The Basics of Inferring Exoplanet Properties
(Originally posted February 28, 2017 on Blogger)
According to the NASA Exoplanet Archive website, 3,453 exoplanets have been confirmed within our Milky Way galaxy, with 577 of them being part of multi-planet systems. In this blog, we'll focus on one of five methods used to detect exoplanets; the transit method. Though, we'll also cover a couple other methods later in this blog, as they apply to inferring some significant physical properties of exoplanets. It's important to know that the transit method, like each of the other stand-alone methods, has its advantages and disadvantages. Up to 40% of transits can turn out to be false readings with regard to having 'found' an exoplanet. Therefore, astronomers often employ more than one method in order to verify a possible discovery. For example, they may detect a possible planet using the transit method, but will also employ the radial velocity method to confirm the finding. We'll get to what these methods are in this blog!
We'll start with a recap of how the transit method works in basic terms. Then we will discuss how some properties of exoplanets are inferred. Their can be different orders in how astronomers proceed, but the order we'll take here will be:
That order is:
- Detect the exoplanet using one of five methods (we'll use the transit method)
- Calculate the distance to its host star
- Calculate the luminosity of its host star
- Calculate the mass of its host star
- Calculate the orbital period & orbital radius of the planet
- Calculate the mass of the planet
- Calculate the radius of the planet
- Calculate the density of the planet
To avoid getting bogged down with too many details, we'll avoid getting into details of the the disadvantages inherit in the calculations listed above. This will be a blanket coverage of how astronomers find exoplanets, and determine some of their, and their star's properties. I hope to do blogs that get into greater detail on each method for calculating these different properties in the future.
The transit method for detecting exoplanets in our Milky Way has become one of the most successful, though one of its major disadvantages is that it requires we find stars whose planets have orbits such that they transit the star edge on with regard to our perspective here on Earth. Otherwise we would not be able to detect dimming in the star's light curve. A star may host a planet or many planets, but if their orbits aren't edge on, then the transit method will not detect them.
There are so many light curve graphs waiting to be poured over by astronomers that NASA has funded a website to crowd source this tedious work! You can help them by signing up as a volunteer here: www.planethunters.org. We covered the basics of detecting an exoplanet using the transit method in the TRAPPIST-1 blog. Here's a rough recap:
a. Astronomers begin plotting the light/IR intensity of a star on a light curve. A light curve is a graph depicting this intensity over time, where the y-axis indicates intensity, and the x-axis indicates time. It starts out looking something like Graph A. >>
b. If a celestial object passes between the star and Earth, then the otherwise relatively steady line will dip. This passing is called a transit. The depth of the dip is directly proportional to either the size of the transiting object, or its nearness to Earth. Astronomers simply don't know for sure at this point. It could be any number of things causing the dip, including a stellar anomaly. A dip on the light curve might look something like the Graph B; a hypothetical continuation of the graph above.
c. Sometimes dips aren't repeated. But if they are, as shown in Graph C, then astronomers carefully monitor the star waiting for a pattern to emerge. Repeating dips could possibly indicate an exoplanet is orbiting the star. However, with only two dips, it is too soon to know the cause. The second dip could be from a planet making a second revolution, or it could be a second planet. It could even be other phenomena. Astronomers simply don't know yet.
d. If an exoplanet is indeed orbiting the star, then a pattern of repeating dips of equal depths plots across the light curve. Once enough data has been collected, astronomers can proceed with a degree of confidence, that they have discovered another planet.
So, even though exoplanet discoveries are announced complete with artist renditions looking like this...
...the reality is, they look more like this...
In the TRAPPIST-1 blog, we covered how astronomers are able to infer some properties of exoplanets. But long before they can infer properties of an exoplanet, they must first infer properties of the exoplanet's host star. Two properties of the host star must be known:
- Its distance from our Sun (stellar distance)
- Its mass
The method commonly used to calculate stellar distance is the parallax method. Once distance is established, astronomers can establish the star's luminosity. With luminosity established, astronomers can use the Hertzsprung-Russell diagram (H-R diagram) to determine the star's mass. Once the star's distance, luminosity, and mass are known, astronomers can proceed with inferring properties of the star's exoplanet(s).
CALCULATING STELLAR DISTANCE
Hold your hand out in front of you at arm's length. Close your left eye while keeping your right eye open. Notice where your hand is in relation to any background objects. Now immediately switch to your left eye, closing your right eye. Notice that your hand now appears to have shifted in relation to any objects in the background. This is parallax. It gets its name from the Greek word, parallaxis (παράλλαξις), which roughly translates to the English word, change.
For all intents and purposes, your hand and the objects in the background did not move. It was your perspective that moved; the movement appeared to be a few centimeters to the left (as you switched from your right eye to your left eye).
The further objects are away from you, the slower they appear to move. Far enough away, & they'll appear not to move at all.
We experience parallax in our daily lives, whether we consciously realize it or not. The next time you're a passenger in a vehicle, look out the window and notice how objects near the road you're on whiz by, while background objects appear to move slower and slower the further from the road they are. Objects that are very far away, such as the moon, Sun (don't look at the Sun!), or stars, don't appear to move at all. They appear "fixed".
Of course they're technically moving; everything is moving relative to something else, however, for purposes of using the parallax method, these distant background objects are fixed. This phenomena is important as we'll see when calculating distance using the parallax method. Before we get to the parallax method, there are two units of measure we should be familiar with:
- The arcsecond
- The parsec
These are used because, as we will see, they make calculations of stellar distances surprisingly simple!
The arcsecond is a unit of measure along an arc. Imagine a circle's 360-degree perimeter as one big arc. An arcsecond would be a small portion of that arc. Very small; in fact only 1/3600 of one degree. Therefore 3,600 arcseconds equals 1 degree.
This may seem like a ridiculously small unit of measure, but it comes in very useful when dealing with very small angles, such as those encountered in astronomical calculations as we'll soon see. There are even smaller units of arc measure; milli- and microarcseconds, also commonly used in astronomical calculations. Using degrees is simply impracticable because this unit of measure is simply too big to be useful.
Where an arcsecond is used to measure very small lengths, the parsec is used to measure very large lengths. One parsec is equal to about 3.26 light years (over 30.8 trillion km).
How did the inventor of the parsec (Herbert Hall) come up with 3.26 light years? It is the parallax of one arcsecond (parsec). But what does this mean?
To understand this, let's recall the quick experiment you just did; holding your hand in front of you while switching between eyes as you look at it. Instead of your hand, let's look at a distant star. And instead of the perspective being between each eye, let's say the perspective is when Earth is on opposite sides of the Sun.
Earth makes one revolution around the Sun every 12 months. Therefore, every 6 months Earth has revolved half way around (opposite sides of the Sun). These two opposing positions are used in the parallax method.
Astronomers will look at a distant star and record its position in the sky against the "fixed" background of more distant stars. Then 6 months later, when our planet has revolved to the opposite side of the Sun, astronomers will take a second recording of that same star's position against the fixed background of more distant stars. This is like shifting between each eye when looking at your hand, but on a much larger scale; on average, about 299.2 million km between the two positions, as opposed to just a few centimeters between your two eyes.
The apparent movement of the star against the fixed background from its initial position as compared to its apparent position 6 months later, is measured in arcseconds. If there were some way we could work with a right triangle, we could figure out the distance to the observed star using basic trigonometry. As it turns out, there is a way!
Think back again to when you looked out your hand and shifted between each eye. The two lines of sight from each eye intersected at your hand and continue to the fixed background objects beyond your hand. If you measure the apparent movement of your hand as you shifted between each eye, and measure the distance between each eye, then we have enough information to determine distance to your hand. I've illustrated the five steps here:
Step 1: two intersecting lines of sight are imagined as you look at your hand. These lines make up two legs of a triangle; leg a and leg c. The third leg is the distance between each eye; leg b. We can imagine this triangle as being mirrored on the back side of your hand; such that whatever angle and leg measurements we find on one triangle, will equal the mirrored angles and legs on the other. In this step we only know the length of "leg b".
Step 2: Here I've simply removed our person and that person's hand to make things look cleaner.
Step 3: We measure the distance of the apparent movement of our hand against the fixed background. When measuring our hand we might use degrees of movement, but for distant stars we'd use arcseconds for this measurement (shown with a red arc). That angle equals the opposite angle beneath it. In geometry, we remember that opposite angles created by two intersecting lines are congruent. In this step we now know the length of leg "b", and now the pair of opposite angles created by the intersection of legs "a" and "c".
Step 4: Here we bisect the opposing triangles thereby creating two right triangles. Doing this cuts the length of leg "b" in half such that it now represents the distance between one of your eyes and your nose, rather than the distance between both eyes. In the example of looking at the parallax of a star, this halved distance now represents the distance of Earth at one of its two 6-month points and the Sun, rather than the distance between both 6-month points in its orbit.
Step 5: We only need to work with one right triangle, so in this step I've simply erased the other. I left the mirrored right triangle up so we can see that the angle we measured in Step 3 has been halved by the bisection. We also see that the length of leg "b" has been halved. In this step we now know two angles, the 90-degree (right) angle, and the halved angle created by the intersecting lines from Step 3. We also know that the length of leg "b" has been halved. Therefore we know the measure of two angles and a leg. This is enough information with which to calculate the length of leg "d"; distance to your hand, or star.
Flipped on its side we see our right triangle. The distance from Earth to the Sun is 1 AU, we have our 6-month angle (denoted as theta here), and we know the right angle given by definition of a right triangle. The tangent equals the opposite leg over the adjacent leg ("TOA" from trigonometry). Since our 6-month angle (aka parallax angle) is known, this can be rearranged to solve for "d".
The reason astronomers use the units of arcseconds and parsecs, is because they work perfectly together such that the only formula we need to know to solve for "d" (distance) is this:
d = 1/p
where, d = distance to star in parsecs
1 = 1 AU (distance of Earth to Sun)
p = 6-month angle in arcseconds
The following graphic shows how the parallax angle is inversely proportional to the star's distance. As the star distance increases, the parallax angle decreases, and vice versa. With the above formula, you can see how that calculates out. Smaller parallax angles equate to greater stellar distances, and vice versa.
Example: We measure a parallax angle of 0.715 arcsecond. We start with the equation:
We plug in 0.715 for p, giving us:
d = 1/0.715
We solve for d, giving us:
d = 1.40 parsecs
Using the parallax method for calculating stellar distance works best for stars between 0.01 - 100 parsecs from our Sun. Our Milky Way galaxy is estimated to be 30,000 parsecs across, so using the parallax method is limited to our small neighborhood in the galaxy. This isn't necessarily a bad thing; the closer we can find extrasolar planets to our Sun, the higher the our chances of one day successfully sending small probes to investigate those systems. If that sounds cool, then you might enjoy reading this article in Scientific American:
DETERMINING STELLAR LUMINOSITY
Determining the distance to a star is the first step astronomers must take before they're able to determine a star's luminosity. Once luminosity is established, then the star's mass can be determined.
A star's total energy output per second is called its luminosity. The more energy a star outputs, the more luminous the star will be, and vice versa. The inverse-square law applies to many fields in physics, including the electromagnetic field. This, of course, includes UV/visible/IR light radiation. As it applies to starlight, the law dictates that the intensity of light emanating from a star (as a point source) is inversely proportional to the square of the distance that light travels from the star. In the utmost simplest terms, this means that the further light travels from the star, the less luminous it appears. Knowing this relationship allows astronomers to determine the luminosity of the star at the star. Though the light is significantly dimmed by the time it reaches Earth, knowing the distance to that star, astronomers can calculate the luminosity of that star.
In mathematics, ∝ is the symbol for proportionality. One of the simplest formulas for calculating stellar luminosity is:
L ∝ 1/r2
where, L is the star's luminosity, and r is the measured distance from the star.
The further light travels from the star, the smaller the fraction of its luminosity.
Over 100 years ago, two astronomers (Ejnar Hertzsprung & Henry Russell) created a diagram plotting stellar luminosity (or equivalent absolute magnitude) against stellar surface temperatures (or equivalent spectral class). The diagram reveals the relationships between luminosity and temperature, as well as spectral class and absolute magnitude.
That diagram, the Hertzsprung-Russell diagram (H-R diagram), has since become one of the most useful tools in astronomy for helping astronomers understand the classifications, evolutions, and ages of different stars.
I took the simplified H-R diagram above, and removed some stars from it to make room for a couple arrows to show the relationships of stellar mass and size.
Luminosity is a function of the star's temperature and surface area as explained by the Stefan-Boltzmann Law:
L = 4πR2 σT4
where L is luminosity, R is stellar radius, and T is the star's surface temperature. The alpha symbol is a constant (5.6703 x 10^-8 W/m^2K^4). The thermal energy radiated by the star per second per unit area is proportional to the fourth power of the absolute temperature (K^4).
From this relationship we see that high-mass stars in the main sequence on the H-R diagram above exhibit higher luminosity. We can also see how surface area can compensate for otherwise lower temperatures if we look at the red giants and supergiants positions on the diagram above. Because of their large radii, despite their lower temperatures, their luminosity is equivalent to smaller high-mass stars in the Main Sequence.
Now that we have established both the star's distance, and subsequently its luminosity, we can proceed to determine the star's mass. Then, at long last, we can proceed to determine the properties of its exoplanet(s)!
DETERMINING STELLAR MASS
There is a linear relationship between the log of a star's luminosity, and the log of it's mass. This relationship is called the Mass-Luminosity Relationship and is described by this proportionality:
~L ∝ M3.5
where, L is the star's luminosity, and M is the star's mass. Luminosity is proportional to about cube of a star's mass. Since, at this point, we will have determined a star's luminosity, we can simply plug that into this proportionality (L) and calculate the star's mass. This Mass-Luminosity Relationship came out of the discoveries made by Hertzsprung and Russell over 100 years ago (H-R diagram).
Finally, we have all the information we need about a star with which to begin looking at certain properties of its planet(s)! Great!
DETERMINING PROPERTIES OF EXOPLANETS
-SIZE (DIAMETER) & ORBITAL PERIOD-
The light curve created out of the transit method explained at the beginning of this blog, reveals more than just the existence of a planet. It also reveals the orbital period of that planet. The orbital period is the time it takes for a planet (or other celestial object) to complete one full orbit around its host star.
The above light curve I drew in MS Paint is very simplified. An actual light curve may look something more like this:
The star's brightness is recorded as points in time, shown above as black dots, rather than as a continuous line as I made for the simplified light curve examples above. Notice the black dots dip at regularly-spaced intervals. In the above example, the dips are marked with vertical red lines for clarity. These dips are caused by the planet transiting its star. The deeper the dip, the more light the planet is blocking, suggesting a large-diameter planet. The above light curve is of a gas giant 1.1 times the diameter of Jupiter. Put simply, dip depth determines diameter (size).
Orbital period can be determined by measuring the distance between each dip on the light curve. Since the x-axis represents time, astronomers can simply look where each dip occurs along the x-axis to determine the orbital period. In the above example, the planet's orbital period is 8.41 Earth days.
Astronomers can also calculate transit duration; the time it takes for the planet to transit the star from one "edge" to the other. However, determining this would depend on at what 'latitude' the planet transits the star from our perspective. Does it transit directly in front of the star across as its "middle" from our vantage point? Or does it transit near the "top" or "bottom" of the star. If it transits across the middle, then the transit duration would be longer relative to the duration it would take that same planet to transit should our perspective change such that it appears to transit at higher latitudes in relation to its star. This is called the impact parameter, and would add a lot of extra detail to this blog that I'm trying to avoid.
I'm not sure how well I'll do this, but I'm trying to find a balance between detail and general concept. As such, I'm also going to skip discussing the effect of limb darkening, where the star's center appears brighter than its edges which has to do with more oblique paths photons take from our perspective.
Once we have the orbital period, and stellar mass figured out, we can proceed to solve for the planet's orbital radius. We can do this by rearranging the equation for Kepler's third law of planetary motion, which explains the relationship between a planet's orbital period, and the distance of that planet from its star. Generally speaking, orbital period (p) is proportional to orbital size (a). generally P ∝ a.
More specifically, the third law states that the square of the orbital period of a planet is directly proportional to the cube of the semi-major axis (see below) of its orbit; so more specifically it is represented by the simple proportion, P^2 ∝ a^3.
The third law can more specifically be expressed as:
...where, T is time in Earth years, a is orbital radius in astronomical units, G is the Newtonian gravitational constant (6.67408 × 10-11 m3 kg-1 s-2), and M is stellar mass given in solar mass units. The denominator of this equation could be expressed G(M+m) instead of simply GM, where the lower-case "m" is the mass of the exoplanet expressed in solar masses.
However, since the planetary mass is relatively insignificant as compared to its star, we can ignore it in our equation. Besides, we don't yet know the exoplanet's mass anyway. The lower-case "m" comes in useful if we're calculating orbits of moons around planets, or orbits of binary stars; celestial objects between which the mass of one is significant to the mass of the other.
At any rate, we can solve for orbital radius (a) by rearranging the formula like this:
The mass of an exoplanet (or any planet for that matter), can be arrived at by employing Newton's Law of Gravitation expressed mathematically as:
where, G is the gravitational constant, m1 is the mass of the host star, m2 is the mass of the planet, r is the orbital radius, and F is the gravitational force.
We want to solve for m2, the mass of the planet, but to do so, we'll need to know all of the other variables. We know the constant G, the stellar mass m1, and the orbital radius r, however, we will also need to know the gravitational "force", F, if we are to solve for the mass of the planet (m2).
We can find F using the radial velocity method, which can also be used to detect exoplanets. Mass and gravity are directly proportional, so a star will have a stronger gravitational effect than a planet. As such we see planets orbiting stars rather than the other way around. However, this isn't to say planets don't have gravitational effects on the stars they orbit. They in fact do! However, this effect is very subtle.
Astronomers are able to detect that subtle effect planets' gravitational fields have on their host stars with the use of sophisticated telescopes. What they see is a wobble in affected stars.
In these exaggerated and simplified animations (above & below), we can see how a planet's gravitational field causes its host star to wobble. In fact, another name for the radial velocity method is the wobble method. If we were able to look at the above system edge on, it would look something like this.
But these stars are so far away that to detect wobble by simply looking to see if the star is wobbling, is next to impossible. Instead, what astronomers look for is Doppler shift of the star's light. As the planet passes behind the star (from our perspective), the star 'wobbles' towards Earth. As it does this, its light blueshifts as the star is pushed towards us so to speak. As the planet transits the star (passes in front of it from our perspective), the star's light redshifts as the star pulls away from us so to speak.
Astronomers can look at these spectral fluctuations to determine the gravitational force of the planet; more pronounced spectral fluctuations indicate stronger gravitational interaction which is a function of mass and distance of and between the star and planet.
If we rearrange the above gravitation equation, and plug in the missing variables, we will be able to solve for the planet's mass (m2). Though, this method doesn't necessarily provide an accurate measure of an exoplanet's mass, it does provide at the very least a minimum mass the planet can have.
If a system has more than one planet, and especially if those planets have orbits near each other (like the TRAPPIST-1 system), then astronomers can look at the star's light curve to detect slight changes in transit periods of each planet. If there is no effect, then transits would be equidistant on the x-axis of the light curve graph. However, if there is an effect, then these transit times would fluctuate in proportion to the strength of gravity affecting them, and show up on the light curve as varying spaces between dips. Astronomers can measure these variances to determine each planet's mass with a higher degree of accuracy.
This is called the transit-timing variation method, and doubles as a method for detecting exoplanets in the first place.
Determining an exoplanet's radius can be done by looking at the depth of dips on the light curve of its host star. The deeper the dip, the more light the planet is physically blocking, therefore the larger the planet's radius. This of course would be the assumption after we've determined the planet's orbital radius. It could be blocking more light simply because it's further from the star. By depth, I mean how much flux has been blocked out compared to some reference.
The depth of the dip on a light curve is directly proportional to the diameter of the planet, and can be measured. Unfortunately there is no simple power law describing the mass-radius relationship of solid exoplanets, but this is not to say the relationship cannot be calculated. To explain it goes beyond the scope of this blog. Even mentioning "solid exoplanet" could have different meanings as solid exoplanets come in different flavors so to speak (solid ice, silicate, iron, etc). For those curious to learn the details, they are explained in detail by this paper out of MIT.
Calculating an exoplanet's density is a simple matter of dividing its mass by its volume; its volume being derived from its radius (assuming a perfect sphere).
ATMOSPHERES & HABITABILITY
I touched upon the topics of exoplanet atmospheres and habitability in my previous blog on the TRAPPIST-1 system. We briefly discussed how astronomers are able to detect general atmospheric composition of many exoplanets via spectroscopy, as well as discussed what the habitable zone is, and more importantly, what it isn't.
I want to save details of these topics for another blog, because a new paper has been published just this week (week of Feb. 25, 2017) titled, "Volcanic Hydrogen Habitable Zone" that I want to read first. It sounds interesting and may have new perspective I'd want to include in my blog (with references of course).
As always, thanks for reading.