Astronomy

Calculate mass of exoplanet from transit method

Calculate mass of exoplanet from transit method


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Is there any way to calculate (or at least estimate) mass of an exoplanet from transit method? I know that mass can be calculated with radial velocity method, however I would like to create program for processing Kepler light curves (without radial velocity data).

The data I have are:

  • radius, mass, surface temperature and luminosity of star,
  • orbital period, semi-major axis and radius of exoplanet,
  • duration and depth of transit.

For the case of a single planet, you can't. The transit method is sensitive to the radius of the planet, not the mass.

For a multi-planet system, you can use variations in the timing of the transits caused by the gravitational interaction between the planets to infer the masses, for example as done in this paper by Nesvorný & Morbidelli (2008). These timing variations can be very significant if the planets are located close to mean-motion resonances (i.e. the ratio between the orbital periods being close to a fraction with small denominator, e.g. 1:2 or 2:3).


Antispinwards is correct. However, for the practical purpose of estimating a mass, you can construct a mass-radius plot for all the exoplanets with known mass, use this to define a mean mass-radius relationship (which has significant scatter), and then use that relationshipto assign a mass to your planet.


Calculate mass of exoplanet from transit method - Astronomy

APS Excellence in Physics Education Award
November 2019


Science SPORE Prize
November 2011


The Open Source Physics Project is supported by NSF DUE-0442581.


Transit Method

This method only works for star-planet systems that have orbits aligned in such a way that, as seen from Earth, the planet travels between us and the star and temporarily blocks some of the light from the star once every orbit.

A planet does not usually block much light from a star, (only 1% or less) but this can be detected. This method will not work for all systems, however, because only about 10% of hot Jupiters are aligned in such a way that we see them transit. Smaller planets in larger orbits are even less likely to be aligned in such a way that we can observe transits. For planets that do transit, astronomers can get valuable information about the planet's atmosphere, surface temperatures and size.

For most sun-like stars, an orbiting planet even as large as a brown dwarf will only cause an observed reduction in brightness of the star of a few percent or less during a transit. Like the radial velocity method, this method has a bias towards discovering large planets orbiting close to their stars, because larger planets block more light and transit more frequently so they are easier to detect. There is also a bias towards finding big planets around small stars. But at the extreme ends of the scale, planets can be almost as big as their stars! There's a lot of current interest in detecting planets around the smaller, cooler, late-spectral type stars such as M-dwarfs. These are just hot enough to sustain the hydrogen burning that distinguishes them from brown dwarfs. But very late-M dwarfs can be tiny, down to about 0.1 the radius of the Sun. At the other end of the scale, brown dwarfs and gas giant planets up to tens of times the mass of Jupiter are all approximately the same size: as large as or a little bit larger than Jupiter. So a gas giant transiting a late-M dwarf blocks a large percentage of the light from the star during a transit and in theory, there could be gas giant planets orbiting brown dwarfs which could be totally eclipsing!

Explore: Find out more by using Agent Exoplanet

Dr. Rachel Street Answers Common Questions About How Astronomers Use The Transit Method To Learn About Exoplanets

What information about a planet can you get by studying transits? Transiting planets are highly prized in exoplanet science because we find out so much more about them. When we discover a new planetary system by measuring its reflex motion (radial velocity), there's always some missing information. This technique can't measure the inclination of the planet's orbit relative to us, and this leads to an uncertainty over the planet's true mass. But if we see the distinctive dip in the lightcurve of a planetary transit, we know the orbit must be nearly edge-on relative to us. So straight away we can accurately measure the system's orbit and its physical properties. But when the planet transits, a small amount of light from the star passes through the atmosphere of the planet, which imprints its signature on the spectrum. With careful analysis, we can extract the spectrum of the planet's atmosphere, and this can tell us a lot about its chemical composition. Transiting planets also pass behind their host stars, and when we detect this in the infrared, we can measure the thermal emission of the planet at different wavelengths and reconstruct the structure of its atmosphere.

How can you tell if there are multiple planets around a star?
There are a couple of ways to tell if a star has more than one planet in its system. One way is to measure the orbital reflex motion of the star over a long period of time - either by radial velocities or astrometry. All planets in the system contribute to the overall detection signature. When the first planet is confirmed, we remove its signature from the measured signal and carefully examine what's left. If there's another planet's signature in the data, it will become clear. Another way is to monitor the star's light over a long period. There's a small chance that more than one planet will transit, and the Kepler mission has found a number of systems this way. We can also measure carefully the time of a series of transits of the same object, and look for any variation relative to the predicted time. If it's not transiting right on schedule, this points to the gravitational pull of another object in the system. In principle, this technique can detect objects even as small as moons!

How do you identify a planet transit from other reasons a star might temporarily dim?
A number of things can make a star appear to become briefly dimmer we call these phenomena "false positive" detections. So when we find a new transiting planet candidate we go to some lengths to check that it's definitely caused by a planet. Here are some of the most common false positives and how we can distinguish them: Eclipsing binary stars. Around 50% of all stars have another star as a companion and sometimes the orbit of the smaller of the two (the secondary) passes across the face of the primary just like in a transit (but it's called an eclipse when it's a star). Normally when this happens the depth of the eclipse is much deeper than a planetary transit because the star is much wider and covers more of the primary. But if the orbit causes the secondary to just barely graze the top of the primary, the depth of the eclipse can be similar to a transit. To rule this out, we look for signs of the additional light from the second star - a planet is much darker. In the spectrum from the object we look for periodic variations in the shape of the spectral lines during the transit. Photometrically, we also measure the eclipse depth of the planet through filters of different colors. Planet transits have virtually the same depth at all optical wavelengths, because the planet isn't contributing significantly to the overall light. But stars do, and differences in the colors of primary and secondary can cause the eclipse depths to vary. Another telltale sign of a stellar binary is a secondary eclipse, as the secondary goes behind the primary. A dark planet will not cause a secondary eclipse at optical wavelengths - it can only be detected this way in the infrared, and even then the signal is tiny. But a stellar secondary will show a detectable secondary eclipse.

Blended stellar binary/ multiple star systems.
Occasionally stars have more than one companion. The extra light from the other stars essentially "wash out" the depth of the eclipse, making it look more like a transit. In most cases, the tests described above can distinguish these cases. A much more common situation is that a binary star happens to appear to be close to another object, along the same line of sight in the sky rather than gravitationally bound. This can also wash out the transit. Again, the tests above come to the rescue, but we also try to observe transits from a telescope with better spacial resolution which can measure the light from the objects separately. If the stars are so close they cannot be separated completely, we will also measure the position of the "photocenter" during the transit - in a planetary transit, the center of the source of light should remain at the position of the primary star, but if the primary is blended with a nearby object, the photocenter can shift towards the neighboring object as light is blocked out in transit.

Stellar variability.
Stars sometimes vary in brightness all by themselves! Some stars pulsate, or have starspots, cooler and therefore darker regions on their surfaces. Pulsations make the star's light vary continuously in a distinctive way, so this is usually easy to spot. Starspots however, are carried across the face of the star as it rotates and could in principle cause a transit-like signature. Generally these are easy to distinguish though. In practice, most stars rotate more slowly than a typical planetary transit, so the timescale is wrong. Starspots also fail the test for different transit depths in different colors. And they are a temporary phenomena, usually dissipating over weeks or months.

What makes studying transits different from the other methods of exoplanet detection?
Transits can tell us so much more about the systems than anything else, but they are rare because they require chance orbital alignment with us. So we have to survey tens of thousands of stars to have a chance of finding just one, but they are worth the effort. The dependence on orbital alignment means that transits are most likely to happen in systems where the planet is close to its host star, so the technique preferentially discovers this type of planet system. The most scientifically valuable transiting planets are those orbiting bright stars because these are easiest to study. and it usually means that the stars are quite close to us. It's a way to discover our neighbors!


How do you calculate the radius of an exoplanet and a star?

Hi! I'm a high school student taking courses in Coursera related to Astrobiology. Currently, we're learning about exoplanets. I don't know how to get the radius of the exoplanet and the star to solve for the density of the exoplanet. I know there is some connection to the Transit method, which is the Transit Depth= radius of the exoplanet^2/radius of the star^2. There was a transit method graph and a radial velocity graph given. I've seen the Stefan-Boltzmann Law, but it needs the star's radius and the temperature (which weren't given). So far, I have data from the 2 graphs, the orbital period, the orbital distance/semimajor axis, velocity of both the exoplanet and the star, and the mass of both the exoplanet and the star. Thank you so much!

EDIT: The purpose of this is to find out if the given exoplanet can be habitable from just those two graphs :) Also, I hope this doesn't break the Homework rule. I'm also genuinely curious! (especially since I'm just learning Astronomy and would like to pursue it in the future)


The transit light curve

For an up to date version of this text please go here.

Introduction

The transit light curve gives an astronomer a wealth of information about the transiting planet as well as the star. It is only for transiting exoplanets that astronomers have been able to get direct estimates of the exoplanet mass and radius. With these parameters at hand astronomers are able to set the most fundamental constraints on models which reveal the physical nature of the exoplanet, such as its average density and surface gravity. As mentioned above the transit events do not just give information about the exoplanet, but quite often also information about the star. With telescopes capable of high precision photometry, transit curve anomalies can say something about the activity of the star. An example of this is when an exoplanet crosses star spots (Fig. 1) [source]. This can be seen in the light curve as a small increase in flux due to the light of a cooler part of the star being blocked out.

With a very high precision light curve with a high Signal to Noise (S/N), the light curve can also be used to infer the presence of other planets in the system. Perturbations in the timing of exoplanet transits may be used to infer the presence of satellites or additional planetary companions [source,source].

Fig. 1: Transit light curves of two transiting exoplanets, TrES-1 (top) and HD 209458 (bottom, offset by -0.007 for clarity). Since TrES-1 has a shorter orbital period and smaller size, the transit duration as well as the duration of ingress and egress are shorter. The hump in the TrES-1 data is likely due to the planet occulting one or more star spots on the surface of the star.

Theory

Kepler’s Third Law

From Newton’s second law of motion and Newton’s law of universal gravitation one can derive an elegant relationship between the semi-major axis (The longest diameter of an ellipse) of the orbit, a, and the period of the exoplanet. This law is known as Kepler’s 3 rd Law. Mathematically the law is written as:

Here is the gravitational constant and the semi major axis of the elliptic orbit. As the period, , is easily determined from observations and using the fact that in most cases the mass of the planet is much less than the mass of the star one can solve for the semi-major axis:

Having both the period and the semi-major axis one can estimate the orbital speed (assuming a circular orbit) to be:

Determining the radius of an exoplanet

The shape of a transit light curve gives astronomers a wealth of information about an exoplanet. One of the simplest things to estimate is the radius of the planet , determined by the amount of blocked star light. As the exoplanet transits in front of the host star, star light is blocked and a dip occurs in the transit light curve. The size of this dip in brightness is estimated by simply looking at the fraction of light that the planet blocks:

is the star flux whilst is the observed change of flux during the transit. This equation assumes the stellar disc has a uniform brightness. As we will see in the section about limb darkening, this is not the case, but as a first estimate this relationship works quite well. To determine an accurate value of the radius of the planet, , one has to fit transit curves (using analytic formulae [source]) which are subject to the estimates of the stars mass and radius (, ) and stellar limb-darkening coefficients.

What is truly special about this estimate is that we immediately have an idea of the size of the exoplanet in terms of the size of the host star. If the radius of the host star is known, one also knows the radius of the planet. For this to work we assume the exoplanet system is viewed from an interstellar distance so great that the distance to the exoplanet or host star can be considered equal.

Determining the transit duration

Once the radius of the star and thus the radius of the exoplanet is known, and having already measured the period and thus inferred the semi-major axis, it is possible to calculate the duration of the full transit . The full transit is measured as the duration of time when any part of the planet obscures the disc of the star. The figures and derivations are adopted from “Transiting Exoplanets“, by Carole A. Haswell.

The total transit duration is heavily dependent on the impact parameter which is defined as the sky-projected distance between the center of the stellar disc and the center of the planet disc at conjunction (The point in the orbit where two objects are most closely aligned, as viewed from Earth). In other words, the distance from the center of the planet to the center of the star at mid-transit as seen by the observer (Fig. 2). For a circular orbit it is mathematically written as:

Fig. 2: Trigonometry showing the impact parameter b

The total transit duration also depends on how the planet crosses the star. If the exoplanet crosses the center of the stellar disc (), the transit duration is the longest. For () the transit duration is shorter. With the help of Fig. 3 and using Pythagoras’s theorem:

Fig. 3: By use of Pythagoras theorem the length l can be expressed in terms of the impact parameter b and the radii of the star and the planet.

Fig. 4: During a transit the planet moves from point A to B on an orbit with inclination i. For an observer far away, the planet coveres the distance 2l. Assuming a circular orbit, the distance around a full orbit is 2*pi a where the planet moves along an arclength alpha a between points A and B. From the triangle formed between A, B and the center of the star, sin(alpha / 2) = l/a.

The length the planet travels across the disc of the star is as seen by the observer. Looking at Fig. 4 we see that the exoplanet moves from to around its orbit, creating an angle (measured in radians) with the center of the host star. With the assumption of a circular orbit, the distance around the entire orbit is , where is the radius of the orbit. The arclength between points and is and the distance along a straight line between and is .

From the triangle formed by , and the center of the star,

From the triangle formed by , and the center of the star,

giving us the full transit duration.

Determining the inclination of the orbit, i.

Radial velocity observations of the host star alone does not give enough information to be able to determine the mass of the exoplanet. Instead it gives a value of known as the minimum mass which is estimated assuming the stellar mass, , is known. During a transit event the orbital inclination, , can be measured directly, thus giving us an estimate of the exoplanet mass. This is done by studying the transit duration, , and ingress and egress times. A transiting exoplanet which does not pass across the center of the disc exactly (, ), will have a shorter transit but longer in- and egress times, if compared to a planet that goes through the center of the disc (, ). Thus the inclination of the orbit can be calculated using the shape of the transit itself together with the equations of Mandel and Agol (2002). Having an estimate of the mass and the radius of the exoplanet, the average density and surface gravity can be estimated, giving hints to the structure and composition of the exoplanet.


Contents

One example of a transit involves the motion of a planet between a terrestrial observer and the Sun. This can happen only with inferior planets, namely Mercury and Venus (see transit of Mercury and transit of Venus). However, because a transit is dependent on the point of observation, the Earth itself transits the Sun if observed from Mars. In the solar transit of the Moon captured during calibration of the STEREO B spacecraft's ultraviolet imaging, the Moon appears much smaller than it does when seen from Earth, because the spacecraft–Moon separation was several times greater than the Earth–Moon distance.

The term can also be used to describe the motion of a satellite across its parent planet, for instance one of the Galilean satellites (Io, Europa, Ganymede, Callisto) across Jupiter, as seen from Earth.

Although rare, cases where four bodies are lined up do happen. One of these events occurred on 27 June 1586, when Mercury transited the Sun as seen from Venus at the same time as a transit of Mercury from Saturn and a transit of Venus from Saturn. [ citation needed ]

Notable observations Edit

No missions were planned to coincide with the transit of Earth visible from Mars on 11 May 1984 and the Viking missions had been terminated a year previously. Consequently, the next opportunity to observe such an alignment will be in 2084.

On 21 December 2012, the Cassini–Huygens probe, in orbit around Saturn, observed the planet Venus transiting the Sun. [3]

On 3 June 2014, the Mars rover Curiosity observed the planet Mercury transiting the Sun, marking the first time a planetary transit has been observed from a celestial body besides Earth. [4]

Mutual planetary transits Edit

In rare cases, one planet can pass in front of another. If the nearer planet appears smaller than the more distant one, the event is called a mutual planetary transit.

Transit of Venus as seen from Earth, 2012

Io transits across Jupiter as seen by Cassini spacecraft

Mercury transiting the Sun, seen from Curiosity rover on Mars (3 June 2014).

The Moon transiting in front of Earth, seen by Deep Space Climate Observatory on 4 August 2015.

The transit method can be used to discover exoplanets. As a planet eclipses/transits its host star it will block a portion of the light from the star. If the planet transits in-between the star and the observer the change in light can be measured to construct a light curve. Light curves are measured with a charged-coupled device. The light curve of a star can disclose several physical characteristics of the planet and star, such as density. Multiple transit events must be measured to determine the characteristics which tend to occur at regular intervals. Multiple planets orbiting the same host star can cause transit-timing variations (TTV). TTV is caused by the gravitational forces of all orbiting bodies acting upon each other. The probability of seeing a transit from Earth is low, however. The probability is given by the following equation.

where Rstar and Rplanet are the radius of the star and planet, respectively, and a is the semi-major axis. Because of the low probability of a transit in any specific system, large selections of the sky must be regularly observed in order to see a transit. Hot Jupiters are more likely to be seen because of their larger radius and short semi-major axis. In order to find earth-sized planets, red dwarf stars are observed because of their small radius. Even though transiting has a low probability it has proven itself to be a good technique for discovering exoplanets.

In recent years, the discovery of extrasolar planets has prompted interest in the possibility of detecting their transits across their own stellar primaries. HD 209458b was the first such transiting planet to be detected.

The transit of celestial objects is one of the few key phenomena used today for the study of exoplanetary systems. Today, transit photometry is the leading form of exoplanet discovery. [5] As an exoplanet moves in front of its host star there is a dimming in the luminosity of the host star that can be measured. [6] Larger planets make the dip in luminosity more noticeable and easier to detect. Followup observations using other methods are often carried out to ensure it is a planet.

There are currently (December 2018) 2345 planets confirmed with Kepler light curves for stellar host. [7]

During a transit there are four "contacts", when the circumference of the small circle (small body disk) touches the circumference of the large circle (large body disk) at a single point. Historically, measuring the precise time of each point of contact was one of the most accurate ways to determine the positions of astronomical bodies. The contacts happen in the following order:

  • First contact: the smaller body is entirely outside the larger body, moving inward ("exterior ingress")
  • Second contact: the smaller body is entirely inside the larger body, moving further inward ("interior ingress")
  • Third contact: the smaller body is entirely inside the larger body, moving outward ("interior egress")
  • Fourth contact: the smaller body is entirely outside the larger body, moving outward ("exterior egress") [8]

A fifth named point is that of greatest transit, when the apparent centers of the two bodies are nearest to each other, halfway through the transit. [8]

Since transit photometry allows for scanning large celestial areas with a simple procedure, it has been the most popular and successful form of finding exoplanets in the past decade and includes many projects, some of which have already been retired, others in use today, and some in progress of being planned and created. The most successful projects include HATNet, KELT, Kepler, and WASP, and some new and developmental stage missions such as TESS, HATPI, and others which can be found among the List of Exoplanet Search Projects.

HATNet Edit

HATNet Project is a set of northern telescopes in Fred Lawrence Whipple Observatory, Arizona and Mauna Kea Observatories, HI, and southern telescopes around the globe, in Africa, Australia, and South America, under the HATSouth branch of the project. [9] These are small aperture telescopes, just like KELT, and look at a wide field which allows them to scan a large area of the sky for possible transiting planets. I addition, their multitude and spread around the world allows for 24/7 observation of the sky so that more short-period transits can be caught. [10]

A third sub-project, HATPI, is currently under construction and will survey most of the night sky seen from its location in Chile. [11]

KELT Edit

KELT is a terrestrial telescope mission designed to search for transiting systems of planets of magnitude 8<M<10. It began operation in October 2004 in Winer Observatory and has a southern companion telescope added in 2009. [12] KELT North observes "26-degree wide strip of sky that is overhead from North America during the year", while KELT South observes single target areas of the size 26 by 26 degrees. Both telescopes can detect and identify transit events as small as a 1% flux dip, which allows for detection of planetary systems similar to those in our planetary system. [13] [14]

Kepler / K2 Edit

The Kepler satellite served the Kepler mission between 7 March 2009 and 11 May 2013, where it observed one part of the sky in search of transiting planets within a 115 square degrees of the sky around the Cygnus, Lyra, and Draco constellations. [15] After that, the satellite continued operating until 15 November 2018, this time changing its field along the ecliptic to a new area roughly every 75 days due to reaction wheel failure. [16]

TESS Edit

TESS was launched on 18 April 2018, and is planned to survey most of the sky by observing it strips defined along the right ascension lines for 27 days each. Each area surveyed is 27 by 90 degrees. Because of the positioning of sections, the area near TESS's rotational axis will be surveyed for up to 1 year, allowing for the identification of planetary systems with longer orbital periods.


Calculate mass of exoplanet from transit method - Astronomy

Please be aware that due to the timing of the final exam this assignment will not have the usual 24hr grace period after the due date. The assignment will be unavailable and the answers will be revealed at 12:01am Monday.

Introduction

Planets that orbit stars other than our Sun are called exoplanets. As you have seen, detecting planets orbiting distant stars is no simple task and it’s taken us over 20 years to really become proficient at it. The discovery rate has increased tremendously since the discovery of the very first exoplanet, 51 Pegasi b in 1995 and has increased to nearly 3-4000 confirmed exoplanets and well over 3000 potential candidates that are currently being vetted. The plot below shows the number of exoplanet discoveries each year through April 2018. The different colors indicate the detection technique. The green coloring are exoplanets discovered using the Radial Velocity Method and the purple coloring are exoplanets discovered using the Transit Method.

The large increase in the discovery rate in 2014 and 2016 corresponds to the release of data from the Kepler space telescope mission. The Kepler mission, launched in 2009, continually monitored the brightness of over 145,000 stars in a single field of view a bit over 10 square degrees on the sky. The brightness of some of these monitored stars dimmed periodically as planets passed in front of the star, blocking out or eclipsing some of the starlight. We call these eclipses transit events. This is how the majority of exoplanets have been discovered to date. The image below shows how a transit looks in our own Solar System. Here, a series of images added together shows Venus transiting or passing between the Earth and the Sun, blocking some of the Sun’s light.

The amount of light that a planet blocks out, known as the transit depth, is related to how big the planets is. If the planet were the same size (had the same radius) as the star it would block out the star light completely when it passed precisely in front (between us and the star), and if the planet were the size of a piece of dust it wouldn’t block out much light at all as demonstrated in the figure below.

In this experiment you will use the light curves of two different exoplanets orbiting a Sun-like star. The light curves will allow you to determine the radius of the planet. This radius when combined with the mass from another method can be used to determine the density of the exoplanets. To obtain the radius of a planet we use the following formula (Equation 1):

Δ A B = ( R p l a n e t R s t a r ) 2

where Δ AB is the change in the apparent brightness of the parent star, R planet is the radius of the exoplanet and R star is the radius of the parent star. To simplify our calculations, we will assume the parent star is Sun-like. In other words, we’ll assume the star has the same mass and radius as the Sun so all of the dynamics in this system are the same as in our own Solar System..

Our goal is to determine the radius of each planet from the change in brightness of the parent star. Using the fact that the Sun has a radius that is 109 times larger than the radius of Earth, we can rearrange Equation 1 to yield the radius of the planet in units of Earth radii (R Earth ) like so (Equation 2):

R p l a n e t = 109 R E a r t h × Δ A B

Keep in mind that the units are R Earth so your final equation should look like R planet = #R Earth .

Assume we measured the brightness of a star about which two planets orbit. We found the following transit events for the two planets, which we will call Exoplanet A and Exoplanet B.

The values in the data table were determined for each Exoplanet in the above light curve. Use the values given in the data table as well as the rest of the information given here to find the missing values and answer the following questions. Keep in mind that there is an associated activity forum, Lesson 10 Activity Forum, for you to work with your classmates on these questions. As usual, do not give out the exact answers, but feel free to help your fellow students. You have only one attempt at this activity so take your time and work carefully.

Image 01 - Transit of Venus. Slovak Union of Amateur Astronomers VT-2004 Team, Jun 8, 2004.


Advantages:

One of the greatest advantages of Transit Photometry is the way it can provide accurate constraints on the size of detected planets. Obviously, this is based on the extent to which a star’s light curve changes as a result of a transit. Whereas a small planet will cause a subtle change in brightness, a larger planet will cause a more noticeable change.

When combined with the Radial Velocity method (which can determine the planet’s mass) one can determine the density of the planet. From this, astronomers are able to assess a planet’s physical structure and composition – i.e. determining if it is a gas giant or rocky planet. The planets that have been studied using both of these methods are by far the best-characterized of all known exoplanets.

In addition to revealing the diameter of planets, Transit Photometry can allow for a planet’s atmosphere to be investigated through spectroscopy. As light from the star passes through the planet’s atmosphere, the resulting spectra can be analyzed to determine what elements are present, thus providing clues as to the chemical composition of the atmosphere.

Artist’s impression of an extra-solar planet transiting its star. Credit: QUB Astrophysics Research Center

Last, but not least, the transit method can also reveal things about a planet’s temperature and radiation based on secondary eclipses (when the planet passes behind it’s sun). On this occasion, astronomers measure the star’s photometric intensity and then subtract it from measurements of the star’s intensity before the secondary eclipse. This allows for measurements of the planet’s temperature and can even determine the presence of clouds formations in the planet’s atmosphere.


Calculate mass of exoplanet from transit method - Astronomy

The Radial Velocity Equation in the Search for Exoplanets
( The Doppler Spectroscopy or Wobble Method )

"Raffiniert ist der Herr Gott, aber Boshaft ist er nicht ( God is clever, but not dishonest - God is subtle, but he is not malicious )", Princeton University’s Fine Hall,
carved over the fireplace in the Common Room with relativity equations as motif imprinted into the leaded glass windows - Albert Einstein ( 1879 - 1955 )

The problem is simply to identify other unseen exoplanets orbiting dimly distant host stars with the acknowledged goal of eventually determining other intelligent SETI life by searching out the bio - chemical "signatures" of life such as carbon, oxygen, phospherous and water molecules throughout the cosmos. But our immediate goal is simply to determine velocity and mass extant in such faintly distant binary, tertiary, quaternary, etc., systems. So we must first begin with the simplest of these, namely, the binary system of one planet as an orbiting companion to one other host star.

As primarily the only realistic tool available to astrophysicists to gauge the "wobbling" light spectrum emanating from a distant host star, binary to an orbiting yet invisible planet gravitationally perturbing the host star, the relativistic red - shift />using doppler spectroscopy to plot the line-of-sight, radial velocity data points for the eventual determination of time period, velocity, mass, and orbital eccentricity for both the host star and its companion binary planet, has been a highly successful method among others. That is, since measurement of distances are not sufficiently precise enough, however the relativistic red - shift />providing velocities along the observer's line-of-sight is fairly well accurate. Additional observations of the host star as regards brightness and color will also provide augmented estimates for the host star's mass and radial distance. It's main drawback is that it's primarily limited to line-of-sight, eclipsing binary, tertiary, etc. systems.

All of this and still yet more, including the chemical compositions of both host star and orbiting planet coming from the light spectrum of the binary system itself, is quite an amazing feat for mathematical physics! As it should really be termed the "Philosophy of Light"!


the common center of mass, and hence motion, is inside the larger host star at the red x-mark

with a line-of-sight, edge-on eclipsing binary system, it is nearly impossible to know the orbital eccentricity - i.e., near circular or elliptical? also the host star will dim when behind the eclipsing exoplanet.

yes, a binary system. however now imagine this as a larger black hole host to a smaller binary companion star, planet, etc.

An Abreviated List of the Mathematical Physics Tools Employed

The Geometry of Elliptical Orbits

The Radial Velocity Equation - Preliminary

Area of One Orbital Revolution

The Radial Velocity Equation - Almost Final Derivation
( this being highly theoretical, not yet practical ! )

Deriving the Velocity Data Points

§ Deriving the velocity data points

The Radial Velocity Semi - Amplitude K of a Wobbling Host Star to a Nearly Invisible Exoplanet
( plotting host star velocity vs. time by a gravitationally effecting exoplanet )

note: is the doppler radial velocity semi - amplitude - i.e., it is both the spectroscopic doppler velocity as well as the semi - amplitude of either the host star or orbiting planet plotted along a sine curve of doppler measured light spectrum frequencies!

The Final Derivation of Phase Velocity

Assuming that the Host Star is Circularly Perturbed

If it is assumed at the outset that the host star is perturbed strictly in a circular fashion without consideration of eccentricity, then the equation for radial velocity is reduced down to a much, much simpler derivation:

The Philosophy of Light
( or how the human mind overcomes narrow solipsistic naïve reality )

Finally, the electromagnetic light spectrum combined with mathematical physics, a creation of the human mind, indeed allows us to pierce the dark starlite veil of the cosmos so that perhaps eventually we can as a human race intelligently communicate with other ETs in the cosmos. And all of this is totally made possible by a speculative sort of "philosophy of light" to be able to imagine beyond our immediate and extremely naïve sense of sight!

Radial Velocity Simulator

source: http://astro.unl.edu
source: http://astro.unl.edu/classaction/animations/extrasolarplanets/radialvelocitysimulator.html

Planet X - Beyond Pluto: 2012 VP113 a new 9th planet?


This animation shows the motion of object 2012 VP113 over 5 hours as recorded in its discovery images. The field of view is about 1 arc-minute wide. This object is currently about 83 astronomical units (7.7 billion miles) from the Sun — nearly as close as it ever gets. By Scott S. Sheppard / Carnegie Inst. of Science.

ESOcast 87: Planet found around closest Star Proxima Centauri to Earth


Proxima b is 1.3 light years away is 1.3 times size of Eart orbits Proxima Centauri star every 11.2 days in a habitable zone for water and orbits closer to its star than Mercury orbits to our Sun being only 5% of the distance between Earth and the Sun.


Calculate mass of exoplanet from transit method - Astronomy

The graph plots the un-calibrated signal minus the average signal from the instrument. When a planet passes in front of the star (making a transit across the star), the total light output drops accordingly. This causes the larger observed dips in the graph.

Note #1: If desirable, the plot may be printed so that the data may be measured more accurately. Simply click on the graph and then print the resulting web page.

Note #2: If no transits are observable in the data, then go beack to the previous page and select a different star.

I. Calculating Orbital Information from the Observational Data

A. Period of the Exoplanet

From the graph above, calculate the average time between transits of the planet across the star face. (Find the day of the first and last transit and divide by the number of time intervals between these transits.) Then enter this period in days in the formula below.

B. Distance of the Exoplanet from Its Parent Star

The third law of planetary motion derived by Johannes Kepler (and modified by Isaac Newton) connects the orbital period of a planet in our solar system, the mass of the Sun and the planet's average distance from the Sun.

Astronomers have been able to estimate the mass of a star if it is a main sequence star (on the H-R diagram) and if its spectral type is known. See the table.

Stellar Masses (in units of solar masses)

Locate the spectral type for this star and read off its mass. Then enter this number in the appropriate empty box below.

Kepler's third law can be written as:

p 2 M = a 3

  • p is the orbital period of the planet in units of years,
  • M is the mass of the star in units of solar masses,
  • a is the average distance the planet is from the star in Astronomical Units.

Explanations

Notes on the Photometric Observations

  • If no significant dips in the signal are observable, then several other possibilities may be at work.
    • There might not be a planet orbiting this star.
    • The planet may be too small or the star too far away for instruments to detect the effect of the planet's transits.
    • The planet mgith be too far away from the star to have made a transit during the length of time the instrument was collecting data.
    • Maybe no planet passes directly in front of this star, even if it has one or more planets orbiting it.

    Notes on Kepler's Third Law

    • While Kepler's third law was derived from data for planets in our solar system it has been found to provide a good description of a planets orbit about any star, if the mass of that planet is small compared to the mass of its star. Essentially all exoplanets discovered to date fit this criteria, and the Earth-size ones which the Kepler Mission will hunt for will definitely match this assumption.
    • The easiest units for mass in this equation are solar masses, where the mass of the Sun is equal to 1 solar mass
    • The average distance between a planet and its parent star is the semi-major axis of the planet's orbit about the star and should be given in Astronomical Units (AU).
    • Use the appropriate buttons. Do not press the "Enter" key.
    • Enter or change numbers only in the appropriate boxes.
    • Complete all "calculations" in order from the top of the page to the bottom.
    • Return to the main Finding Exoplanets page to learn more about the search for planets outside of our solar system.

    Simulation Authors: Richard L. Bowman (Bridgewater College) and David Koch (Kepler Mission)

    Maintained by: Richard L. Bowman (2002-2011 last updated: 14-Sep-11)



Comments:

  1. Carmi

    Completely I share your opinion. It seems to me it is excellent idea. Completely with you I will agree.

  2. Thurstan

    I think no.

  3. Devine

    You can't even find fault!

  4. Nikinos

    What an interesting idea ..

  5. Aloeus

    the result will be good

  6. Suzu

    This argument only incomparably



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