Yesterday’s post about Earth’s changing mass raised similar questions about the Sun. In the Sun’s case we know that it’s losing mass, and at a rate fast enough that it’s forced us to change the way we measure astronomical distances.

The Sun loses mass in two major ways. The first is through solar wind. The surface of the Sun is hot enough that electrons and protons boil off its surface and stream away from the Sun, generating a “wind” of ionized particles. When those particles strike Earth’s upper atmosphere they can produce aurora. The solar wind varies a bit in intensity, but from satellite observations we know that the Sun loses about 1.5 million tonnes of material each second due to solar wind.

The second way the Sun loses mass is through nuclear fusion. The Sun fuses hydrogen into helium in its core, producing its life-giving glow over billions of years. The production of helium transforms some of the hydrogen’s mass into energy, which radiates away from the Sun in the form of light and neutrinos. By observing just how much energy the Sun radiates, and using Einstein’s equation relating mass and energy, we find the Sun loses about 4 million tonnes of mass each second due to fusion.

So the Sun loses about 5.5 million tonnes of mass every second, or about 174 trillion tonnes of mass every year. That’s a lot of mass, but compared to the total mass of the Sun it’s negligible. The Sun will keep shining for another 5 billion years, and by that time it will have lost only about 0.034% of its current mass.

While the amount of mass loss is negligible, it isn’t zero, and it has an effect on Earth’s orbit. As the Sun loses mass its gravitational pull on the Earth weakens over time. As a result, Earth is receding slightly from the Sun. Because of solar mass loss the Earth’s distance from the Sun increases by about 1.6 centimeters per year. In astronomy, one of the ways we measure distance is through the astronomical unit, which has traditionally been defined as the distance from the Sun to the Earth. For most of astronomical history the changing distance of Earth was too small to consider, and so the astronomical unit could be considered a constant. But over time our measurement of this distance has become astoundingly accurate, and currently has a precision of about 3 parts per billion. This is accurate enough to observe the gradual increase in distance. So in 2012 the astronomical unit was defined as fixed constant. As a result the Earth is slightly more than 1 astronomical unit away from the Sun.

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We generally think of the Earth as having a constant mass. On a basic level that’s true, but the Earth’s mass does change very slightly. So is it’s mass increasing or decreasing?

Earth gains mass through dust and meteorites that are captured by its gravity. If you watched the recent meteor shower you know this can occur on a regular basis. In fact from satellite observations of meteor trails it’s estimated that about 100 – 300 metric tons (tonnes) of material strikes Earth every day. That adds up to about 30,000 to 100,000 tonnes per year. That might seem like a lot, but over a million years that would only amount to less than a billionth of a percent of Earth’s total mass.

Earth loses mass through a couple of processes. One is the fact that material in Earth’s crust undergoes radioactive decay, and therefore energy and some subatomic particles can escape our world. Another is the loss of hydrogen and helium from our atmosphere. The first process only amounts to about 15 tonnes per year, but the loss from our atmosphere amounts to about 95,000 tonnes per year.

So it’s most likely that Earth is losing a bit of mass each year, but if the rate of meteors is on the higher end of estimates, then it could be gaining a bit of mass.

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How do you weigh a galaxy?

With planets we can measure their distance from the Sun and their orbital speed. By observing their motion in detail we can calculate their mass very precisely. For binary stars we can use a similar method. Observe the size of their mutual orbits and their orbital period, and by Kepler’s laws you can determine their mass. We can use the same method to calculate the mass of the supermassive black hole in the center of our galaxy.

But all of these methods rely on observing the change of an object’s speed over time (its acceleration) either directly or indirectly. For galaxies you can’t really do that. Galaxies aren’t a single object moving in a simple way, but rather a complex system of stars, gas and dust all moving and interacting.

One way that we can calculate the mass of a galaxy is to observe the motion of particular stars in the galaxy. Their accelerations are too small to observe, but by looking at how the speeds of stars closer to the center compare to speeds further from the center we can get an idea of a galaxy’s mass. But because much of a galaxy’s mass is due to dark matter, it is difficult to determine the total mass of a galaxy. We can infer the distribution of dark matter in regions where there are stars, but how do we determine how far beyond the stars the dark matter extends (known as the dark matter halo).

In a recent paper in the Astrophysical Journal, a team looked at a computational approach to determining the mass of galaxies, in particular the masses of the Milky Way and the Andromeda galaxy. Their method was to look at the gravitational attraction between the two galaxies.

While the Milky Way and Andromeda are gravitationally attracting each other, that attraction isn’t very large. Sure, they both have billions of stars worth of mass, but they are also more than 2 million light years apart, and gravitational attraction is weaker at larger distances. So there isn’t a way to measure the acceleration due to gravity. We do, however, have a measure of the speeds of the galaxies relative to each other, and it is this data that the team analyzed.

Credit: Phelps, et al.

Basically, the team ran a computer simulation of the two galaxies, along with other members of the local group, using a method known as the numerical action method. This method assumes a mass for both galaxies, then calculates their velocities due to gravity following the principle of least action (which makes the velocities easier to calculate). By calculating the galactic velocities for a range of masses, you can compare the result with the actual observed motion. The better the statistical match, the more likely your assumed masses are the actual masses.

You can see the computational results in the figure above. Each of the four images starts with different data from the local group, and the more blue regions are closer matches to observation. By combining these different results, the best match is a mass of 2.5 trillion solar masses for the Milky Way, and 3.5 trillion solar masses for Andromeda (give or take a trillion solar masses with 95% confidence).

So it turns out we can weigh a galaxy, it just takes some computational physics to do it.

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Photons are massless. We know they are massless because particles with mass can’t move at the speed of light. We know that special relativity works, and the speed of light is the same in all frames of reference, and special relativity only works if photons are massless. Except…

What if photons had a tiny mass. If it was small enough, then light could travel at almost the speed of light, and special relativity would almost be true. How confident are we that the photon doesn’t have mass? After all, we once thought neutrinos didn’t have mass and we now know they do have mass.

Just to be clear, we know a lot more about light than we do about neutrinos. Neutrinos are notoriously difficult to detect, while photons are quite easy to observe (you’re probably doing it now!) We’ve conducted tests on light in lots of different ways, and in every experiment the “massless photon” model works. We have no reason to assume that the mass of a photon is anything other than zero.

Still, it is worth testing that assumption. We know that if photons have mass it would have to be very small. There are theoretical models that allow for photon mass. For very small masses these models look very similar to the ones we use (such as Maxwell’s equations and the constant speed of light), but at larger masses they predict effects we would have seen by now.

One of these effects would be seen in the cosmic microwave background. As outlined in a recent paper in Physical Review Letters, the cosmic background wouldn’t match a blackbody curve if the photon had mass. (I’ve talked about the blackbody curve before, and you can see the curve in the figure below.)

The energy of photons depends upon its color. Photons at the blue end of the spectrum have more energy than photons at the red end of the spectrum. If photons have mass, then they don’t all move at the constant “speed of light”, but instead the speed of light would depend on the wavelength of light. Large wavelength (reddish) light has less energy than short wavelength (bluish) light. This means instead of being a perfect blackbody, the cosmic microwave background would be brighter than expected at long wavelengths and dimmer than expected at short wavelengths. The larger the photon mass, the bigger this effect would be.

The cosmic background matches a blackbody so perfectly that the photon mass can be no larger than a hundredth the mass of an electron. That’s pretty tiny, but other optical experiments require that the photon have a mass no larger than a trillionth of a trillionth of the mass of an electron. That’s a septillionth of an electron mass. So if the cosmic background puts less of a limit on photon mass than other experiments, what’s the big deal?

If the photon has mass, even a septillionth of an electron mass, it is possible that the lightest neutrino has an even smaller mass. We know that neutrinos have mass. There are actually three types of neutrinos, and we know the sum of their masses can be no larger than 2 millionths of an electron mass. But we don’t know what their masses actually are. If the lightest neutrino has a mass even smaller than the photon, then it would be possible for photons to decay into neutrinos.

That would mean photons wouldn’t last forever. Instead they would have a half life. This would have huge cosmological consequences, because it would mean that over billions of light years some of the light would decay into neutrinos. The distant regions of the universe would appear dimmer than they should, and since the brightness of things like supernovae are used to measure distance, the universe would appear larger than it actually is.

But the photons wouldn’t decay equally. Because bluish photons would have a greater speed than reddish photons, the time dilation effect of special relativity would mean that bluish photons would last longer than reddish photons. And that means the light from the cosmic microwave background would again be distorted

You can see this in the image above, where the observed blackbody spectrum (the dots) is plotted against different half-lifes for the photon. (Just as a side note, those error bars are actually 1000 times larger than the actual ones, which would be too small to see on the graph). The lines are plotted for different half-life values.

Based on observation of the cosmic microwave background, the half-life of a photon must be at least three years. That doesn’t seem like a long time, but light travels so fast that time dilation would make them last much longer. In the visible spectrum their effective half-life would be about a quintillion years, or about a billion times longer than the age of the universe.

In other words, we can experimentally demonstrate that if photons have mass, then it is so extremely tiny that it is effectively massless, and if they can decay into neutrinos their average lifetime is so incredibly long that it is effectively forever. So the assumption that the photon is massless and never decays is a good one.

Again, just to be clear, there is absolutely no evidence that photons are anything other than massless, and there are important theoretical reasons to presume photons are massless. But experiments like these are good because they test the limits of our assumptions, and help keep us honest.

Because if we stopped testing our ideas, that really would be a dying of the light.

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There’s news on the web that cosmologists have proven the existence of negative mass. The news is based upon an article that recently appeared on the preprint arxiv, and has not yet been peer reviewed. The article in no way proves the existence of negative mass, but rather demonstrates the theoretical possibility of a form of negative mass within general relativity. In other words, it is an interesting “what if” paper rather than applied astrophysics.

Usually when we talk about the mass of an object we think of it as a basic property of an object. Mass is always positive in quantity, so negative mass must be the same thing but opposite. But in fact there are three types of mass than an object can have. Theres’s inertial mass, which determines how easy or difficult it is to move an object; passive gravitational mass, which interacts with a local gravitational field (and determines the weight of an object), and the active gravitational mass, which creates the gravitational field.  In this paper the focus is on negative active mass, so that it creates a repulsive gravitational field.

Various forms of negative mass are sometimes referred to as exotic matter, and they are often invoked to create science fiction things like warp drive and wormholes. Of course this leads to effects such as time travel and violations of relativity and causality.  For this reason (and the fact that we’ve never observed anything with negative mass), this kind of repulsive gravity matter is considered impossible.

In general relativity we often describe what is possible for matter by what are known as energy conditions. One of these is known as the dominant energy condition, which basically requires that matter doesn’t move faster than light, which is a big no-no in relativity. What the authors of this new paper have shown is that negative matter can be described in general relativity without violating the dominant energy condition. From this, they’ve found a solution within general relativity that looks like a negative mass object within an inflating universe. It is basically a “toy model” within general relativity.

This doesn’t in any way prove (or even suggest) that negative mass exists. It is interesting, though, because it models an inflating universe such as might have existed in the earliest moments after the big bang. It even makes a prediction, though not a very satisfying one. If the early universe were filled with a sea of both mass and negative mass, then gravitational waves could be damped. If the BICEP2 results are found to be false, and Planck also fails to detect primordial gravitational waves, then negative mass could explain how inflation could exist even though we don’t see primordial gravitational waves.

Of course, using the lack of evidence for gravity waves as evidence for inflation and negative mass is hardly scientific. But again, this is a “what if” paper, pushing the limits of theory to see what useful ideas might come out of it.

Paper: Saoussen Mbarek and M. B. Paranjape. Negative mass bubbles in de Sitter space-time. arXiv:1407.1457 [gr-qc] (2014).

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If you’ve ever stepped on a scale to measure your weight, you’ve done a basic physics experiment.  The number on the scale represents the force of the Earth’s gravity on your mass.  The greater your mass, the greater your weight.  But this number also represents the weight of the Earth in your gravitational field.  Because (as per Newton’s third law) the force of the Earth on you and the force of you on the Earth must be the same, stepping on the scale determines how strongly you and the Earth are attracted to each other.  This give and take between masses is also important in astrophysics, such as the determination of stellar masses.

When we look at a solitary star, it is a bit difficult to determine its mass.  We can infer its mass from things such as temperature and brightness, but how do we know such an inference is correct?  It turns out we can determine the masses of some stars through the fact that they gravitationally attract each other.

The first known binary stars were observed in the 1800s, but it wasn’t until the 1900s with the introduction of the filar micrometer that decently accurate measurements could be made.  This device allowed you to center your telescope on the primary star, and then measure the position of the secondary star relative to the primary.  By taking measurements over time (sometimes years or decades) you can start to see the companion star trace its path.  In the figure below, I’ve plotted observations of the star Xi Ursae Majoris.

The good news is that the motion of a binary companion is an ellipse, so with a bit of mathematics you can fit your observation to an elliptical path.  The bad news is that the positions we observe are within the plane of our field of view.  The actual orbit is usually tilted relative to us, so we have to do more math to determine its actual orbit.  In times past this had to be done by hand, and was really laborious.  With modern computers we can fit the data and calculate orbits pretty easily.

Of course if you know the orbit of the companion star you can also calculate their masses.  Knowing their separation distance and orbital period you can calculate their masses via Kepler’s laws.  Once we’ve determined their masses, we can look at other properties such as their brightness and temperature to see how they correlate to mass.

In modern times we have a much better handle on stellar evolution, so there are ways of determining stellar mass that don’t require a comparison to binary stars.  But the basic gravitational property of give and take seen in binary stars provided a much needed step toward that level of understanding.

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We all know that many objects (atoms, cats, us) have mass.  What you probably don’t know is that there are multiple different types of mass, and this has real physical (and astrophysical) consequences.

The most familiar type of mass is probably the “quantity” version.  That is, an object like a car has a certain amount of “stuff” (metal, plastic, glass), and that quantity of matter can be measured by mass.  Of course, some things are small and dense, while others are large and light.   So with quantity mass it’s not just the size that matters, but the density.

In our daily lives, we usually determine mass by an object’s weight.  Heavier things have more mass.  The weight of an object depends on how it interacts with a gravitational field, such as the Earth’s gravitational field.  This interaction is due to a type of mass known as passive gravitational mass.  The more passive gravitational mass, the heavier an object will be in the Earth’s gravitational field.

Of course, a passive gravitational mass can interact with the Earth because the Earth has a gravitational field.  This field is produced by the active gravitational mass of Earth.  The active and passive gravitational masses are what allow gravitational interaction (at least in classical physics).

Another type of mass determines how easy or difficult it is to move an object.  This is known as inertial mass.  The inertial mass is the m in Newton’s second law (F = ma), and is why a baseball is easier to throw than a bowling ball.

So all massive objects have a certain quantity of matter, a passive gravitational mass that interacts with gravitational fields, an active gravitational mass that produces a gravitational field, and an inertial mass that determines how the object will move when forces act on it.  In Newtonian physics, all these different types of masses are the same.  When we use the term mass, we generally mean the Newtonian version, which is why we don’t distinguish between them.

But this simple, Newtonian view of mass can lead to some confusion when it comes to things like special relativity.  Special relativity is derived from the fact that the speed of light is a constant.  This means that if I’m travelling at 90% of the speed of light relative to you, and shine a flashlight ahead of me, I will see the light move from me at the speed of light, not 10% of the speed of light.  This strange behavior leads to things like time dilation.  From your vantage point my time appears to move more slowly.  But it also means that my mass appears to get larger.

This is sometimes referred to as relativistic mass, and is a very real effect.  For example, in the Large Hadron Collider protons are accelerated to *nearly* the speed of light.  They are moving so fast that their relativistic mass is much larger than their regular Newtonian mass.  This means we have to push them harder to keep them moving in a circular path. So as the protons are sped up, the magnetic fields used to keep them moving in a circle have to be strengthened.  The closer the protons get to the speed of light, the bigger their relativistic mass, and the harder they are to move.  This is also, by the way, why they can’t be accelerated to the speed of light.

In physics we tend to avoid the term “relativistic mass”, because it isn’t really the same as the other masses.  Relativistic mass is an object’s apparent inertial mass from your vantage point.  So if you see a proton zip past you at a large fraction of the speed of light, the proton would appear to have a large inertial mass.  But if I’m zipping along with the proton, I would say its mass is a normal proton mass.  Relativistic mass is dependent on who’s doing the observing, while the other masses are an inherent property of the object.

In Newtonian physics, the equivalence of the inertial and gravitational masses is why everything falls at the same rate.  Even though a baseball and a cannonball have different masses, they fall at the same speed (barring air resistance).  Bigger masses are harder to move, but they also feel a stronger gravitational force.  This has been known since the time of Galileo, but it was used by Einstein as the foundation of general relativity.

In order to formulate general relativity in terms of general covariance, Einstein later strengthened this argument to yield what is known as the strong equivalence principle:  The ratio between the inertial mass of a particle and its gravitational mass is a universal constant.

Einstein saw a parallel between the relative nature of motion in his theory of special relativity and the relative nature of gravity, and so he worked to generalize relativity to include both gravity and motion. This theory of general relativity is what he published in 1915. It was a radical proposal. In his theory Einstein argued that gravity was not a force in the way Newton had thought. Instead, gravity was an effect of a curvature of space and time.

In general relativity, the active gravitational mass of an object curves space around it.  But this leads to another type of mass, what you might call “curvature mass”.  As an example, consider the mass of a black hole.  A black hole is so dense that any matter it once had is now trapped forever behind its event horizon.  We can’t observe that matter, but we can determine the mass of a black hole by measuring the curvature of space around it.  So a black hole has mass even though it isn’t made of “stuff”.

In our daily lives we can treat mass as a single property of an object.  But in astrophysics that simple view can lead to massive problems.

Cartoon of (likely mythical) Galileo dropping objects from the tower of Pisa. Credit: San Diego State University

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Yesterday I mentioned how we could use the motion of hydrogen gas in our galaxy to determine the distribution of mass in our galaxy.  Since that calculation on is based on gravity, what we really measure is the total amount of gravitational mass there is in our galaxy.  When we do this we find something very strange: our galaxy (and other galaxies) have more mass than they should.

There are other ways we can measure the amount of matter in our galaxy.  In particular we can measure the amount of stars, gas and dust we can see, which gives us a good estimate.  However when we do that we find the mass we measure gravitationally is much higher than the directly observed mass.  If we subtract the observed mass from the total gravitational mass, what we have is a different kind of mass.  A dark, unseen kind known as dark matter.  In the figure below I’ve plotted a comparison of the the amount of regular visible matter vs the amount of dark matter in our galaxy.

We’re not yet sure what dark matter is, though we have some ideas.  What we do know is that it has gravitational mass and it is not any kind of regular matter such as gas, dust, planets or stars.  Dark matter is something different, and there is about four times more dark matter than regular matter in the universe.  What dark matter actually is remains to be seen.

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I’ve been preparing for an intro physics class Monday, and that means covering Newton’s laws of motion. Since it is an introductory class I don’t discuss the nature of mass too deeply. Essentially I tell my students that there is inertial mass, given by the second law of motion, and a gravitational mass given by Newton’s law of gravity. I then go on to say that since everything near the Earth falls at the same rate, the gravitational mass in the law of gravity must be proportional to the inertial mass in Newton’s second law. That is, the two masses are equivalent, which is the heart of the equivalence principle.

But things are never quite as simple as they seem, and the concept of mass is no exception. In Newtonian physics there are not two types of mass, but three. There is the inertial mass, which determines the acceleration due to an applied force; there is the passive gravitational mass, which interacts with the local external gravitational field; then there is the active gravitational mass, which creates the external gravitational field in which other particles interact. Newton assumed that all three types of mass were one and the same, and it is generally assumed that Newton’s was correct, but nothing in general relativity requires it, and there is (as yet) no experimental evidence to validate it.

Acceleration “looks” like gravity.

When Einstein first proposed the principle of equivalence as a foundation to general relativity, his basic argument was that, without some external point of reference, a free-floating observer far from gravitational sources and a free-falling observer in the gravitational field of a massive body each have the same experience. Likewise an observer standing on the surface of a massive body and an observer which uniformly accelerates at a rate equal to the body’s surface gravity have identical experiences. Thus, the free-float and free-fall frames can be considered equivalent. In the same manner, the uniform acceleration frame and the surface frame are equivalent. This is known as the weak equivalence principle:

All effects of a uniform gravitational field are identical to the effects of a uniform acceleration of the coordinate system.

In order to formulate general relativity in terms of general covariance, Einstein later strengthened this argument to yield what is known as the strong equivalence principle:

The ratio between the inertial mass of a particle and its gravitational mass is a universal constant.

It is this latter principle which was experimentally validated by the classic Eötvös experiment, which determined that objects fall at the same rate regardless of their material consistency.

The strong equivalence principle does not require that all masses are equal. It only requires that an object’s inertial and passive masses are proportional. Although the equivalence principle says nothing about active mass, conservation of momentum does. If you apply conservation of momentum to two gravitationally interacting objects, you find that momentum is only conserved is if the active mass of an object is proportional to its inertial and passive masses. Thus in order to relate all three masses, we need not only the equivalence principle, but also the conservation of energy-momentum.

Matter vs. antimatter. Source LBNL.

The constants of proportionality can be wrapped into the gravitational constant, so it would seem we can simply follow Newton, set all three types of mass equal to each other and be done with it. There is, however, a catch. Although we can arbitrarily set the magnitudes of active and passive mass equal to each other, it is possible for them to be opposite in sign. In other words, if there was some weird type of matter that gravitationally repelled other masses, the equivalence principle and conservation of momentum would still hold true. The equivalence principle has been tested between regular matter, which requires all three masses to be the same. Since ordinary matter is mutually attractive we can say that Newton’s assertion is correct for matter.

But what about anti-matter? No one has been able to test this assumption, so we can’t say for certain. It is possible that active mass is negative for antimatter, which would mean it falls upward in a gravitational field. If that is the case, then although general relativity would still apply to regular matter, it wouldn’t apply to matter + antimatter. Since general relativity is a powerful and experimentally validated theory, it is generally assumed that Newton’s assertion would hold for anti-matter as well. But the only way to know for sure is to test it.

Recently we’ve been able to create usable quantities of anti-hydrogen, which will finally give us the chance to put antimatter to the test. It’s generally thought that antimatter will fall downward just like regular matter, but if it doesn’t, it will be time for some new ideas for gravity.

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