The Physics of Starflight: Proxima Centauri b or Bust!

The ability to travel to the stars has been one of mankind’s deepest desires. Ever since we learned that we are just one world in a vast universe of limitless worlds, we have yearned to visit some of those others. Yet nature has thrown up an almost insurmountable barrier to that desire–the speed of light. Only by traveling at or near the speed of light may we venture to far-off worlds, and even then, decades or centuries will pass during the voyage. The vast distances of space keep all the worlds isolated–possibly for the better.

Yet the closest worlds are not so far away that they will always remain out of reach. The very limit of the speed of light provides ways of getting there within human lifetimes. The non-intuitive effects of special relativity come to our rescue, and we may yet travel to the closest exoplanet we know of.

Proxima Centauri b

The closest habitable Earth-like exoplanet is Proxima Centauri b, orbiting the red dwarf star Proxima Centauri that is about 4.2 lightyears away from Earth. The planet has a short orbital period of only about 11 Earth days, but the dimness of the red dwarf puts the planet in what may be a habitable zone where water is in liquid form. Its official discovery date was August 24, 2016 by the European Southern Observatory in the Atacama Desert of Chile using the Doppler method. The Alpha Centauri system is a three-star system, and even before the discovery of the planet, this nearest star system to Earth was the inspiration for the Hugo-Award winning sci-fi trilogy The Three Body Problem by Chinese author Liu Cixin, originally published in 2008.

It may seem like a coincidence that the closest Earth-like planet to Earth is in the closest star system to Earth, but it says something about how common such exoplanets may be in our galaxy.

Artist’s rendition of Proxima Centauri b. From WikiCommons.

Breakthrough Starshot

There are already plans to send centimeter-sized spacecraft to Alpha Centauri. One such project that has received a lot of press is Breakthrough Starshot, a project of the Breakthrough Initiatives. Breakthrough Starshot would send around 1000 centimeter-sized camera-carrying laser-fitted spacecraft with 5-meter-diameter solar sails propelled by a large array of high-power lasers. The reason there are so many of these tine spacecraft is because of the collisions that are expected to take place with interstellar dust during the voyage. It is possible that only a few dozen of the craft will finally make it to Alpha Centauri intact.

Relative locations of the stars of the Alpha Centauri system. From ScienceNews.

As these spacecraft fly by the Alpha Centauri system, possibly within one hundred million miles of Proxima Centauri b, their tiny HR digital cameras will take pictures of the planet’s surface with enough resolution to see surface features. The on-board lasers will then transmit the pictures back to Earth. The travel time to the planet is expected to be 20 or 30 years, plus the four years for the laser information to make it back to Earth. Therefore, it would take a quarter century after launch to find out if Proxima Centauri b is habitable or not. The biggest question is whether it has an atmosphere. The red dwarf it orbits sends out catastrophic electromagnetic bursts that could strip the planet of its atmosphere thus preventing any chance for life to evolve or even to be sustained there if introduced.

There are multiple projects under consideration for travel to the Alpha Centauri systems. Even NASA has a tentative mission plan called the 2069 Mission (100 year anniversary of the Moon landing). This would entail a single spacecraft with a much larger solar sail than the small starshot units. Some of the mission plans proposed star-drive technology, such as nuclear propulsion systems, rather than light sails. Some of these designs could sustain a 1-g acceleration throughout the entire mission. It is intriguing to do the math on what such a mission could look like, in terms of travel time. Could we get an unmanned probe to Alpha Centauri in a matter of years? Let’s find out.

Special Relativity of Acceleration

The most surprising aspect of deriving the properties of relativistic acceleration using special relativity is that it works at all. We were all taught as young physicists that special relativity deals with inertial frames in constant motion. So the idea of frames that are accelerating might first seem to be outside the scope of special relativity. But one of Einstein’s key insights, as he sought to extend special relativity towards a more general theory, was that one can define a series of instantaneously inertial co-moving frames relative to an accelerating body. In other words, at any instant in time, the accelerating frame has an inertial co-moving frame. Once this is defined, one can construct invariants, just as in usual special relativity. And these invariants unlock the full mathematical structure of accelerating objects within the scope of special relativity.

For instance, the four-velocity and the four-acceleration in a co-moving frame for an object accelerating at g are given by

The object is momentarily stationary in the co-moving frame, which is why the four-velocity has only the zeroth component, and the four-acceleration has simply g for its first component.

Armed with these four-vectors, one constructs the invariants

and

This last equation is solved for the specific co-moving frame as

But the invariant is more general, allowing the expression

which yields

From these, putting them all together, one obtains the general differential equations for the change in velocity as a set of coupled equations

The solution to these equations is

where the unprimed frame is the lab frame (or Earth frame), and the primed frame is the frame of the accelerating object, for instance a starship heading towards Alpha Centauri. These equations allow one to calculate distances, times and speeds as seen in the Earth frame as well as the distances, times and speeds as seen in the starship frame. If the starship is accelerating at some acceleration g’ other than g, then the results are obtained simply by replacing g by g’ in the equations.

Relativistic Flight

It turns out that the acceleration due to gravity on our home planet provides a very convenient (but purely coincidental) correspondence

With a similarly convenient expression

These considerably simplify the math for a starship accelerating at g.

Let’s now consider a starship accelerating by g for the first half of the flight to Alpha Centauri, turning around and decelerating at g for the second half of the flight, so that the starship comes to a stop at its destination. The equations for the times to the half-way point are

This means at the midpoint that 1.83 years have elapsed on the starship, and about 3 years have elapsed on Earth. The total time to get to Alpha Centauri (and come to a stop) is then simply

It is interesting to look at the speed at the midpoint. This is obtained by

which is solved to give

This amazing result shows that the starship is traveling at 95% of the speed of light at the midpoint when accelerating at the modest value of g for about 3 years. Of course, the engineering challenges for providing such an acceleration for such a long time are currently prohibitive … but who knows? There is a lot of time ahead of us for technology to advance to such a point in the next century or so.

Figure. Time lapsed inside the spacecraft and on Earth for the probe to reach Alpha Centauri as a function of the acceleration of the craft. At 10 g’s, the time elapsed on Earth is a little less than 5 years. However, the signal sent back will take an additional 4.37 years to arrive for a total time of about 9 years.

Matlab alphacentaur.m

% alphacentaur.m
clear
format compact

g0 = 1;
L = 4.37;

for loop = 1:100
    
    g = 0.1*loop*g0;
    
    taup = (1/g)*acosh(g*L/2 + 1);
    tearth = (1/g)*sinh(g*taup);
    
    tauspacecraft(loop) = 2*taup;
    tlab(loop) = 2*tearth;
    
    acc(loop) = g;
    
end

figure(1)
loglog(acc,tauspacecraft,acc,tlab,'LineWidth',2)
legend('Space Craft','Earth Frame','FontSize',18)
xlabel('Acceleration (g)','FontSize',18)
ylabel('Time (years)','FontSize',18)
dum = set(gcf,'Color','White');
H = gca;
H.LineWidth = 2;
H.FontSize = 18;

To Centauri and Beyond

Once we get unmanned probes to Alpha Centauri, it opens the door to star systems beyond. The next closest are Barnards star at 6 Ly away, Luhman 16 at 6.5 Ly, Wise at 7.4 Ly, and Wolf 359 at 7.9 Ly. Several of these are known to have orbiting exoplanets. Ross 128 at 11 Ly and Lyuten at 12.2 Ly have known earth-like planets. There are about 40 known earth-like planets within 40 lightyears from Earth, and likely there are more we haven’t found yet. It is almost inconceivable that none of these would have some kind of life. Finding life beyond our solar system would be a monumental milestone in the history of science. Perhaps that day will come within this century.


Further Reading

R. A. Mould, Basic Relativity. Springer (1994)

D. D. Nolte, Introduction to Modern Dynamics : Chaos, Networks, Space and Time, 2nd ed.: Oxford University Press (2019)

Life in a Solar System with a Super-sized Jupiter

There are many known super-Jupiters that orbit their stars—they are detected through a slight Doppler wobble they induce on their stars [1].  But what would become of a rocky planet also orbiting those stars as they feel the tug of both the star and the super planet?

This is not of immediate concern for us, because our solar system has had its current configuration of planets for over 4 billion years.  But there can be wandering interstellar planets or brown dwarfs that could visit our solar system, like Oumuamua did in 2017, but much bigger and able to scramble the planetary orbits. Such hypothesized astronomical objects have been given the name “Nemesis“, and it warrants thought on what living in an altered solar system might be like.

What would happen to Earth if Jupiter were 50 times bigger? Could we survive?

The Three-Body Problem

The Sun-Earth-Jupiter configuration is a three-body problem that has a long and interesting history, playing a key role in several aspects of modern dynamics [2].  There is no general analytical solution to the three-body problem.  To find the behavior of three mutually interacting bodies requires numerical solution.  However, there are subsets of the three-body problem that do yield to partial analytical approaches.  One of these is called the restricted three-body problem [3].  It consists of two massive bodies plus a third (nearly) massless body that all move in a plane.  This restricted problem was first tackled by Euler and later by Poincaré, who discovered the existence of chaos in its solutions.

The geometry of the restricted three-body problem is shown in Fig. 1. In this problem, take mass m1 = mS to be the Sun’s mass, m2 = mJ to be Jupiter’s mass, and the third (small) mass is the Earth. 

Fig. 1  The restricted 3-body problem in the plane.  The third mass is negligible relative to the first two masses that obey 2-body dynamics.

The equation of motion for the Earth is

where

and the parameter ξ characterizes the strength of the perturbation of the Earth’s orbit around the Sun.  The parameters for the Jupiter-Sun system are

with

for the 11.86 year journey of Jupiter around the Sun.  Eq. (1) is a four-dimensional non-autonomous flow

The solutions of an Earth orbit are shown in Fig.2.  The natural Earth-Sun-Jupiter system has a mass ratio mJ/mS = 0.001 for Jupiter relative to the Sun mass.  Even in this case, Jupiter causes perturbations of the Earth’s orbit by about one percent.  If the mass of Jupiter increases, the perturbations would grow larger until around ξ= 0.06 when the perturbations become severe and the orbit grows unstable.  The Earth gains energy from the momentum of the Sun-Jupiter system and can reach escape velocity.  The simulation for a mass ratio of 0.07 shows the Earth ejected from the Solar System.

Fig.2  Orbit of Earth as a function of the size of a Jupiter-like planet.  The natural system has a Jupiter-Earth mass ratio of 0.03.  As the size of Jupiter increases, the Earth orbit becomes unstable and can acquire escape velocity to escape from the Solar System. From body3.m. (Reprinted from Ref. [4])

The chances for ejection depends on initial conditions for these simulations, but generally the danger becomes severe when Jupiter is about 50 times larger than it currently is. Otherwise the Earth remains safe from ejection. However, if the Earth is to keep its climate intact, then Jupiter should not be any larger than about 5 times its current size. At the other extreme, for a planet 70 times larger than Jupiter, the Earth may not get ejected at once, but it can take a wild ride through the solar system. A simulation for a 70x Jupiter is shown in Fig. 3. In this case, the Earth is captured for a while as a “moon” of Jupiter in a very tight orbit around the super planet as it orbits the sun before it is set free again to orbit the sun in highly elliptical orbits. Because of the premise of the restricted three-body problem, the Earth has no effect on the orbit of Jupiter.

Fig. 3 Orbit of Earth for TJ = 11.86 years and ξ = 0.069. The radius of Jupiter is RJ = 5.2. Earth is “captured” for a while by Jupiter into a very tight orbit.

Resonance

If Nemesis were to swing by and scramble the solar system, then Jupiter might move closer to the Earth. More ominously, the period of Jupiter’s orbit could come into resonance with the Earth’s period. This occurs when the ratio of orbital periods is a ratio of small integers. Resonance can amplify small perturbations, so perhaps Jupiter would become a danger to Earth. However, the forces exerted by Jupiter on the Earth changes the Earth’s orbit and hence its period, preventing strict resonance to occur, and the Earth is not ejected from the solar system even for initial rational periods or larger planet mass. This is related to the famous KAM theory of resonances by Kolmogorov, Arnold and Moser that tends to protect the Earth from the chaos of the solar system. More often than not in these scenarios, the Earth is either captured by the super Jupiter, or it is thrown into a large orbit that is still bound to the sun. Some examples are given in the following figures.

Fig. 4 Orbit of Earth for an initial 8:1 resonance of TJ = 8 years and ξ = 0.073. The Radius of Jupiter is R = 4. Jupiter perturbs the Earth’s orbit so strongly that the 8:1 resonance is quickly removed.
Fig. 5 Earth orbit for TJ = 12 years and ξ = 0.071. The Earth is thrown into a nearly circular orbit beyond the orbit of Saturn.

Fig. 6 Earth Orbit for TJ = 4 years and ξ = 0.0615. Earth is thrown into an orbit of high ellipticity out to the orbit of Neptune.

Life on a planet in a solar system with two large bodies has been envisioned in dramatic detail in the science fiction novel “Three-Body Problem” by Liu Cixin about the Trisolarians of the closest known exoplanet to Earth–Proxima Centauri b.

Matlab Code: body3.m

function body3

clear

chsi0 = 1/1000;     % Earth-moon ratio = 1/317
wj0 = 2*pi/11.86;

wj = 2*pi/8;
chsi = 73*chsi0;    % (11.86,60) (11.86,67.5) (11.86,69) (11.86,70) (4,60) (4,61.5) (8,73) (12,71) 

rj = 5.203*(wj0/wj)^0.6666

rsun = chsi*rj/(1+chsi);
rjup = (1/chsi)*rj/(1+1/chsi);

r0 = 1-rsun;
y0 = [r0 0 0 2*pi/sqrt(r0)];

tspan = [0 300];
options = odeset('RelTol',1e-5,'AbsTol',1e-6);
[t,y] = ode45(@f5,tspan,y0,options);

figure(1)
plot(t,y(:,1),t,y(:,3))

figure(2)
plot(y(:,1),y(:,3),'k')
axis equal
axis([-6 6 -6 6])

RE = sqrt(y(:,1).^2 + y(:,3).^2);
stdRE = std(RE)

%print -dtiff -r800 threebody

    function yd = f5(t,y)
        
        xj = rjup*cos(wj*t);
        yj = rjup*sin(wj*t);
        xs = -rsun*cos(wj*t);
        ys = -rsun*sin(wj*t);
        rj32 = ((y(1) - xj).^2 + (y(3) - yj).^2).^1.5;
        r32 = ((y(1) - xs).^2 + (y(3) - ys).^2).^1.5;

        yp(1) = y(2);
        yp(2) = -4*pi^2*((y(1)-xs)/r32 + chsi*(y(1)-xj)/rj32);
        yp(3) = y(4);
        yp(4) = -4*pi^2*((y(3)-ys)/r32 + chsi*(y(3)-yj)/rj32);
 
        yd = [yp(1);yp(2);yp(3);yp(4)];

    end     % end f5

end



References:

[1] D. D. Nolte, “The Fall and Rise of the Doppler Effect,” Physics Today, vol. 73, no. 3, pp. 31-35, Mar (2020)

[2] J. Barrow-Green, Poincaré and the three body problem. London Mathematical Society, 1997.

[3] M. C. Gutzwiller, “Moon-Earth-Sun: The oldest three-body problem,” Reviews of Modern Physics, vol. 70, no. 2, pp. 589-639, Apr (1998)

[4] D. D. Nolte, Introduction to Modern Dynamics : Chaos, Networks, Space and Time, 1st ed. (Oxford University Press, 2015).

The Doppler Universe

If you are a fan of the Doppler effect, then time trials at the Indy 500 Speedway will floor you.  Even if you have experienced the fall in pitch of a passing train whistle while stopped in your car at a railroad crossing, or heard the falling whine of a jet passing overhead, I can guarantee that you have never heard anything like an Indy car passing you by at 225 miles an hour.

Indy 500 Time Trials and the Doppler Effect

The Indy 500 time trials are the best way to experience the effect, rather than on race day when there is so much crowd noise and the overlapping sounds of all the cars.  During the week before the race, the cars go out on the track, one by one, in time trials to decide the starting order in the pack on race day.  Fans are allowed to wander around the entire complex, so you can get right up to the fence at track level on the straight-away.  The cars go by only thirty feet away, so they are coming almost straight at you as they approach and straight away from you as they leave.  The whine of the car as it approaches is 43% higher than when it is standing still, and it drops to 33% lower than the standing frequency—a ratio almost approaching a factor of two.  And they go past so fast, it is almost a step function, going from a steady high note to a steady low note in less than a second.  That is the Doppler effect!

But as obvious as the acoustic Doppler effect is to us today, it was far from obvious when it was proposed in 1842 by Christian Doppler at a time when trains, the fastest mode of transport at the time, ran at 20 miles per hour or less.  In fact, Doppler’s theory generated so much controversy that the Academy of Sciences of Vienna held a trial in 1853 to decide its merit—and Doppler lost!  For the surprising story of Doppler and the fate of his discovery, see my Physics Today article

From that fraught beginning, the effect has expanded in such importance, that today it is a daily part of our lives.  From Doppler weather radar, to speed traps on the highway, to ultrasound images of babies—Doppler is everywhere.

Development of the Doppler-Fizeau Effect

When Doppler proposed the shift in color of the light from stars in 1842 [1], depending on their motion towards or away from us, he may have been inspired by his walk to work every morning, watching the ripples on the surface of the Vltava River in Prague as the water slipped by the bridge piers.  The drawings in his early papers look reminiscently like the patterns you see with compressed ripples on the upstream side of the pier and stretched out on the downstream side.  Taking this principle to the night sky, Doppler envisioned that binary stars, where one companion was blue and the other was red, was caused by their relative motion.  He could not have known at that time that typical binary star speeds were too small to cause this effect, but his principle was far more general, applying to all wave phenomena. 

Six years later in 1848 [2], the French physicist Armand Hippolyte Fizeau, soon to be famous for making the first direct measurement of the speed of light, proposed the same principle, unaware of Doppler’s publications in German.  As Fizeau was preparing his famous measurement, he originally worked with a spinning mirror (he would ultimately use a toothed wheel instead) and was thinking about what effect the moving mirror might have on the reflected light.  He considered the effect of star motion on starlight, just as Doppler had, but realized that it was more likely that the speed of the star would affect the locations of the spectral lines rather than change the color.  This is in fact the correct argument, because a Doppler shift on the black-body spectrum of a white or yellow star shifts a bit of the infrared into the visible red portion, while shifting a bit of the ultraviolet out of the visible, so that the overall color of the star remains the same, but Fraunhofer lines would shift in the process.  Because of the independent development of the phenomenon by both Doppler and Fizeau, and because Fizeau was a bit clearer in the consequences, the effect is more accurately called the Doppler-Fizeau Effect, and in France sometimes only as the Fizeau Effect.  Here in the US, we tend to forget the contributions of Fizeau, and it is all Doppler.

Fig. 1 The title page of Doppler’s 1842 paper [1] proposing the shift in color of stars caused by their motions. (“On the colored light of double stars and a few other stars in the heavens: Study of an integral part of Bradley’s general aberration theory”)
Fig. 2 Doppler used simple proportionality and relative velocities to deduce the first-order change in frequency of waves caused by motion of the source relative to the receiver, or of the receiver relative to the source.
Fig. 3 Doppler’s drawing of what would later be called the Mach cone generating a shock wave. Mach was one of Doppler’s later champions, making dramatic laboratory demonstrations of the acoustic effect, even as skepticism persisted in accepting the phenomenon.

Doppler and Exoplanet Discovery

It is fitting that many of today’s applications of the Doppler effect are in astronomy. His original idea on binary star colors was wrong, but his idea that relative motion changes frequencies was right, and it has become one of the most powerful astrometric techniques in astronomy today. One of its important recent applications was in the discovery of extrasolar planets orbiting distant stars.

When a large planet like Jupiter orbits a star, the center of mass of the two-body system remains at a constant point, but the individual centers of mass of the planet and the star both orbit the common point. This makes it look like the star has a wobble, first moving towards our viewpoint on Earth, then moving away. Because of this relative motion of the star, the light can appear blueshifted caused by the Doppler effect, then redshifted with a set periodicity. This was observed by Queloz and Mayer in 1995 for the star 51 Pegasi, which represented the first detection of an exoplanet [3]. The duo won the Nobel Prize in 2019 for the discovery.

Fig. 4 A gas giant (like Jupiter) and a star obit a common center of mass causing the star to wobble. The light of the star when viewed at Earth is periodically red- and blue-shifted by the Doppler effect. From Ref.

Doppler and Vera Rubins’ Galaxy Velocity Curves

In the late 1960’s and early 1970’s Vera Rubin at the Carnegie Institution of Washington used newly developed spectrographs to use the Doppler effect to study the speeds of ionized hydrogen gas surrounding massive stars in individual galaxies [4]. From simple Newtonian dynamics it is well understood that the speed of stars as a function of distance from the galactic center should increase with increasing distance up to the average radius of the galaxy, and then should decrease at larger distances. This trend in speed as a function of radius is called a rotation curve. As Rubin constructed the rotation curves for many galaxies, the increase of speed with increasing radius at small radii emerged as a clear trend, but the stars farther out in the galaxies were all moving far too fast. In fact, they are moving so fast that they exceeded escape velocity and should have flown off into space long ago. This disturbing pattern was repeated consistently in one rotation curve after another for many galaxies.

Fig. 5 Locations of Doppler shifts of ionized hydrogen measured by Vera Rubin on the Andromeda galaxy. From Ref.
Fig. 6 Vera Rubin’s velocity curve for the Andromeda galaxy. From Ref.
Fig. 7 Measured velocity curves relative to what is expected from the visible mass distribution of the galaxy. From Ref.

A simple fix to the problem of the rotation curves is to assume that there is significant mass present in every galaxy that is not observable either as luminous matter or as interstellar dust. In other words, there is unobserved matter, dark matter, in all galaxies that keeps all their stars gravitationally bound. Estimates of the amount of dark matter needed to fix the velocity curves is about five times as much dark matter as observable matter. In short, 80% of the mass of a galaxy is not normal. It is neither a perturbation nor an artifact, but something fundamental and large. The discovery of the rotation curve anomaly by Rubin using the Doppler effect stands as one of the strongest evidence for the existence of dark matter.

There is so much dark matter in the Universe that it must have a major effect on the overall curvature of space-time according to Einstein’s field equations. One of the best probes of the large-scale structure of the Universe is the afterglow of the Big Bang, known as the cosmic microwave background (CMB).

Doppler and the Big Bang

The Big Bang was astronomically hot, but as the Universe expanded it cooled. About 380,000 years after the Big Bang, the Universe cooled sufficiently that the electron-proton plasma that filled space at that time condensed into hydrogen. Plasma is charged and opaque to photons, while hydrogen is neutral and transparent. Therefore, when the hydrogen condensed, the thermal photons suddenly flew free and have continued unimpeded, continuing to cool. Today the thermal glow has reached about three degrees above absolute zero. Photons in thermal equilibrium with this low temperature have an average wavelength of a few millimeters corresponding to microwave frequencies, which is why the afterglow of the Big Bang got its name: the Cosmic Microwave Background (CMB).

Not surprisingly, the CMB has no preferred reference frame, because every point in space is expanding relative to every other point in space. In other words, space itself is expanding. Yet soon after the CMB was discovered by Arno Penzias and Robert Wilson (for which they were awarded the Nobel Prize in Physics in 1978), an anisotropy was discovered in the background that had a dipole symmetry caused by the Doppler effect as the Solar System moves at 368±2 km/sec relative to the rest frame of the CMB. Our direction is towards galactic longitude 263.85o and latitude 48.25o, or a bit southwest of Virgo. Interestingly, the local group of about 100 galaxies, of which the Milky Way and Andromeda are the largest members, is moving at 627±22 km/sec in the direction of galactic longitude 276o and latitude 30o. Therefore, it seems like we are a bit slack in our speed compared to the rest of the local group. This is in part because we are being pulled towards Andromeda in roughly the opposite direction, but also because of the speed of the solar system in our Galaxy.

Fig. 8 The CMB dipole anisotropy caused by the Doppler effect as the Earth moves at 368 km/sec through the rest frame of the CMB.

Aside from the dipole anisotropy, the CMB is amazingly uniform when viewed from any direction in space, but not perfectly uniform. At the level of 0.005 percent, there are variations in the temperature depending on the location on the sky. These fluctuations in background temperature are called the CMB anisotropy, and they help interpret current models of the Universe. For instance, the average angular size of the fluctuations is related to the overall curvature of the Universe. This is because, in the early Universe, not all parts of it were in communication with each other. This set an original spatial size to thermal discrepancies. As the Universe continued to expand, the size of the regional variations expanded with it, and the sizes observed today would appear larger or smaller, depending on how the universe is curved. Therefore, to measure the energy density of the Universe, and hence to find its curvature, required measurements of the CMB temperature that were accurate to better than a part in 10,000.

Equivalently, parts of the early universe had greater mass density than others, causing the gravitational infall of matter towards these regions. Then, through the Doppler effect, light emitted (or scattered) by matter moving towards these regions contributes to the anisotropy. They contribute what are known as “Doppler peaks” in the spatial frequency spectrum of the CMB anisotropy.

Fig. 9 The CMB small-scale anisotropy, part of which is contributed by Doppler shifts of matter falling into denser regions in the early universe.

The examples discussed in this blog (exoplanet discovery, galaxy rotation curves, and cosmic background) are just a small sampling of the many ways that the Doppler effect is used in Astronomy. But clearly, Doppler has played a key role in the long history of the universe.


References:

[1] C. A. DOPPLER, “Über das farbige Licht der Doppelsterne und einiger anderer Gestirne des Himmels (About the coloured light of the binary stars and some other stars of the heavens),” Proceedings of the Royal Bohemian Society of Sciences, vol. V, no. 2, pp. 465–482, (Reissued 1903) (1842)

[2] H. Fizeau, “Acoustique et optique,” presented at the Société Philomathique de Paris, Paris, 1848.

[3] M. Mayor and D. Queloz, “A JUPITER-MASS COMPANION TO A SOLAR-TYPE STAR,” Nature, vol. 378, no. 6555, pp. 355-359, Nov (1995)

[4] Rubin, Vera; Ford, Jr., W. Kent (1970). “Rotation of the Andromeda Nebula from a Spectroscopic Survey of Emission Regions”. The Astrophysical Journal. 159: 379


Further Reading

D. D. Nolte, “The Fall and Rise of the Doppler Effect,” Physics Today, vol. 73, no. 3, pp. 31-35, Mar (2020)

M. Tegmark, “Doppler peaks and all that: CMB anisotropies and what they can tell us,” in International School of Physics Enrico Fermi Course 132 on Dark Matter in the Universe, Varenna, Italy, Jul 25-Aug 04 1995, vol. 132, in Proceedings of the International School of Physics Enrico Fermi, 1996, pp. 379-416