## Stars and the Doppler Effect

When you take white light, and sort it by wavelength through the use of a diffraction grating (a piece of glass, water, a CD-Disc, etc), a rainbow appears. On one end, the bluer, shorter wavelengths, and on the other, the redder, longer wavelengths. The redder wavelengths, being longer, did not interact with the diffraction grating enough to be bent as severely as the other wavelengths, and therefore isn’t angled as severely. The opposite is true of the shorter, bluer wavelengths. A device that splits light into the spectrum is called a spectroscope.

Due to reasons that are outside the scope of this writing, atoms absorb and emit specific wavelengths of light depending on their electron structure, being unique to each element. Absorbing atoms in the photospheres of stars leave these absorption lines in the stellar spectrum, producing a pattern of dark lines. Below is the spectrum of the star Rigel. These spectral lines shed light on the composition of the photosphere of a star and allow us to classify them based on these spectral lines into “spectral types.” A device that measures these spectral lines is called a “spectrometer.”

A main reason spectral lines are commonly seen in prisms is because the light entering the prism is unregulated. A narrow light beam is needed to detect these spectral lines (a wider beam will cause the spectral lines to be submerged in the overlapping spectra). As such, a narrow slit is placed between the light source and the diffraction grating.

In a two-body system (for example, a star and a planet), the barycentric motion of each component will have a velocity component toward or away from us, called the radial velocity, assuming that the system is not inclined face on (i = 0° = 180°). Because the Doppler Effect works on all waves, light included, we would therefore expect to be able to detect the radial velocity of a star by paying attention to how the wavelengths of a stars light change over the orbital period of the system. For a light source receding from an observer, all wavelengths would shift to the redder end of the spectrum (called the redshift). For a light source approaching an observer, all wavelengths would shift to the bluer end of the spectrum (called the blueshift, or negative redshift). Specifically, the redshift, z, of a wavelength λ will result in that wavelength being observed as wavelength λ0, when the emitting source has a velocity v,

$\displaystyle z=\frac{\lambda-\lambda_0}{\lambda_0} = \frac{1+v_r/c}{\sqrt{1-v^2/c^2}} - 1$

Where c is the speed of light, and vr is the radial velocity of the light source, in this case the star.

This is where absorption lines become important, because they give us markers to trace the redshift of the spectrum. Watching them move across the spectrum tells us the changing radial velocity of the target star. The accuracy to which this is done is currently the main limiting factor in the search for extrasolar planets through spectroscopy. It is not difficult to detect redshifts for light sources moving at velocities of several tens of kilometres per second, but the barycentric motion of stars due to the influence of a planet is typically far less.

Eggenberger and Udry (2009) give us a nice overview of the challenges facing precise Doppler spectrometry.

• The motion of the centre of the light source at the spectrograph slit must be accounted for. Errors larger than 1 m s-1 will occur if the photocentre moves away from the slit by a few thousandths of the slit width. Photocenter motions due to telescope guiding errors, focus, seeing fluctuations, and atmospheric refraction usually amount to a tenth of the slit width, leading to errors of several kilometres per second. The usual solution to this is to use an optical fibre (occasionally with a scrambling device) to scramble the starlight from the telescope to the spectrograph. This produces a nearly uniformly illuminated disk at the spectrograph entrance.
• Changes in the environment can cause severe errors in Doppler measurements. For the CORALIE spectrograph, a temperature change of 1 K produces a total velocity drift of 90 m s-1, while a pressure change of 1 mbar produces a net velocity drift of ~300 m s-1. The favoured way to deal with this is to stabilize and control the entire spectrograph in temperature and pressure, but small wavelength shifts can’t be avoided, and often a simultaneous wavelength calibration is done so that instrumental effects can be detected and corrected.
• The motion of Earth. Earth orbits around the sun at +/- 30 km s-1 per year, and has a diurnal rotation of 1-2 km s-1. To obtain precise barycentric radial velocities, one must have precise Solar System ephemeris and needs to know the photon-weighted midpoint of each observation to better than 30 s.

Intrinsic sources of error arise from stellar phenomena. A rapidly-rotating star will blur its spectral lines due to the range of radial velocities of the rotating photosphere relative to the observer. Photospheric jitter can lead to a few metres per second of noise as well, but can be accounted for by taking a time-averaged observation, assuming the jitter is random and uncorrelated.

Detecting planets around other stars is hard. One of the easiest ways is to detect the apparent dimming of a star as a planet crosses between the observer and the star. The planet will block some of the photons from reaching the telescope, resulting in an apparent dimming of the star. This event is known as a “transit,” and the planets that do this are called “transiting planets.” The amount of light that is blocked is easy to calculate when one considers the problem from the perspective of simple 2-dimensional geometry. The amount of light blocked will be directly proportional to the amount of surface area of the star covered by the planet. For a star of radius $R_{star}$, and a planet of radius $R_{pl}$, the amount of the change in flux, $\Delta F$, is simply ratio of the area of the two bodies:

$\displaystyle \Delta F = \left( \frac{R_*}{R_pl} \right) ^2$

Clearly, then, larger planets will produce larger ΔF than smaller planets, which directly translates to a more detectable planet. But the difficulty does not scale linearly with the radius of the planet, as one can tell from the exponent in the above equation. Consider two planets, b and c, with radii $R_b$ and $4_c$, such that $R_b$ = 2$R_c$, essentially planet b is twice the radius of planet c. With the above equation it is seen that $\Delta F_b = (2^2)\Delta F_c = 4\Delta F_c$ – One-half the planet produces one-fourth the dip in brightness. Detecting Earth-sized planets can be understood to be far more difficult than detecting Jupiter-sized planets. The typical Jupiter-sized planet will be about 10 times the radius of the typical Earth-sized planet. This translates to 100 times the $\Delta F$. Can you detect a transiting Jupiter-sized planet? You need a hundred-fold increase in sensitivity to detect Earths.

Not all planets will transit. Assuming a random planetary inclination, the geometric probability that a planet will transit its star may be expressed (somewhat over-simplistically*) as

$\displaystyle P_{transit} = \frac{a}{2R_*}$

Where a is the semi-major axis of the planet’s orbit. We can see, therefore, that planets which orbit closer to their stars are more likely to transit than those further away. For Mercury, this works out to a ~1.19% transit probability. For Earth, this is an even more disappointing ~0.5%. This also means that if we assume that other stars are randomly distributed across the sky (which is not completely unreasonable out to distances where the structure of the galaxy does not become apparent), then we can say that ~0.5% of stars will have the right perspective to view Earth as a transiting planet. Similarly, we might also say that if all solar-radius stars have Earth-radius planets, then 0.5% of them are detectable through this method.

An important feature in determining how long transits last, aside from their orbital period, is the “impact parameter,” b, which is a measure of how far the transit chord is from the centre of the stellar disc, as measured in units of the stellar radius. A transit with b = 0 will be a perfectly edge-on orbit with the transiting planet passing straight through the centre of the stellar disk, with higher values of b being less dead-on transits.

Values of b > 1 imply a non-transiting planet, with the maximum attainable value for b being achieved for a face-on orbit (i = 0° or 180°) at $a/R_*$. There is a very small range of values of b > 1 for which transits still occur, depending on the radius of the planet. While the centre of the planetary disc may not intrude upon the stellar disc for these values of b, planets are not points, and so have a radius of their own. Transits with values of b close to 1 are called “grazing” transits, because the planet just grazes the stellar disk.

Mathematically, the impact parameter may be calculated by $b \equiv a \cos{i}$.

These transits may be plotted as brightness as a function of time, leading to a “transit light curve.” A transit event will have four events called “contacts.” The “first contact” is when the planetary disc first reaches the stellar disc. The “second contact” occurs when the entire planetary disc has moved onto the stellar disc. “Third contact” occurs when the planetary disc has reached the other edge of the stellar disc on its way out of the transit. Finally, “fourth contact” occurs when the planetary disc has moved completely off the stellar disc. The time from first to second contact is characterised by a significant drop flux, followed by a comparatively constant flux from second to third contacts. From third contact to fourth contact, the flux jumps up to the pre-transit value. The time between first and fourth contacts is the total transit duration, tt, while the time between second and third contacts is the “full transit duration,” tf, denoting the amount of time the planet is fully transiting the star.

For the image below, two transits are shown, one with a high impact parameter and one with an impact parameter of b = 0. Notice how differing the value of b changes both the duration and shape of the transit light curve. For both, all four contacts are labelled as vertical lines.

How does this look for the case of a real planet transiting a real star? Below is the transit light curve of the planet HD 189733 b.

The transit of HD 189733 b

Notice that the light curve between second and third contact is curved. This is because of stellar limb darkening, where the light coming from the limb of the star is darkened compared to the light from the centre of the stellar disc.

* Taking the radius of the planet, the eccentricity of its orbit, and the longitude of periastron into account, the geometric probability that a planet with a randomly oriented orbit will transit is expressed as

$\displaystyle P_{transit} = \left( \frac{a}{2(R_*+R_{pl})} \right) \left( \frac{1-e \cos{\pi/2 - \omega}}{1-e^2} \right)$

This dependence on the longitude of perihelion can be understood from the consideration of eccentric orbits. In reality, what is the dominant driver as to the probability that a planet will transit is more of the planet’s distance from the star during the transit window, as opposed to the planet’s distance from the star as measured by the semi-major axis. Below shows two planets with identical orbits, except for the latter having a higher value of ω. Note that because of this, the latter planet’s distance from the star during the transit window, shown in light-grey, is much further away than the first planet, and so its transit probability is considerably lower.

## The Path of the Wanderers

The Heliocentric Model of the Solar System

While the ancients did not understand the motions of the “wandering stars,” Newton and Kepler made great strides toward understanding them.

Assuming the simple case of a single body, A, orbiting another, B, the two bodies will orbit around a common barycentre that represents the centre of mass of the system. Thus while it is acceptable to say that one body orbits another, it is more physically accurate to say that both bodies are in a mutual orbit about each other.

Assume the existence of two objects, A and B, with masses MA and MB, respectively. Let aA be the distance from A to the barycentre of a system, and let aB be the distance from B to the barycentre. The ratio of aA / aB is determined purely by inverse ratio of the two masses. For example if MA = MB, then aA = aB, and if MA = 3MB, then aA = (1/3)MB. Physically, this is because A is more massive than B, and so it has more gravity and is thus B needs to be farther from the centre of mass to balance the system.

In the latter case, the gravitational acceleration of A on B is also greater, causing B to have a higher velocity than A. Thus despite B having a wide orbit, it will complete the orbit at the same time as A, and thus, the orbital period, P, of the two will be equal. The fact that is true for all values of is important for constraining the orbital period of one object if the other is not visible for whatever reason. The orbital period of both bodies may be defined as

$\displaystyle P = 2\pi \sqrt{a^3 / \mu}$

Where a is the semi-major axis, roughly the average distance between A and B (and is defined as half the length of the orbits longest axis, the “major axis”), and μ is defined as μ = GM, the product of the gravitational constant, G, multiplied by the total masses of the two bodies, M. It is necessary to consider the sum of the masses because the gravitational pull of both objects on each other will contribute to the orbital period. In cases where MA > MB, the contribution of B may be considered negligible and the orbital period determined more or less accurately without taking it into account (for example, calculating the orbital period of Earth about the sun may be done reasonably well without taking Earth’s gravitational pull on the sun into account).

Not all orbits are perfect circles. Some may be eccentric. Mathematically, an orbit should never be thought of as a circle, but rather as an ellipse with the barycentre at one focus. However an orbit with an eccentricity of e = 0 is physically indistinguishable from circular.

For an eccentric orbit (any orbit for which e > 0), there exists a point in the orbit where the bodies are closest to each other, this is referred to as the periapsis, ap. The point in the orbit where the two bodies are most separated is the apoapsis aa, though these terms vary depending on which body is orbited. Aphelion and perihelion are used for bodies orbiting the sun, and apastron and periastron are used for bodies orbiting other stars.

If e = 0, the orbit may be considered circular. If 0 < e < 1, then the orbit is eccentric. If e = 1, then the orbit is parabolic, and not bound to the orbited body. If e > 1, then the orbit is hyperbolic and is not bound to the orbited body. In the case of e ≥ 1, there exists a periapsis, but no apoapsis.

Regardless of the eccentricity of an orbit, the orbital period will remain constant for a constant value of a. However the orbital velocity will change in such a way so that a line drawn from the planet to the star will cover a constant amount of area per unit time. As such, a planet will orbit faster at periastron, and slower at apastron.

Conveniently, the eccentricity of a planet around the barycenter will be equal to the eccentricity of the star’s orbit around the barycentre. Note that if a body in a circular orbit accelerates (be it by its own power like a spacecraft, or by a gravitational pull from another body), its orbit will become more eccentric.

Inclination for the planets in our solar system is defined as the angle between the plane of the planet’s orbit and the plane of Earth’s orbit, which defines the “ecliptic.” However this is invalid for other planetary systems for which Earth is not presently orbiting to provide such a reference. As such, we use a different reference plane upon which the sky is. We view the sky as if in the centre of an all-encompassing sphere. When looking at a planetary system from the viewpoint of Earth, we may consider the background sky to be a perpendicular plane, and determine the inclination of an orbit as the angle between that body’s orbital plane and the plane of the sky.

Inclination of a planetary system

Thus an edge-on orbit around some distant star would be perpendicular to the plane of the sky, and would be considered to have an inclination of 90°, and face-on orbits are parallel to the plane of the sky and have an inclination of 0° or 180°. As with the orbital period and eccentricity, both the planet and the star will have an equal inclination (and indeed their orbital planes are identical).

One may rotate an orbit about the orbited body within the orbit plane, such that the rotation axis is perpendicular to the orbit plane. This is called the longitude of perihelion, ω, because it rotates the perihelion longitudinally around the orbit, and is measured in degrees. For an orbit with e = 0, this does not make a noticeable change in the orbit, and in such cases, it is customary to set ω = 0°. The reference angle for ω is the line between the observer and the star. There is a 180° difference between ωplanet and ωstar.

The longitude of the ascending node, Ω, is the angle around the reference plane where the orbital plane of the planet in an inclined orbit crosses the reference plane. For the specific case of the ascending node, the planet rises above the reference plane (more northward). For the specific case of when the planet dips below the reference plane, this is called the longitude of the descending node. From visual observation alone, it is impossible to discern the ascending node from the descending node for a distant planetary system. The value for Ω is equal for both the planet and the star. For non-inclined orbits we let Ω = 0°.

The mean anomaly of an orbit, M, is the position of the planet around the orbit, in degrees, and is unrelated to the shape of the orbit. It is typically given with an ephemeris, or a date (or more scientifically, ‘epoch’) for which the value of M is true. There is a 180° offset between the values of M for the star and the planet for the same epoch.

The elements of an orbit

## Reflecting Back (The Mirror is Foggy)

For the entirety of human history, up until just under two decades ago, it was genuinely unknown whether or not other stars possessed planets. There was no reason to suspect they didn’t, but by the 1980’s, there was some evidence to suggest they did. Among the two most important observations were that 1) The most well studied star to date (an unremarkable G dwarf by the name of “Sol”) was known to harbour nine eight planetary companions, and most importantly, 2) Young stars were observed to be hosts to circumstellar material suggestive of an ongoing planet-formation process, much as the limited understanding of planet formation predicted should exist.

As to the properties of these so-called “extrasolar planets” (or “exoplanets”), nothing was known, but one might infer some of their basic properties of their planetary systems from observations of our solar system. The basic rules one can derive are

1. Gas giant planets form far from their stars.
2. Terrestrial planets form close to their stars.

The reasoning behind this can also be understood from the context of planetary formation, as was understood at the time. A gas giant planet (that is, a planet for which the majority of its mass are “gases,” like H and He) will more likely form where mass is abundant in the disk. After a large core coalesces together and reaches a critical mass, it begins runaway accretion of gases around it, accreting several times its mass worth of H and He. But gas does not last terribly long in a planet-forming disk. Radiation pressure from the new star will drive out the light gases in a time-scale on the order of a few million years (Myr). So gas near the star is likely to be short lived, preventing gas giant formation in the inner regions of a planetary system. Solid cores, however, could form there and go on to be terrestrial planets that is, planets made dominantly of solids or “metals”).

The challenge of detecting extrasolar planets at the time was therefore contemplated in the context of the difficulty of detecting solar planets if they orbited nearby stars. Of all of them, Jupiter would be the easiest to find through the same techniques used for discovering stellar companions to known stars, but the instrumentation lacked the required sensitivity.

In 1952, Otto von Struve proposed an idea that did not get much attention at the time. He proposed that gas giant planets might occasionally be found so close to their stars that they might be commonly detectable in transit, where the planet blocks light from the star. I quote brief snippets here, but the page-and-a-half summary of the problem is a very interesting read from the perspective of hindsight.

“One of the burning questions of astronomy deals with the frequency of planet-like bodies in the galaxy which belong to stars other than the sun. … But how should we proceed to detect them? … There seems to be at present no way to discover objects of the mass and size of Jupiter; nor is there much hope that we could discover objects ten times as large in mass as Jupiter, if they are at distances of one or more AU from their parent stars. … But there seems to be no compelling reason why the hypothetical stellar planets should not, in some cases, be much closer to their parent stars than is the case in the solar system. It would be of interest to test whether there are any such objects. We know that stellar companions can exist at very small distances. It is not unreasonable that a planet might exist at a distance of 1/50 AU, or about 3,000,000 km. Its period around a star of solar mass would then be about 1 day.”

He then goes on to argue how such planets might actually be comparatively easy to detect with the instrumentation of the late 1950’s. The ‘burning question’, however, would remain unanswered for another 40 years.

## The Internet’s n-th Blog; where n > 156,000,000

Yep, my blog is the one in the 3rd column from the left, 5th from the front.

I will be posting my thoughts on extrasolar planet detection, characterisation and exploration on this blog. Some of it will be inspired by papers I’ve read on arXiv, while others will be a bit random. Still others will focus on specific extrasolar planet discoveries. While I will try to relay information in as simple a way as I can see, the nature of the topic is scientific, and therefore it will assume at least a high school education for the target audience. Math and graphs will be no stranger to this blog.