. It doesn’t change with time, even if the body is changing in lots of other ways. The classic example is the twirling figure skater who pulls in her arms, effectively reducing her *r*, which means her *d-theta/dt* must increase, to compensate. That’s why she twirls faster. Angular momentum is conserved whenever there’s no net torque; no force in the *theta *direction. This is certainly true for the planets; the only force is gravity, which is in the radial direction, toward the Sun.

So, as a planet goes around, even though *r* and *v* are always changing, *r*–squared *d-theta/dt* is a constant, and therefore, *d-theta/dt* varies as one over *r*-squared. That’s Kepler’s second law! So, we now see that Kepler’s second law is a consequence of the conservation of angular momentum. Just as the ice skater twirls faster when she pulls in her arms, the planets twirl faster when they approach the Sun.

This is an important result, with implications beyond planetary motion. It helps explain why material speeds up as it spirals into a black hole, why a star spins faster when it contracts in size, and why a young star is surrounded by a spinning disk of material, within which the planets are formed.

Why is the orbital period proportional to the 3-halves power of the semimajor axis? We’ll prove it for a circular orbit, where we can see the physics very clearly.

Imagine a planet moving in a circle of radius *a*, with some constant speed *v*. Over a full orbital period, *P*, the planet travels all the way around the circle. Therefore, *v *must equal the circumference of the circle, 2*pi* *a* divided by *P*. Or equivalently, *P* equals 2*pi* *a* over *v*. So, we already see one reason why *P *increases with *a*: the circumference of the circle gets bigger. There’s a longer way to go. In addition, when *a* is larger, *v* is lower; the planet moves more slowly because the gravitational attraction is weaker. This increases *P* even more, so that at the end of the day *P* goes like *a* to the 3/2.

In a time *dt*, the planet advances by a small angle *d-theta*, which corresponds to an arc length of *a d-theta*. So, *v *equals *a d-theta/dt*. During that same time interval, the velocity vector rotates by the same angle *d-theta*. The change in the velocity vector is *v *times *d-theta*. And so, the magnitude of the acceleration, the rate of change of velocity, must be *v d-theta/dt*. We can combine the equations, by solving the first one for *d-theta/dt*, and then inserting the answer, *v *over* a*, into the second equation. This gives an acceleration of *v*-squared over *a*.

We have just proven something you might have already known: to keep a body moving at speed *v* in a circle of radius *a*, you need to supply an inward acceleration, a centripetal acceleration, of *v*-squared over *a*. In the case of a planet, that acceleration is provided by the Sun’s gravitational force, *G *time big-*M* over *a*-squared. We set that equal to *v*-squared over *a*, and solve for *v*, finding *v* equals the square root of *GM* over *a*. Which we can then insert into our earlier expression for the period and, presto, we see that *P* is proportional to *a* to the 3/2 power. We also see that the proportionality constant is 2*pi* over the square root of *GM*. It goes down with the mass of the attracting body.

Let’s rewrite Kepler’s third law in a more convenient form, a so-called scaling relation. We know that when *M* equals one solar mass, and *a* equals 1 AU, the period will come out to be a year; that’s the situation for the Earth. And we also know that *P* is proportional to *M* ^{to }the minus-1/2 power, and *a *to the 3/2 power. So, we can rewrite Kepler’s third law as *P* equals one year times *M* over *M-*dot to the minus 1/2 times *a* over 1 AU to the 3/2 power.

Newton’s law of motion is that the force acting on a body equals the mass of that body times its acceleration.

Momentum is mass times velocity, *p *equals* mv*.

Angular momentum is conserved whenever there’s no net torque. Kepler’s second law is a consequence of the conservation of angular momentum.

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As the planet moves, the line joining the planet and the star, the planet’s radius vector, sweeps out area at a steady rate. If we would color the area swept out by the radius vector, and switch the color every second, we would notice that when the planet is near the star, the radius is shorter, but the planet moves faster, so the colored sector has a large angular width, and when the planet is far away, the radius is longer but the planet is slower, making a tall skinny sector.

What Kepler found is that all the sectors have the same area: equal areas in equal times. However, this can mathematically be put it another way.

This article comes directly from content in the video seriesIntroduction to Astrophysics.Watch it now, on Wondrium.

Let’s imagine and put the planet at an arbitrary position and watch it move for an infinitesimal time interval *dt*. It sweeps out a thin sector spanning an angle *d-theta*, with an area *dA*. And what’s the area of that sector? Well, the planet moves in both the radial direction, the *r-*direction, and the perpendicular direction, which we’ll call the *theta*-direction—the direction of increasing *theta*. It’s the motion in the *theta*-direction that sweeps out area. Purely radial motion doesn’t sweep any area.

In time *dt*, the planet moves in the *theta-*direction by an amount *r *times *d-theta*, using the usual small angle approximation. So, the swept-out sector is basically a skinny right triangle with sides of *r* and *rd-theta*. The area of that triangle is 1/2 *r* times *rd-theta*. That does leave out a tiny little corner piece of the sector, its dimensions of *dr* and *rd-theta*, that’s the product of 2 tiny numbers. So, in the limit of infinitesimal *dt*, that little piece is vanishingly small compared to the rest of the triangle. All of which means we can write *dA* equals 1/2 *r* times *rd-theta*. And *dA/dt*, the rate at which area is swept out, is 1/2 times *r*-squared *d-theta/dt*.

If that rate is a constant, as Kepler observed, then *dtheta/dt* must be proportional to one over *r*-squared. That is the modern way to state Kepler’s second law. *Theta* advances at a rate that varies as one over *r*-squared.

Then there’s Kepler’s third law. This one is about total time required to go all the way around, the orbital period – the bigger the orbit, the longer the period.

To be quantitative, let’s think of a logarithmic chart, where the horizontal axis shows the semimajor axis, in AU, and the vertical axis shows the period, in Earth years. So, the point representing the Earth is at 1 AU and one year, and the other points are for the other planets. Strikingly, they all fall on a single straight line! It has a slope of 3/2; if we move 2 units to the right, the line goes up 3 units. Since this is a log plot, it means the log of *P* is 3/2 times the log of *a*, plus some constant. Which in turn means that *P* in proportion to *a *to the3/2 power. That’s Kepler’s third law.

Kepler’s third law is the most reliable way we have to measure the mass of just about anything in astrophysics. The basic idea is that to measure an object’s mass, we need to watch other things moving in response to its gravity. It works for stars, planets, black holes, neutron stars, entire galaxies, and even, in a sense, for measuring the mass of the entire universe.

Let’s write Kepler’s third law in a scaling relation. We know that when M equals one solar mass, and *a* equals 1 AU, the period will come out to be a year; that’s the situation for the Earth. And we also know that P is proportional to M to the minus-1/2 power, and *a* to the 3/2 power. We can write Kepler’s third law as *P* equals one year times *M* over *M-*dot to the minus 1/2 times *a* over 1 AU to the 3/2 power.

What we’ve done is to evaluate the equation for a benchmark case, the Earth, and then specified how the answer scales with the inputs, *M* and *a*. The scaling relation is handy because as long as we use units of years, AU, and solar masses, we won’t have to look up the numerical values of *G* and the mass of the Sun , or worry about the 2*pi*. We can just plug the numbers directly into the scaling relation.

We can make a scaling relation of out just about any equation because it helps give us a sense of the numbers. It keeps us from getting lost in the orders of magnitude. Let’s illustrate with Jupiter. We know that Europa has a *P* of 0.01 years, and the value of *a* is about 0.0045 AU. If we plug those numbers into our scaling relation, and solve for *M* over *M-*dot, we get 0.0009; we just learned that Jupiter is a bit less than a thousandth the mass of the Sun.

Kepler’s second law is about how fast the planets move; when they’re close to the Sun, they move faster, in a specific way.

Kepler’s third law is about total time required to go all the way around, the orbital period – the bigger the orbit, the longer the period.

Kepler’s third law is the most reliable way we have to measure the mass of just about anything in astrophysics. The basic idea is that to measure an object’s mass, we need to watch other things moving in response to its gravity. It works for stars, planets, black holes, neutron stars, entire galaxies, and even, in a sense, for measuring the mass of the entire universe.

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Mathematically, a circle is defined as the set of all points that are the same distance from some chosen center. Put a pin in a corkboard, then take a piece of string and tie both ends to the pin, making a loop. Then, put a pen in the loop, stretch the string tight, and sweep around the board, drawing as you go. You’ve just made a circle, centered on the pin.

For an ellipse, you need 2 pins, and you tie the ends of the string to the pins, with some slack. And stretch the string tight with the pen, and sweep around the board, drawing as you go. What you get in that case is an ellipse. The 2 pins are the focus points, or foci, of the ellipse. For all the points on the ellipse, the distance to the first focus, plus the distance to the second focus, is a constant, that’s the length of the string, in our construction.

The length of that string is also equal to the length of the long axis of the ellipse, the major axis. If we unpinned the string and straightened it out, it would reach exactly across the major axis. We conclude that for any point on an ellipse, the distance to one focus plus the distance to the other focus equals the length of the major axis.

This article comes directly from content in the video seriesIntroduction to Astrophysics.Watch it now, on Wondrium.

To make a circle, we only needed to make one choice: the radius, which we’ll call *a*. For an ellipse, the equivalent is the radius along the major axis, which we’ll also label *a*, and we’ll call that the semi-major axis. But with an ellipse, we also have to choose the distance between the center and either focus.

We can choose whatever distance we want, as long as it’s smaller than *a*. Tradition dictates that we express that distance as *a *times* e*, where *e *is a number smaller than one, which is called the eccentricity. When *e* is zero, the foci coincide at the center, and we go back to a circle of radius *a*; as *e* gets larger, and closer to one, the foci separate and we get a more elongated ellipse.

For a circle, we know area equals *pi* *a*-squared. For an ellipse, it turns out to be *pi* *a*-squared times the square root of one minus-*e*-squared.

To understand the mathematical equation for an ellipse, in polar coordinates, let’s start by introducing a coordinate system. We’ll put the origin at one of the foci, and we’ll lay down *x* and *y* axes along the major and minor axes. To specify the points on the ellipse, we would use polar coordinates: *r* is the distance from the origin, and *theta* is the angle measured counter-clockwise from the *x* axis.

First, let’s think about what we expect. It’s going to be a rising and falling function, for *theta* equals zero, *r* has its minimum value of *a* minus *ae*, that’s *a *times one minus-*e*. As we dial *theta* up to higher values, *r* increases, and at *theta *equals *pi*, 180°, *r* achieves its maximum value of *a* times one plus-*e*. Then it shrinks back down as *theta* wraps around to 2*pi*.

To find the equation, we start with the fact that at any point on the ellipsis, the sum of distances to the foci is equal to the major-axis length, 2*a*. We can write that as *r* plus *r*-prime equals 2*a*, where *r* is the distance to the focus at the origin, and *r*-prime is the distance to the other focus. But that’s not such a convenient equation. We want it purely in terms of *r* and *theta*, not *r*-prime, so how do we get rid of the *r*-prime? We use the Law of Cosines.

The Pythagorean theorem says *c*-squared is equal to *a*-squared plus *b*-squared, where *a*, *b*, and *c* are the lengths of the sides of a right triangle. The Law of Cosines is the generalization to any triangle. It says that for any triangle, *c*-squared equals *a*-squared plus *b*-squared minus 2*ab* times the cosine of *gamma*, where *gamma* is the angle across from the *c*-side.

We’ll apply it to our triangle, with *r*-prime as our *c*-side. So, we have *r*-prime squared equals *r*-squared plus 2*ae* -squared minus 2*r* times 2*ae* times the cosine of angle opposite *r*-prime, which is *pi* minus *theta*. And the cosine of *pi* minus *theta* is minus the cosine of *theta*. At the end, we find *r* equals *a* times one minus *e*-squared divided by one plus *e *cos *theta*. That’s our equation!

So, does it make sense? When the eccentricity is zero, the equation reduces to *r* equals *a* times one over one, that’s just *r* equals *a*; that’s a circle. And when *e* is not zero, and we dial *theta* around the clock, cos *theta* goes from one to minus one and back to one, the denominator starts big and gets small, then big again. That, too, makes sense; it says *r *oscillates between a minimum value at *theta* equals zero and a maximum at *pi*.

In particular, if we plug in *theta *equals 0, we get *a *times one minus* e*-squared over one plus *e*. And since one minus* e*-squared is one plus *e* times one minus *e*, the one plus *e*’s cancel out, and we’re left with *a *times one minus*-e*, which is what we expected. That’s the minimum distance from the focus. Likewise, when *theta *equals *pi*, we get the expected maximum distance of *a *times one plus-*e*.

So, what Kepler noticed, his ‘first law’, is that all the planets move on ellipses, with the Sun not at the center but rather at one of the foci. Each planet has its own value for the semi major axis and eccentricity.

Kepler’s first law is that the planets trace out ellipses as they go around the Sun.

Mathematically, a circle is defined as the set of all points that are the same distance from some chosen center.

The Pythagorean theorem says *c*-squared is equal to *a*-squared plus *b*-squared, where *a*, *b*, and *c* are the lengths of the sides of a right triangle.

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In 1923, Edwin Hubble spotted some Cepheids in the Andromeda Nebula, which at that time, was a subject of debate, over whether it was a distant galaxy of billions of stars, or some kind of swirly gas cloud or star cluster that was inside the Milky Way. Hubble used the Cepheids to find the distance to Andromeda, and he showed it was a good fraction of a megaparsec away. Therefore, its angular size of a few degrees corresponds to a true size comparable to that of the Milky Way.

Interestingly though, even to this day, we don’t know exactly why Cepheids are such good standard candles. There’s been a lot of progress, but it’s fair to say that we can’t calculate the period-luminosity relationship from first principles. Of course, we don’t always need to understand a tool for it to be useful. We just need to confirm it works reliably. But it is a strange situation.

Cepheids are these powerful beacons that have been used for more than a century to map out our galactic neighborhood. And yet, what’s going on inside the beacons, remains an area of ongoing research.

With our best telescopes, Cepheids can be seen out to a distance of around 50 megaparsecs. But if we want to go further, if we want to reach out to gigaparsecs, we need to find something else. Ideally, we’d find another type of standard candle, one that’s orders of magnitude more luminous than a Cepheid, so we could see it all the way across the universe. Unfortunately, nobody has found such a standard candle. But it turns out there is something equally good, maybe even better: a standard explosion.

In the 1980s, astronomers realized that a certain category of exploding stars—supernovas—produce fireballs that all have nearly the same peak luminosity. They’re called Type Ia supernovas, and they all explode with about the same energy. Or, at least, nearly the same energy; they’re predictable enough that if we measure the color and duration of the afterglow, we can determine its luminosity to within a few percent. And how do we know that? It’s because we’ve spotted Type Ia supernovas in nearby galaxies that also have Cepheids in them.

This article comes directly from content in the video seriesIntroduction to Astrophysics. Watch it now, on Wondrium.

A chart showing data for some nearby Type Ia collection illustrates how we can use these supernovas as standard explosions. The horizontal axis, in the chart shows time, in days, and the vertical axis shows the measured luminosity of the explosion. In the chart, they all rise to about the same level, with differences that correlate with the duration. Keeping in mind that the faster the explosion fades, the lower the peak luminosity, when we measure the rise and fall of flux from a really distant type Ia supernova, we can match the observed duration of the event to one on this chart, and then read off the peak luminosity.

That’s how we can use Type Ia supernovas as standard explosions. Their great advantage is that they’re as bright as 5 billion Suns, bright enough to see even when they happen in galaxies, far, far away.

These are stupendous explosions which we don’t know the cause of. What we do know is that they’re almost certainly exploding white dwarfs but the trigger for the explosion remains a topic of active research. What is clear, though, is that they work. We can use them to measure cosmological distances. They’re the last step in our quest.

The chart which illustrates different methods for figuring out distances to astronomical objects, is referred to as the distance ladder. As each rung is an order of magnitude in distance, it’s a logarithmic chart. The bars joining the rungs are the different measurement techniques that work over that range of distances. Out to a few AU, we can use radar ranging. To go beyond the solar system, we use parallax, which works out to kiloparsecs; that takes us most of the way around the Milky Way.

From there, we rely on standard candles: Cepheid variables. We measure their pulsation periods, deduce their luminosities, and use the flux/luminosity relation to get the distance. That works out to around 50 megaparsecs, a big part of the universe, containing hundreds of thousands of galaxies.

Even farther away, we rely on standard explosions: Type Ia supernovas. We watch the light from the fireball rise and fall, deduce the peak luminosity, and again use the flux/luminosity relation to get the distance which gets us out to gigaparsecs.

Edwin Hubble used the Cepheids to find the distance to Andromeda, and he showed it was a good fraction of a megaparsec away. Therefore, its angular size of a few degrees corresponds to a true size comparable to that of the Milky Way.

In the 1980s, astronomers realized that a certain category of exploding stars—supernovas—produce fireballs that all have nearly the same peak luminosity. They’re called Type Ia supernovas, and they all explode with about the same energy.

Out to a few AU, we can use radar ranging. To go beyond the solar system, we use parallax, which works out to kiloparsecs; that takes us most of the way around the Milky Way.

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In astronomy, when we talk about units, as is the case with the diffraction limit, alpha equals 1 AU over d works when alpha is expressed in radians. If an object of size S is located a distance d from Earth, its angular size is S over d radians. But what if we want to use arcseconds? Well, one radian works out to be 206,265 arcseonds. So, if we’re expressing alpha in arcseconds, the right side of our parallax equation becomes 206,265 AU over d.

The tradition at this point is to also do a little sleight of hand and define a new unit of distance, the parallax-second, or parsec, equal to 206,265 AU. That way the numbers are easier: d equals one parsec divided by alpha in arcseconds. For example, the bright star Sirius shows a parallax of 0.38 arcseconds, so its distance in parsecs is 1 over 0.38, or 2.6.

Out of these, the parsec is a handy unit for measuring the distances between stars, and it happens to have the same order of magnitude as the light-year. One parsec is about 3.3 light years. Over the course of a year, nearby stars appear to move on the celestial sphere in little ellipses of with a maximum angular size, in arcseconds, equal to one divided by the distance in parsecs. So, a star 10 parsecs away has a parallax angle of a 10th of an arcsecond.

When we talk about units for angles: lambda over D is a dimensionless number, it’s a length divided by a length. That means the angle is measured in ‘natural units’ or radians. In ordinary life, we measure angles in degrees, with a right angle being 90°, and 360° going around the whole circle. However, it’s simpler to calculate in radians, the system in which a right angle is pi over 2, and the whole circle is 2pi. For example, if we’re observing visible light with a 10-centimeter telescope, then the diffraction limit comes out to be 6 times 10 to the minus-6 radians. This, according to the Babylonian system, would be about 1.2 arcseconds.

To put that into perspective, our eyes probably have an angular resolution of around 50 or 100 arcseconds. And the full Moon has an angular diameter of half a degree, or 1800 arcseconds.

However, when it comes to measuring distances to stars, parallax is by far the most reliable method. Yet, as the distance gets larger, eventually the parallax angle becomes too small to measure, if for no other reason than the diffraction limit. Right now, our best parallax measurements come from a space telescope, called *Gaia*, which was launched by the European Space Agency in 2013.

*Gaia* measured parallaxes as small as a ten-thousandth of an arcsecond; that’s good enough to make maps of the galaxy out to 10,000 parsecs, 10 kiloparsecs. That is impressive. But to go beyond our galaxy—and there’s a lot beyond our galaxy—we need to take another step in the quest to measure distances.

This quest is actually a long story, and it’s central to the history of astronomy. As of now, there are two best ways to measure the distances to very remote objects. They both rely on the flux-luminosity relation: F equals L over 4pi d-squared, wherein, L is the power that an object emits, and F is the power per unit area measured by Earthlings. So, we can measure F, but we can’t figure out L unless we also know the distance, d.

But suppose there were some light source out there for which we already knew L. In that case, one could calculate the distance by re-arranging the flux-luminosity equation: d equals square root L over 4pi F.

This article comes directly from content in the video seriesIntroduction to Astrophysics. Watch it now, on Wondrium.

Unfortunately, stars don’t come with manufacturer’s labels! Yet, in some special cases, we can figure out the wattage even without a label. Let’s take the example of a famous star called *RS Puppis*, surrounded by a cloud of material that was ejected by the star. It’s an example of a category of stars called Cepheid variables. They are called ‘Cepheid’ because the first known example was in the constellation of Cepheus, and ‘variable’ because these stars vary in brightness.

They pulse, they get brighter and fainter, in an endless cycle, with a period—the time for a full oscillation—that’s typically a few weeks. The important thing is that the average luminosity of a Cepheid can be predicted accurately from the period of the pulsations. This was first discovered by Henrietta Leavitt, in 1912. Stars that pulse more slowly are intrinsically more luminous.

One reason we know this to be true is that some Cepheids are close enough for parallax measurements, so we can determine their luminosities. And among that collection, one can easily observe that L is linked to the pulsation period, P. A schematic chart of luminosity versus period shows this increasing relationship.

So, if we spot a Cepheid a megaparsec away, in some other galaxy, we can’t measure its parallax, but we can measure its pulsation period. We just monitor the flux, and see how long it takes to rise and fall. We can then use the period-luminosity relationship to determine L, and boom, we’ve got both F and L, and we can calculate d, figuring out the distance to the galaxy where the Cepheid resides.

The parsec is a handy unit for measuring the distances between stars, and it happens to have the same order of magnitude as the light-year.

Our best parallax measurements come from a space telescope, called *Gaia*, which was launched by the European Space Agency in 2013.

Cepheid variables are called ‘Cepheid’ because the first known example was in the constellation of Cepheus, and ‘variable’ because these stars vary in brightness.

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Imagine a situation in which we’re on the Earth, and an object is located a distance d from the Earth. The object has a true size of S, and an angular size alpha, that is, the rays arriving from opposite sides of the object have an angle alpha between them. Using the small angle approximation, alpha approximates to S divided by d, or equivalently, S equals alpha times d. The problem is that if all that we know is alpha, we can’t figure out S without knowing d.

A similar situation arises when we measure the brightness of a source. For a given brightness, we can’t tell if the source is intrinsically luminous, and very far away, or if it’s actually intrinsically faint and happens to be nearby. Suppose the luminosity is L. That’s the power, the energy per unit time, that our luminous object is pouring out into space. We can measure L in units of watts, for example. And if all that power spreads out as the radiation goes farther from the source.

The Earth is far away, at a distance of d. So, by the time the light reaches the Earth, it’s been spread out over a huge sphere of radius d. We can’t measure L directly. Instead, we’ve got a telescope with a certain collecting area, and we measure the power received by the telescope. We can then calculate the power per unit area, which we call the Flux, whereby F equals L divided by the surface area of the sphere, 4pi d-squared. That’s an important equation, the flux-luminosity relation: F equals L divided by 4pi d-squared. It’s another example of an inverse square law: The flux goes down as the inverse square of the distance.

If we measure F, we need to know d in order to deduce L. And we really want to know L, the true luminosity, if we want to figure out what’s physically going on to produce the radiation. The message again is that measuring the distance to an astronomical object is a crucial problem that we need to solve. And like any tough problem, it’s useful to break it down into manageable parts.

The first step is using a radar. Build a giant radio transmitter, aim it at a nearby planet, and fire. If we hit the target, and our receiver is sensitive enough, we can detect the echo, the reflected radio waves. The echo will be delayed by a time interval Delta-t equals 2d over c, where 2d is the round-trip distance, and c is the speed of radio waves, that is, it’s the speed of light. And since we know the speed of light, we can calculate d.

This article comes directly from content in the video seriesIntroduction to Astrophysics. Watch it now, on Wondrium.

With the world’s biggest radio transmitters, we can measure the distances to Mercury, Venus, Mars, and even some asteroids. That allows us to make maps of the solar system with a precision of a few parts in 10 billion. Even though it’s an astonishing level of detail, unfortunately, this method is limited to relatively nearby objects. We can show that the amplitude of the echo falls off like 1 over d to the 4th power. So, to go beyond the solar system, we need other techniques.

This brings us to the next step in our solution: parallax. Parallax is based on simple geometry. It can be understood by simply holding out our arm and raising a finger. Now, if we close our left eye and look at our finger and the scene in the background and then repeat the same after switching eyes and closing the right one, it will look like our finger just jumped! That’s because our right eye views from a slightly different angle, so it sees your finger projected against a different part of the background scene. That’s parallax. And if we measure that shift in angle, as well as the distance between our eyes, we could use trigonometry to calculate the distance to our finger.

The parallax equation in astronomy takes advantage of the Earth’s motion around the Sun. In order to make the calculations, we take a picture of a nearby star, and then we wait six months for the Earth to go halfway around, and then we take another picture. That’s like closing our left eye and opening our right eye. The nearby star, like our finger, will appear to have shifted in position relative to the much more numerous background stars.

Following the the parallax equation, over the course of a year, nearby stars appear to move on the celestial sphere in little ellipses. These elipses are of a maximum angular size, in arcseconds, equal to one divided by the distance in parsecs. So, a star 10 parsecs away has a parallax angle of a 10th of an arcsecond.

If the star is directly above the plane of Earth’s orbit or off to the side somewhere, that doesn’t change the basic idea; it just means that the star will appear to move in an ellipse, rather than a circle, and the parallax angle is the semimajor axis of the ellipse. And if the star is right on the ecliptic—the projection of the Earth’s orbit onto the celestial sphere—it’ll go back and forth along a straight line.

The power per unit area is what we call the Flux, whereby F equals L divided by the surface area of the sphere, 4pi d-squared.

With the world’s biggest radio transmitters, we can measure the distances to Mercury, Venus, Mars, and even some asteroids.

The parallax equation in astronomy takes advantage of the Earth’s motion around the Sun.

Understanding the Distinctive Properties of the Solar System

Getting Data about the Solar System: Telescopes and Spacecrafts

The Outer Region of the Solar System

In astrophysics, the location of a body is specified using latitude, longitude, and elevation. This is the spherical polar coordinate system. It makes sense for the Earth because it is a sphere. And it makes sense for astronomers, too, not because the universe is a sphere, but because we’re trapped on a sphere. That makes it natural to use an Earth-centered coordinate system and to extend our concepts of latitude, longitude, and elevation up into the heavens.

Our ancient ancestors imagined the sky to be a glass ‘celestial sphere’ upon which the stars and planets were painted. Of course, this is not physically correct; the stars that we see forming constellations are actually at very different distances from the Earth. But the celestial sphere is still a useful fiction.

So, let’s imagine a giant, transparent sphere, centered on the Earth, marked with grid lines of latitude and longitude. The latitude lines tell us how far we are from the celestial equator, that’s the projection of the Earth’s equator up into the sky. And the longitude lines tell us how far east or west we are from the celestial equivalent of the prime meridian. That way when we look at a distant star, we can read off the star’s angular coordinates by seeing where it appears relative to the grid. That leaves only the third dimension, the distance to the star, the celestial equivalent of elevation, which is much trickier to measure.

Let’s start with the angular coordinates and take an example of two stars that happen to be located along nearly the same line of sight, from the Earth. They’ll appear close together on the celestial sphere. However, if they’re too close, they will blend together and appear as a single point of light, rather than two. So, what determines whether we can perceive the double star? As one might guess, it depends on how good our telescope is. But we can be more specific than that, and it’s worth going into detail because the question gets right at the basic dilemma of astronomy, which is that all we have is light.

With few exceptions, our only source of knowledge is the electromagnetic radiation that happens to hit the spinning ball of rock we live on. So, we need to understand the physics of light.

Imagine a telescope as a big lens pointed at star straight overhead, which focuses the starlight into a tight spot on our camera. If there’s another star in a slightly different direction, then ideally, the lens would focus its light onto a different spot in the image. So, our image shows two dots, star 1 and star 2. The problem, though, is we can’t focus light into as small a point as we might like. Moreover, the stars blend together when the angle between them, Delta-theta, is on the order of lambda over D, where lambda is the wavelength of light, and D is the diameter of our lens, or mirror, or whatever we’re using to collect and focus light.

The reason why the stars blend together and, inevitably, blurr is owing to the phenomenon called diffraction, a consequence of the wave nature of light. Light is an electromagnetic wave, a traveling pattern of oscillating electric and magnetic fields. So, we can imagine the light from star 1 as an ocean wave, a traveling pattern of crests and troughs of electromagnetic energy, with a wavelength—a separation between crests—of lambda.

This wave then passes through the diameter D of our telescope and then a lens or a mirror responds to that pattern of energy by redirecting it toward a camera.

Introduction to Astrophysics. Watch it now, on Wondrium.

The energy gets redirected to a different position on the camera, if the starlight comes in from a different angle, tilted by Delta-theta. But if Delta-theta is really tiny, the wave energy is smeared out with a spatial extent of lambda. So, the telescope still sees a crest filling the opening. The optical system responds by directing the energy to the same spot on the detector.

As we increase Delta-theta, at what point do the waves from star 2 start to look different from star 1? It’s when we no longer have a crest extending across the opening. The tilt is large enough that there’s a crest at one end, and a trough at the other end. For this minimum value of Delta-theta, it’s at least possible for the optics to distinguish between the two waves.

Since the diffraction limit is proportional to lambda over D, the way to improve our angular resolution is to increase D, build a bigger telescope. And that does work, to a point.

But in practice, there are lots of other reasons why our images might be blurry, besides diffraction. Maybe our lens isn’t perfectly polished, or our mirror has defects. And then there’s the constantly fluctuating atmosphere, which scrambles the directions of light rays at the level of about an arcsecond, even at our best mountaintop observatories. So, even with a large telescope we usually can’t achieve the ultimate diffraction limit. That’s one reason why we launch telescopes into space, above the atmosphere.

However, the problem remains that even with a perfect telescope and no atmosphere, we can’t resolve details on angular scales smaller than of order lambda over D radians, where lambda is the wavelength of light, and D is the diameter of the telescope. Moreover, turbulence in the Earth’s atmosphere usually limits us to about an arcsecond, regardless of the size of our telescope.

The latitude lines tell us how far we are from the celestial equator, that’s the projection of the Earth’s equator up into the sky. And the longitude lines tell us how far east or west we are from the celestial equivalent of the prime meridian. That way when we look at a distant star, we can read off the star’s angular coordinates.

The reason why the stars blend together and, inevitably, blurr is owing to the phenomenon called diffraction, a consequence of the wave nature of light.

Turbulence in the Earth’s atmosphere usually limits us to about an arcsecond, regardless of the size of our telescope.

Mapping the Distribution of Galaxies in the Universe

Methods for Determining Distances to Galaxies

How Surveys Help Studying the Night Sky

Let’s understand what dataset looks like on the logarithmic set of axes.

There are 4 different groups, differing in the relationship between mass and radius. For the lowest-mass objects, we can draw a straight line through the points up until about twice the mass of the Earth, at which point the pattern changes; the size starts going up more rapidly with mass. The line connecting the dots has a steeper slope. Then at around a 100 Earth masses, this relationship changes again; it flattens out. And finally, at around 20,000 Earth masses, the size starts rising again with mass, more rapidly than in any other part of the chart.

To get oriented, consider the data points for all the planets in our solar system. Together, they span the first 3 groups, and the Sun is in the fourth group. In each of these 4 zones, we can fit the data, at least approximately, with a straight line.

On a regular *x-y* chart, a straight line means that *y = ax + b*, where *a* is the slope of the line, and *b* is a constant, the *y*-intercept. That’s a linear relationship.

But on a logarithmic chart, a straight line means that there’s a linear relationship between the logs of the variables: log* x = a *log* y + b*; or in this case, the log of the radius equals *a* times the log of the mass, plus a constant. And what does that tell us? It’ll be clearer if we solve for *R*. We can do that by taking the inverse log of both sides, that is, we’ll take 10 to the power of each side. On the left, we have 10 to the log *R* which is just *R*, as we wanted. On the right, we have 10 ^{a}^{ log M + b}.

This article comes directly from content in the video seriesIntroduction to Astrophysics.Watch it now, on Wondrium.

When we have a sum in the exponent, we can split it into a product. We can rewrite the right side as 10^{a}^{ log M} x 10* ^{b}*. And another fact is that

That kind of relationship is called a power law. One variable is proportional to another one raised to some power.

If we measure the slope, the value of *a*, in each of these 4 zones, we note that for the lowest mass objects, the slope is about 1/3. That means *R* is proportional to *M* to the 1/3 power: the cube root of *M*.

This low-mass regime is closest to the one where we have direct experience: small things, like rocks and boulders. So, what do we expect for the relation between radius and mass of everyday objects? Well, it should depend on the density of whatever material it’s made of.

Let’s use the Greek letter *rho* for density. Rocks have a certain characteristic density, around 2 or 3 grams per cubic centimeter. And mass is equal to density times volume. For a spherical object, the volume is 4/3 *pi* times the cube of the radius. So, we expect *M* to be proportional to *R *cubed, or, inverting that, *R* is proportional to *M *to the 1/3 power. Which is what we see in the chart!

In the second zone, we find a slope of about a 1/2; *R* is proportional to *M *to the 1/2 power, which is bigger than 1/3. That means that the more massive objects have larger radii than we would have expected, if they all had the same density. So, it must be that the more massive objects are less dense.

And this is true. These are gaseous planets, like Uranus, Neptune, and Saturn. In this zone, the planets have a lot of hydrogen and helium gas, very lightweight elements, in addition to rock.

In the third zone, the line is horizontal: the slope is zero. The size hardly changes at all, even when we increase the mass by a factor of 100. In everyday life, when we pack more mass onto a ball, the ball gets bigger. But apparently, this is not the case for balls between 100 and 10,000 Earth masses.

Our own Jupiter is in this zone, as are lots of exoplanets. What must be happening here is that the more massive objects are much denser than the less massive versions. Part of the reason that they’re increasingly dense is gravitational compression. These objects are so massive that their own gravity compresses them to higher densities than usual.

There’s no single name for all the objects in this zone. Sometimes we call them ‘Jovian planets’, although toward the higher mass end, the traditional term is ‘brown dwarfs’.

Finally, we have the highest mass objects, for which we get a slope of about one, which means that for these guys, radius is proportional to mass.

These are stars—objects for which the gravitational compression is so strong that nuclear fusion ignites at the center, creating lots of heat and pressure. This same nuclear fusion also produces the light that stars are famous for; it’s what makes stars shine.

Gaseous planets have a lot of hydrogen and helium gas, very lightweight elements, in addition to rock.

The more massive objects are much denser than the less massive versions. Part of the reason that they’re increasingly dense is gravitational compression. These objects are so massive that their own gravity compresses them to higher densities than usual. Thus, their size hardly changes at all, even when we increase the mass by a factor of 100.

Stars are objects for which the gravitational compression is so strong that nuclear fusion ignites at the center, creating lots of heat and pressure. This same nuclear fusion also produces the light that stars are famous for; it’s what makes stars shine.

The Voyager Golden Records

Saturn, Uranus, and Neptune: The Other Gas Giants

Searching for Exoplanets: Methods and Techniques

Astrophysics began in the 17th century with Isaac Newton, who explained the motion of the planets with his new equations relating force, mass, acceleration, and gravity. The actual word ‘astrophysics’ is more recent. It’s from the mid-19th century, after the invention of photography and spectroscopy. Those two techniques were crucial because they allowed us to go beyond looking through telescopes with our eyes. Now, we could make more objective records, detect fainter sources, and connect our observations to laboratory experiments with light, heat, and atoms.

On the other hand, astronomy is a cultural activity dating back thousands of years, which only gradually became scientific. The ancient Babylonians, the Chinese, and the Mayans were all accomplished astronomers, but they weren’t astrophysicists.

There are many reasons to love astrophysics, one of which is that no other science spans such a vast range of scales, from nanometers to billions of light years, from the radiation of a single electron to the output of trillions of suns.

If we make a scale model of the solar system, with everything shrunk down by a factor of almost a billion, the Sun would be a sphere as tall as a man, and the Earth would be like a grape, located 2.5 blocks away. This works well to put the solar system into perspective

But this approach fails when we try to put the whole universe into perspective. Even if we scale everything down by a factor of a billion, the very nearest star to the Sun would be 25 thousand miles away and our next-door neighboring galaxy would be 20 billion miles away.

This article comes directly from content in the video seriesIntroduction to Astrophysics.Watch it now, on Wondrium.

No matter how much we try to scale things down to a manageable size, we would still get mind-boggling numbers. The problem is there’s no one scale factor that can put all the different phenomena into a mentally comprehensible map.

All the natural scales are separated by factors of order a hundred, or a thousand, except for the parsec, the scale in between the stars, which is 206,265 times bigger than the AU (Astronomical Unit), the scale of planetary systems.

What this means is that stars and their planets are unusually isolated—the spacing between them, relative to their size, is larger than for other types of astronomical objects. This has the fascinating implication that collisions between galaxies are common, but collisions between stars are vanishingly rare. Even when 2 entire galaxies are colliding, there’s not much chance that any 2 stars inside those galaxies will hit each other.

There is a tactic astrophysicists use to cope with all these orders of magnitude. They make logarithmic maps. Taking the logarithm of a number means expressing the number as a power of 10, and then plucking out the exponent. For example, 1000 is 10^{3} power. So, the logarithm of 1000 is 3. The log of a million is 6. And this works for numbers smaller than 10, too. The number one is equal to 10 to the zeroth power, so the log of one is zero. One-tenth is 10^{–1} power, so the log of a 10th is negative one. And so on.

So, what’s a logarithmic map? An ordinary map is based on a single scale factor. One inch on the map might be one kilometer in real-life. But on a logarithmic map, the scale factor changes as we move from one end to the other. The first inch might correspond to one meter in real-life, but then the second inch is 10 meters, then 100 meters, a 1000, 10^{4}, 10^{5}, and so on. Mathematically, with every inch, we increase the logarithm of the scale factor by one unit.

There are logarithmic charts, besides maps. Astrophysicists use logarithmic time lines, and they make more abstract logarithmic charts to help understand things that range over many orders of magnitude.

For example, our galaxy is full of objects ranging widely in mass, and in size. Among other things, there are asteroids, moons, planets, and stars. Let’s say we go around the galaxy and measure the mass and radius of everything smaller than the Sun. To compare all these things, there is made a chart of mass versus radius.

On a regular chart, the horizontal axis is mass, in units of Earth masses, and the vertical axis is radius, in units of Earth radii. Each data point shows the mass and radius of a single object. There’s clearly a relationship between radius and mass; the more massive the object, the bigger the radius, which makes sense. But there’s a problem with this chart. Because we need to make the axes range high enough to encompass the very largest objects—millions of Earth masses—the more numerous smaller objects end up crammed in close to the origin, and we can’t make out the details.

If we remake this chart with logarithmic axes, then the horizontal axis still tells us the mass, but now each tick mark represents a factor of 10. So, the Earth is at one Earth mass, 10 to the zero. And the Sun is a few hundred thousand Earth masses, which is 10^{5.5} power, so the data point for the Sun is 5.5 ticks to the right. Likewise, the vertical axis still tells us the radius, but on a logarithmic scale.

Astrophysics is the application of the laws of physics to understand celestial phenomena. Meanwhile, astronomy is the careful observation of heavenly bodies which only gradually has become scientific.

Parsec, the scale in between the stars, is 206,265 times bigger than the AU (Astronomical Unit), the scale of planetary systems.

Unlike an ordinary map, which is based on a single scale factor, on a logarithmic map, the scale factor changes as we move from one end to the other. The first inch might correspond to one meter in real-life, but then the second inch is 10 meters, then 100 meters, a 1000, 10^{4}, 10^{5}, and so on.

Riccardo Giacconi: Pioneering X-ray Astronomy

The Golden Age of Astronomy

Some Heroes of Astronomy and Their Legacies

Let’s begin by zooming out. Let’s say we start with a map of the Wondrium headquarters in Chantilly, Virginia, where the scale bar represents 100 meters. If we expand our field of view by a factor of 10, the scale bar is then a 1000 meters, a kilometer, and we can take in the whole city.

Let’s zoom again, another factor of 10. Now, we start to see regional features. Taking another step, our scale bar is 100 kilometers, and we can see the entire mid-Atlantic seaboard. Another factor of 10, and we can see the entire Earth, hanging in empty space. At this point we’ve zoomed out from hundreds to millions of meters.

This brings up the issue of units: units of measurement. Sometimes, we can use standard metric units for lengths: millimeters, meters, kilometers, and so on. The scale bar we just discussed is a million meters long, or, one mega-meter. Another way to write that is with scientific notation, 10 to the 6 meters, since 10 to the 6th power is a million.

However, there are times we want to use units that are more appropriate for the scale we’re interested in. When we think about entire planets, meters are not so convenient. It’s better to measure things in units of, say, the radius of the Earth.

One Earth radius is defined as 6378 kilometers. That way we can say, the planet Neptune has a radius of about 4 Earth radii, and Jupiter’s is about 11. Four and 11 are much easier to comprehend than however many millions of meters. That’s why the Earth radius is a handy unit. We write it as R with a subscript, a little plus sign inside a circle—the astronomical symbol for Earth.

The next useful unit that we need is for the size of stars, rather than planets. So, let’s zoom out and zip inward to the center of the solar system, so we can take in the entire Sun. Its radius is about 700 million meters, or a little more than 100 times bigger than the Earth. The solar radius is our unit of choice when we discuss stars. We write it as R with another astronomical symbol—this time it’s a dot inside a circle.

This article comes directly from content in the video seriesIntroduction to Astrophysics.Watch it now, on Wondrium.

Let’s keep zooming out. Once we get to the scale of 10 to the 13 meters, most of the other planets come into view. We’ve reached the scale of the solar system.

On this scale, the scale of planetary systems, the traditional unit is the radius of Earth’s orbit around the Sun. In fact, that’s such a useful unit, it’s called the Astronomical Unit, or AU. It’s about 215 solar radii, or 150 billion meters. And with the AU, we can easily describe the solar system. Mercury is about 2/5 of an AU from the Sun, and Jupiter is out at 5.2 AU.

We’re still not done zooming out, though. When we expand our scale again, beyond the solar system, we find ourselves in empty space for quite a while, until we get to 10 the 16th meters. Then, at last, some of the neighboring stars come into view. A good unit to use on this scale is the light year—the distance light travels in one year, which is just short of 10^{16} meters. For example, the nearest star, Proxima Centauri, is 4.2 light years away.

But in practice, astrophysicists don’t use light years, except when giving public lectures. The preferred unit is called the parsec, and it’s about 3.3 light years. The typical distance between stars is a parsec or 2.

From here, we need to zoom out 4 more factors of 10—4 more orders of magnitude—until the architecture of the Milky Way galaxy comes into view, at around 10^{20} meters. At this stage, we stop inventing units with different names and just keep using parsecs, but with metric prefixes, like kilo, for a thousand. The diameter of a typical spiral galaxy is 10 or 20 kiloparsecs.

It takes a couple more orders of magnitude to start seeing neighboring galaxies. The typical spacing between galaxies is a few megaparsecs, millions of parsecs. After another step, we see that the galaxies themselves group together to form clusters of galaxies, joined by what look like filaments, or webs of galaxies. And when we keep going, when we keep increasing our scale bar all the way to 10^{26} meters, the universe starts to look like random static, nowhere different from anywhere else.

The natural scale here is the gigaparsec, billions of parsecs. That’s the end of the line. The largest spatial scales about which we have any direct knowledge. By zooming out 26 orders of magnitude, we have a view of the entire observable universe.

Astronomical Unit, or AU, is the radius of the Earth’s orbit around the Sun. With the AU, we can easily describe the solar system. Mercury is about 2/5 of an AU from the Sun, and Jupiter is out at 5.2 AU.

Light year is the distance light travels in one year, which is just short of 10^{16} meters. For example, the nearest star, Proxima Centauri, is 4.2 light years away.

Parsec is the preferred unit of measurement by astrophysicists. It’s about 3.3 light years. The typical distance between stars is a parsec or 2.

The Discoveries and Images from the Hubble Space Telescope

The Challenges of Building, Launching, and Operating a Space Telescope

Margaret Burbidge: Breaking Barriers in Astrophysics