# Can we calculate the orbit of exoplanets?

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I'm not an astronomer, but the question came up and I'm interested if we are able (or how accurate we are able) to calculate the orbital parameters of an exoplanet. Since the transit method must yield some properties of its orbit, we must know some of it. I also found some information here regarding the measurement of the orbital plane (Can we know the orbital planes of extraterrestrial planetary bodies?).

But do we have a clear picture of its orbit? Like we have of the planets in our system?

Could we, for example, tell if the planet was born in this system or has been caught by it at some point (giving it a different orbit I suppose)?

The characteristics of the orbit of an exoplanet can be determined very accurately using measurements of the reflex motion (the "Doppler wobble") of its parent star. These measurements yield the orbital period and eccentricity of the orbit. If we can estimate the mass of the parent star, then the orbital semi-major axis is known through Kepler's 3rd law. Orbital periods, and hence semi-major axes, can also be measured very precisely from transit data, and eccentricities can also be estimated provided one has a good idea of the mass and size of the parent star (e.g. van Eylen & Albrecht 2015).

Additional details can be gleaned from transiting exoplanets, such as the inclination of the orbit to the line of sight and any difference between the orbital axis and the spin axis of the star it orbits. These measurements exploit something called the "Rossiter-McLaughlin effect", as explained in an answer to the question you linked to.

The answer to your broader question is yes, of course people are looking at the distribution of orbital parameters: the behaviour of eccentricities versus planet mass and semi-major axis; the prevalence of planets of particular mass at certain distances around different types of stars; the frequency of occurence of multiple planets and orbital resonances, the misalignment between the spin of a star and the orbits of its planets, and so-on, in order to try and understand how planetary systems form and evolve.

Some resources:

The HEASARC Exoplanet orbital parameter database

exoplanets.org

If you have:

• Distance to the host star - d;

• Luminocity of the host star - I;

• orbital period p, measured by Transit method.

You can determine:

• Star's mass (M) from d and I

• From p and M of host star then you can measure semimajor axis (a) of exoplanet's orbit by Third Kepler's law:

$$p^2 = a^3/M$$

## Why We May Not Want To Visit Some Exoplanets

Image Credit: NASA Ames/JPL-Caltech/T. Pyle

The image above shows an artists’ impression of the exoplanet Kepler-452b, the first exoplanet that was found to orbit a Sun-like star. Dubbed a super-Earth, the planet is about 60% bigger than Earth, which is the main reason why we may not want to send a manned mission there.

Although the total number of known exoplanets is now approaching about 4,000 or so, most of them are known to be gas giants. However, a small percentage of exoplanets are now thought to be rocky in nature, and an even smaller sub-set of exoplanets may very well be Earth-like in the sense that they are in stable orbits in the habitable zones around their host stars, have atmospheres, and have liquid water on their surfaces. Nevertheless, to date no Earth-like exoplanets with masses comparable to that of Earth have been positively identified.

In a recently published paper by German “citizen scientist” Michael Hippke, who is affiliated with the Sonnenberg Observatory in Germany, the author uses the exoplanet Kepler-452b as an example to explain why it may not be desirable to send a manned mission to super-Earth planets that typically have masses of several times that of Earth.

Essentially, the paper describes the engineering challenges the inhabitants of super massive rocky planets would have to overcome in attempts to achieve meaningful space flight capabilities. Of principal importance is the fact that even on Earth, getting a rocket carrying a small payload into orbit is exceedingly expensive in terms of fuel and energy expenditure. For instance, in order to achieve an Earth orbit, a launch vehicle has to have a mass of between 50 and 150 times that of the payload, with most of the weight of the launch vehicle being taken up by fuel.

Using the exoplanet Kepler-452b that is 9.7 times as massive as Earth as an example, Hippke used simple, high-school level math to calculate that a rocket requiring 9,000 tons of fuel to reach a stable Earth orbit, would require at least 55,000 tons of fuel to reach a similar orbit around Kepler-452b. As a practical matter, a rocket equivalent to the Saturn V rocket that carried the Apollo missions, would require at least 400,000 tons of fuel to lift a payload similar to the weight of the lunar landing modules into an orbit around Kepler-452b. Clearly, building rockets that can routinely carry fuel loads that compare favorably to the mass of a 100-storey skyscraper is beyond anything that current Earth-based technology and engineering skills can accomplish.

Of course, Hippke concedes that Kepler-452b is an extreme example, but he also points out that less-massive rocky exoplanets will present the same, but appropriately scaled engineering challenges. Moreover, in a closely related paper published by Abraham Loeb (Harvard University), the author makes the point that it may be extremely difficult, if not impossible to leave the surface of a planet that is orbiting close to red dwarf stars with chemically fueled rockets, since the gravitational pull of the host star would have to be overcome as well. While Kepler-452b does not orbit a red dwarf, it does orbit its host star in only 3.7 days at a distance of 0.5 astronomical units, which is 50% of the average distance between Earth and the Sun, and in turn, greatly increases the energy required to leave the surface of the planet.

Of particular note is the fact that Hippke’s paper provides one possible solution to the Fermi Paradox, in the sense that we don’t see alien life in the Universe because the inhabitants of planets that host life may not be able to leave those planets.

For this reason, visiting massive exoplanets may be a very bad idea from our perspective. Even if we did manage to get there, we may not be able to leave again simply because we could never carry sufficient fuel to escape from the combined gravitational pull of the massive planet and its host star, and like the residents of the Hotel California, we would be stuck there forever.

## Planetary Formations

Although planets orbiting other stars (exoplanets, sometimes called extra solar planets) had long been theorised, there was no evidence for their existence until the mid-1990s.

Since then several hundred have been discovered and confirmed. The most common are normally described as ‘Hot Jupiters’, large bodies orbiting very closely to their parent star. More of these bodies have been discovered because they are easier to detect.

At least 2 exoplanets, Gliese 581c and d lie within a habitable zone where water can exist as a liquid. Exoplanet MOA-2007-BLG-192Lb w is also notable as it has a mass of 3.3 of Earth. These are known as ‘Super-Earths’.

Detecting exoplanets is difficult for a number of reasons:

• They orbit stars and the brightness of the star may prevent us from finding them.
• Planets may orbit too far away from our line of sight for some methods to detect them e.g. when their ecliptic is too eccentric for them to be noticed.

Different methods exist to detect them:

Transit Methods
As a planet moves in front of a star minute changes in light occur. Scientist can then work out the size and orbit of a planet

Astrometry
By measuring the position of a star very accurately, any minute wobble in the stars position can be due to the tiny pull of a planet on the parent star.

Changes in the movement of a star towards and away from Earth can be detected using the Doppler effect.

There are other methods such as optical detection which involves observing a star but cancelling out different wavelengths of light to observe it. There is also gravitational lensing which uses the bending of light by gravity around stars to detect planets in that direction.

## How Many Planets Can Orbit a Star?

Question: How many planets a stellar-planetary system can accommodate safely into a stable orbit around that star? What are the factors it depends on?

For Example: Currently our solar system has 8 planets in a stable orbit. I want to know the maximum possible number of planets our system can accommodate so that added planets(beyond the orbit of Neptune) also follow the orbital pattern of 8 existing planets?
Suppose all the planets in our solar system are exact replica like Earth, then how many earth-like planets that our current sun can get hold off? — Vinod

Answer: With no other constraints on the star or the planets that orbit the star, the only requirement for stable orbits of planets around the star is that the total mass of the planets be less than the mass of the star. Therefore, one could in principal have a nearly infinite number of very small planets that orbit a star. In reality there are other constraints, such as merging of small planets that are near each other in the early phases of the formation of a planetary system, that reduces the final configuration of a planetary system. These additional constraints, though, conspire in a complex way to produce the final orbital configuration for an exoplanetary system.

## Detecting life

In addition to searching for an Earth twin, direct imaging could help scientists to find potentially habitable worlds.

According to Elisa Quintana, a Kepler research scientist with the SETI Institute and the NASA Ames Research Center, direct imaging could reveal not only the atmosphere of a planet, but also potential biomarkers. It could even reveal life beyond the solar system.

"There's three ways we're going to know if there's other life," she said.

The first is by searching the skies for artificial radio signals. The second is hunting for microbial life within the solar system, in places like Mars or the icy moons of Jupiter and Saturn. And the third method is direct imaging.

According to Quintana, direct imaging "might very well be the first time we detect some form of life."

#### Uncertainties

We have recorded literature uncertainties in stellar masses, but when estimating uncertainties in msini and a we conservatively assume a minimum uncertainty in stellar mass of 5%. We do this to account for likely systematic errors in model estimates of stellar masses (limits in their accuracy) for most planet-bearing stars.

We now report asymmetric error bars throughout the database in all fields. For quantities with asymmetric uncertainties from the literature, we record the uncertainty field as half of the span between the upper and lower limits. We store the asymmetry in an additional field, which ends in D, as the value of the upper uncertainty. For instance, e = 0.5 +0.1 -0.2 would be stored as three fields: ECC = 0.5 , UECC = 0.15 , and UECCD = 0.1 .

## Exoplanetary Systems

As we search for exoplanets, we don&rsquot expect to find only one planet per star. Our solar system has eight major planets, half a dozen dwarf planets, and millions of smaller objects orbiting the Sun. The evidence we have of planetary systems in formation also suggest that they are likely to produce multi-planet systems.

The first planetary system was found around the star Upsilon Andromedae in 1999 using the Doppler method, and many others have been found since then (almost 700 as of early 2020). If such exoplanetary system are common, let&rsquos consider which systems we expect to find in the Kepler transit data.

A planet will transit its star only if Earth lies in the plane of the planet&rsquos orbit. If the planets in other systems do not have orbits in the same plane, we are unlikely to see multiple transiting objects. Also, as we have noted before, Kepler was sensitive only to planets with orbital periods less than about 4 years. What we expect from Kepler data, then, is evidence of coplanar planetary systems confined to what would be the realm of the terrestrial planets in our solar system.

By 2020, astronomers gathered data on nearly 700 such exoplanet systems. Many have only two known planets, but a few have as many as five, and one has eight (the same number of planets as our own solar system). For the most part, these are very compact systems with most of their planets closer to their star than Mercury is to the Sun. The figure below shows one of the largest exoplanet systems: that of the star called Kepler-62 (Figure (PageIndex<5>)). Our solar system is shown to the same scale, for comparison (note that the Kepler-62 planets are drawn with artistic license we have no detailed images of any exoplanets).

Figure (PageIndex<5>) Exoplanet System Kepler-62, with the Solar System Shown to the Same Scale. The green areas are the &ldquohabitable zones,&rdquo the range of distance from the star where surface temperatures are likely to be consistent with liquid water.

All but one of the planets in the K-62 system are larger than Earth. These are super-Earths, and one of them (62d) is in the size range of a mini-Neptune, where it is likely to be largely gaseous. The smallest planet in this system is about the size of Mars. The three inner planets orbit very close to their star, and only the outer two have orbits larger than Mercury in our system. The green areas represent each star&rsquos &ldquohabitable zone,&rdquo which is the distance from the star where we calculate that surface temperatures would be consistent with liquid water. The Kepler-62 habitable zone is much smaller than that of the Sun because the star is intrinsically fainter.

With closely spaced systems like this, the planets can interact gravitationally with each other. The result is that the observed transits occur a few minutes earlier or later than would be predicted from simple orbits. These gravitational interactions have allowed the Kepler scientists to calculate masses for the planets, providing another way to learn about exoplanets.

Kepler has discovered some interesting and unusual planetary systems. For example, most astronomers expected planets to be limited to single stars. But we have found planets orbiting close double stars, so that the planet would see two suns in its sky, like those of the fictional planet Tatooine in the Star Wars films. At the opposite extreme, planets can orbit one star of a wide, double-star system without major interference from the second star.

## Possible to calculate an Exoplanet's Magnitude?

There's a nice list here on all exoplanets discovered up to date, was just curious if it's possible to calculate an exoplanet's magnitude and how.

WIll it ever be possible for humans to observe an exoplanet larger than a pinhead?

If you had to choose one of these planets to observe, which one would you pick? One with the closest star and a big mass >
5 Jupiters, I guess?

Think you'd have a chance with a 60 inch aperture scope?

### #2 brianb11213

if it's possible to calculate an exoplanet's magnitude and how.

You know the primary star's magnitude & the distance & radius of the exoplanet. Make an assumption about the albedo (0.5 is probably reasonable for a planet with an atmosphere) & you can get a good value.

Here's an illustration. Suppose the planet is 0.1 times the star's radius and is 100 stellar radii from the star. Light reaching the planet is 0.1^2 * 0.01^2 times the radiation from the star . 0.000001 - 1 part in a million, or 15 magnitudes fainter than the primary. Add another 0.7 mags for the albedo assumed to be 0.5.

Think you'd have a chance with a 60 inch aperture scope?

### #4 thrawn

The big problem is not the magnitude of planets or even resolution, because we have pretty big telescopes already. The big problems are the glare from the star, which totally washes out the planet image, and atmospheric seeing.

We use a coronagraph http://en.wikipedia. iki/Coronagraph for the first problem, and adaptive optics or a space-telescope for the second.

The Keck telescope (400") has imaged extrasolar planets: http://en.wikipedia.org/wiki/HR_8799

You have no chance without a nice coronagraph and adaptive optics. BUT you don't need a 400" telescope, read this: http://www.msnbc.msn. _science-space/

As for getting moar than pinpricks, we're hoping to get spectra for planets when the ELT's get going: http://en.wikipedia. Large_Telescope

But no one knows when the funding or even the technology will be available to get pixels of imaging for teh planets. We're talking many decades though.

### #7 robin_astro

if it's possible to calculate an exoplanet's magnitude and how.

You know the primary star's magnitude & the distance & radius of the exoplanet. Make an assumption about the albedo (0.5 is probably reasonable for a planet with an atmosphere) & you can get a good value.

Here's an illustration. Suppose the planet is 0.1 times the star's radius and is 100 stellar radii from the star. Light reaching the planet is 0.1^2 * 0.01^2 times the radiation from the star . 0.000001 - 1 part in a million, or 15 magnitudes fainter than the primary. Add another 0.7 mags for the albedo assumed to be 0.5.

Think you'd have a chance with a 60 inch aperture scope?

This does raise an interesting possibility to detect non transiting exoplanets, as the amount of light they would add would vary round the orbit depending on the phase. I suspect the main stumbling block though would be that even the most stable of stars most likely vary in brightness by much more than 1 part in 1 million so the natural variability would swamp any due to the exoplanet

### #8 btieman

This does raise an interesting possibility to detect non transiting exoplanets, as the amount of light they would add would vary round the orbit depending on the phase. I suspect the main stumbling block though would be that even the most stable of stars most likely vary in brightness by much more than 1 part in 1 million so the natural variability would swamp any due to the exoplanet

Hehe. ok, this was a transiting exoplanet and the phase variation from a non-transiting exoplanet would likely be smaller--approaching 0 phase variance as the planet's orbit approached 90 degrees to our line of sight, but Kepler certainly has a chance!

There are better links to this, but it was the first one I could google easily.

### #9 btieman

This does raise an interesting possibility to detect non transiting exoplanets, as the amount of light they would add would vary round the orbit depending on the phase. I suspect the main stumbling block though would be that even the most stable of stars most likely vary in brightness by much more than 1 part in 1 million so the natural variability would swamp any due to the exoplanet

Hehe. ok, this was a transiting exoplanet and the phase variation from a non-transiting exoplanet would likely be smaller--approaching 0 phase variance as the planet's orbit approached 90 degrees to our line of sight, but Kepler certainly has a chance!

There are better links to this, but it was the first one I could google easily.

### #10 brianb11213

This does raise an interesting possibility to detect non transiting exoplanets, as the amount of light they would add would vary round the orbit depending on the phase.

Interesting idea but I think the amplitude of the light variation would be impossible even without the atmosphere to muck things up.

How about taking a very high resolution spectrograph & hoping to see a faint line (reflected light) moving from one side to the other of the main line depending on the radial velocity of the planet? Even a periodic assymetry in the lines would do.

### #11 btieman

This does raise an interesting possibility to detect non transiting exoplanets, as the amount of light they would add would vary round the orbit depending on the phase.

Interesting idea but I think the amplitude of the light variation would be impossible even without the atmosphere to muck things up.

How about taking a very high resolution spectrograph & hoping to see a faint line (reflected light) moving from one side to the other of the main line depending on the radial velocity of the planet? Even a periodic assymetry in the lines would do.

### #12 gavinm

This does raise an interesting possibility to detect non transiting exoplanets, as the amount of light they would add would vary round the orbit depending on the phase.

Interesting idea but I think the amplitude of the light variation would be impossible even without the atmosphere to muck things up.

How about taking a very high resolution spectrograph & hoping to see a faint line (reflected light) moving from one side to the other of the main line depending on the radial velocity of the planet? Even a periodic assymetry in the lines would do.

No, they're just measuring simple radial velocities of the star.

## Gravitational Microlensing

Microlensing was first proposed by Einstein in the 1930s, and is particularly useful for discovering low-mass exoplanets orbiting stars near the centre of our galaxy. The technique will be used by the upcoming Nancy Grace Roman Telescope, due to launch in the mid-2020s.

It operates based on the premise that mass bends space-time, so light from a distant star is magnified and brightened by the gravitational pull of a closer star orbiting in front of it, as observed from Earth.

The background star can actually appear to get up to 1,000 times brighter than it actually is.Variations in that brightness can indicate whether there is a planet orbiting that closer ‘lens’ star.

This method is much the same as gravitational lensing, but on a smaller scale.

## Going Beyond

To learn about some unusual exoplanets, check out Scientific American's Top 10 Exoplanets, keep your exoplanet apps up-to-date and watch the science news as new discoveries are announced all the time.

If you want to help in the search for exoplanets, you can sign up for the Planet Hunters program run by Yale University. After a brief training exercise, the online portal will display light curves for you to label with any exoplanet dips you may find. Your answers are compared with other individuals' and, if a consensus is reached, the star will be flagged for professional analysis. It's fun, and you are contributing to real science.

There's plenty more to come in future editions of mobile astronomy, including astro-gadget reviews, advanced smartphone photography and columns about using apps to teach astronomy. Send me your ideas, too. Until then, keep looking up!

Editor's note: Chris Vaughan is an astronomy public outreach and education specialist, and operator of the historic 1.88 meter David Dunlap Observatory telescope. You can reach him via email, and follow him on Twitter as @astrogeoguy, as well as on Facebook and Tumblr.