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What is the relation between the wavelength and the radial velocity? I have a data: wavelength and intensity and I would like to plot intensity with respect to radial velocity - similarly as in the picture (source)

EDIT

I have a file of alpha H with 2 columns - wavelength (in angstrom) and intensity (normalized for 1) and I would like to plot intensity with respect to radial velocity. I use in gnuplot:

c = 299792458 lambda_0 = 6562.817*10**(-10) plot 'file.asc' using (c*(\$1*10**(-10)-lambda_0)/lambda_0):2

Why I get the wrong range on the x-axis?

$$v_r = cleft( frac{lambda - lambda_0}{lambda_0} ight),$$ where $$lambda$$ is the observed wavelength and $$lambda_0$$ is the wavelength at rest. This gives an equivalent velocity with respect to the central wavelength of the line.

In your plot, what is shown is a line profile where the wavelength separation from $$lambda_0$$ for that line (or from the measured centre of the line, I can't tell which) has been converted into a velocity.

Note that it makes little sense to plot more than a single line profile in this way.

## A connection between radial velocity and distance

If we send the light from a star or galaxy through a prism, it breaks up into a spectrum, with short wavelength (blue light) at one end, and long wavelengths (red light) at the other:

Superimposed on the spectrum of a star (or galaxy) are a series of dark lines. These absorption lines mark wavelengths at which gases in the star's outer atmosphere have absorbed light. Different gases absorb light of different wavelengths. In fact, one can identify particular elements in the spectrum of a star (or galaxy) by the wavelengths of light which are absorbed.

• material moves towards us: shift to shorter wavelengths (blue)
• material moves away from us: shift to longer wavelengths (red)

This is called the Doppler effect. The size of the shift in the lines depends on the size of the radial velocity. For relatively small velocities,

Thus, for the calcium H and K lines,

#### Motions of stars and galaxies -- the early days

We can measure the speed of stars in our own Milky Way, as they orbit around the center of the galaxy. Most of the nearby stars follow paths which are similar to the Sun's. Like cars all going the same way along the highway, they appear to move past us relatively slowly: we measure speeds of around 30 to 50 km/sec, relative to the Sun. Occasionally, we find a star with a much higher speed: up to 220 km/sec. These "speeders" are not part of the disk, but part of our galaxy's halo. They fly far above and far below the plane of the disk, and zoom through the disk at very high speeds.

In the early twentieth century, the construction of big telescopes at Mount Wilson (the 60-inch and 100-inch) allowed astronomers to determine the motions of galaxies for the first time.

Measuring these radial velocities wasn't easy. Even nearby galaxies are relatively faint, and, unlike stars, their light is spread out over an extended area. Milton Humason used the 100-inch telescope on Mt. Wilson to obtain a spectrum of NGC 7619 in 1929, which yielded the largest radial velocity ever seen (up to that time): 3779 km/sec away from us. His paper describes the difficulties in an understated manner:

#### Hubble's diagram

Edwin Hubble studied galaxies, using the biggest telescopes in the world (then at Mount Wilson). He photographed nearby galaxies and tried to find ways to determine their distances. In the nineteen-twenties, his best method (and it wasn't a great one) was based on the brightest stars in a galaxy. He assumed that the brightest stars were all of the same absolute luminosity, or standard candles by comparing the apparent brightness of the brightest stars in different galaxies, he could calculate the distances.

When Hubble plotted the velocities of nearby galaxies against their distances, he found a strong correlation:

Hubble used the slope of the line in this diagram to determine what has become known as the Hubble constant:

More recent measurements, using the Surface Brightness Fluctuation technique, show a similar correlation, but yield a much smaller value of H0.

#### Indirect evidence for distance/velocity connection far away

When we look at distant galaxies, we can see the absorption lines shift by very large amounts:

This tells us that the galaxies are moving away from us at very large speeds. Unfortunately, at these very large distances, we can't measure distances accurately.

But if we are willing to accept a few extra assumptions, we can look much farther into the universe to see if the radial velocity of objects continues to increase linearly with their (assumed) distance.

If all Type Ia supernovae have the same absolute luminosity, and if the radial velocities of their host galaxies rise linearly with distance, then a plot of their magnitude (logarithm of brightness) versus (logarithm of) redshift should be a straight line. A big survey of supernovae by astronomers in Chile (Hamuy et al., AJ 112, 2398, 1996) shows the expected correlation:

If all Brightest Cluster Galaxies have the same absolute luminosity, and if the radial velocity of a galaxy rises linearly with distance, then a plot of their magnitude (logarithm of brightness) versus (logarithm of) redshift should be a straight line. Recent work by Collins and Mann (MNRAS 297, 128, 1998) shows this to be the case, at least roughly:

Astronomers often use radial velocity as a stand-in for distance. For example, you may hear an astronomer say,

"Hey, Jane, how distant is the Coma Cluster?"

"Rats, that's too far for us to use the TRGB technique . "

But there's another quantity, related to radial velocity, which astronomers use even more frequently: redshift. We use it because it is what we actually measure from spectra: the factor by which absorption or emission lines are shifted from their rest values.

So, for example, if we know that the K-line of calcium has a rest wavelength of 393.3 nanometers, and we observe it in the spectrum of a faint galaxy at a wavelength of 550.0 nanometers, then

For very distant objects, moving at very high speeds away from us, we sometimes see redshifts greater than one. How is this possible? Is the galaxy moving faster than the speed of light?

No. At very high speeds, the Doppler shift formula changes slightly. The low-speed formula can be re-arranged to give the radial velocity for some observed shift in wavelength:

For large shifts, one must include the effects of special relativity, which result in

So, for example, the most distant quasar yet measured has a spectrum which shows the Lyman-alpha emission line (normally at 121.6 nm = 1216 Angstroms) at a shifted wavelength of about 8300 Angstroms:

The quasar thus has a redshift of

And its radial velocity is therefore

For further information, see

• A scientific biography of Edwin Hubble written by Allan Sandage, his successor at Mount Palomar.
• An experiment in which you (yes, you) can measure the Hubble constant using galaxies in the Hubble Deep Field. This is way cool, but does require Java and may be a bit slow, since it's hosted in England. Thanks to Ian Smail, at the University of Dunham.
• The LCRS galaxy redshift survey has a nice picture showing how astronomers now use the redshift of a galaxy as a stand-in for its distance.

## Wavelength and radial velocity - 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!

source: http://astro.unl.edu

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.

The component of velocity along the line of sight to the observer. Objects with a negative radial velocity are travelling towards the observer whereas those with a positive radial velocity are moving away.

radial velocity, in astronomy, the speed with which a star moves toward or away from the sun. It is determined from the red or blue shift in the star's spectrum.

Radial velocity follow-up of GJ1132 with HARPS★,★★
A precise mass for planet b and the discovery of a second planet .

Method
This method uses the fact that if a star has a planet (or planets) around it, it is not strictly correct to say that the planet orbits the star. Instead, the planet and the star orbit their common center of mass.

(measured in km/s) is the velocity along the line of sight away from (considered a positive velocity) or toward (negative velocity) the observer.

Method
Detecting Other Worlds: The Wobble Method .

is the velocity with which an object travels towards or away from an observer. It is measured by examining the Doppler shift of features in the spectrum of astronomical objects.

The first exoplanets were discovered by what is called the "wobble." This sounds low tech, but this is very significant (the "wobble" is also associated with astrometric detection).

: The velocity of an celestial object along the line of sight to the observer.
Right Ascension (RA): It is the celestial equivalent of terrestrial longitude in the equatorial coordinate system. It divides the celestial equator into 24 hours, each of 60 minutes.

When we measure the spectrum of a star, we determine the wavelength of each of its lines. If the star is not moving with respect to the Sun, then the wavelength corresponding to each element will be the same as those we measure in a laboratory here on Earth.

The movement of an object either towards or away from a stationary observer.

of an object with respect to a given point is the rate of change of the distance between the object and the point.

- Acceleration of an object going away from or headed towards an observer.
Radiant - Location in the sky where meteors belonging to a meteor shower appear to come from. 2. Very bright and shining.
Radiation - Electromagnetic waves as it relates to astronomy.

Normally the component of the velocity of a celestial object along the line of sight from the Earth. It is positive when the object is moving away from the Earth, and negative when it is moving towards us.

of a body can be determined by the Doppler shift of its spectral lines
Radiant - The point in the sky from which the meteors in a meteor shower seem to originate .

. The movement of a celestial body either away from (a positive value) or toward (a negative value) the observer.
Radiant. The point in the sky where meteors of a given shower seem to originate or radiate from.

method of exoplanet detection uses the fact that a star-planet pair orbit around a common center of gravity, called the barycenter. The text states that it is possible to measure a star's motion as little as 3 m s-1.

method
Transit method
Primary and secondary eclipse
Transit timing variation method
Gravitational microlensing
Pulsar timing .

variations would suggest that star B has a substellar companion of about eight times the mass of Jupiter in an orbit of about 30 to 100 years to complete. However, the highly elliptical orbit of the binary pair which brings stars A and B as close together as 6.

, the speed of light, observed and rest wavelengths
Definition of an element
number of protons and neutrons .

Component of velocity directed along the line of sight. Radial velocities are found from the Doppler effect, in which the spectrum lines from an approaching body are shifted to shorter wavelengths ('to the blue') and those from a receding body to longer wavelengths ('to the red').

plot, spectrum and Earth-view of spectroscopic binary system.
a) Which star is more massive?
b) Explain why the spectral lines periodically split then move together in a system such as this.
c) Why do the stars have different ranges in their radial velocities?

: The velocity of a star towards or away from Earth.
Redshift: When light has been shifted towards longer, redder wavelengths due to an object moving away from the observer.
Relative Magnitude: See "Magnitude, Apparent." .

or Doppler method
As a planet orbits a star, the star also moves in its own small orbit around the system's center of mass.

- (n.)
The velocity of an object along a line (the radius) joining the object and the observer the component of velocity toward or away from the observer.

: the rate of change of the distance to an object.

Refraction, astronomical: the change in direction of travel (bending) of a light ray as it passes obliquely through the atmosphere. As a result of refraction the observed altitude of a celestial object is greater than its geometric altitude.

The velocity component along the line of sight toward or away from an observer. Recession is positive approach is negative.

of a galaxy in a different mathematical manner, like this: .

technique relies on the fact that a planet's gravity causes tiny wobbles in the orbit of its star. The technique is based on the fact that not only does the star's gravity affect any orbiting planets, but those planets can also affect the star, albeit to a much lesser degree.

method),
study the position of a star and search for periodic changes in its position (the astrometric method), .

, and _____ must all be measured. (Hint) .

technique has been the most successful. Its first catch came in 1995, when Swiss astronomers Michel Mayor and Didier Queloz discovered a planet orbiting the star 51 Pegasi. The planet, which is about half the mass of Jupiter, takes only 4.2 days to orbit the star.

In radar, that vector component of the velocity of a moving target that is directed away from or toward the ground station. radian The angle subtended at the center of a circle by an arc equal in length to a radius of the circle.

of around 200 km/s, making it one of the fastest runaway stars in our Milky Way Galaxy.
This Hubble image shows vast arcs of glowing gas around WR 124, which are resolved into filamentary, chaotic substructures.

rotational velocity If you examine an object's spectrum, you will often see that the star is redder or bluer than you would expect. This is caused by the Doppler shift: the object is moving toward or away from the Earth.

. Red dwarf star The smallest and dimmest stars.

: The movement of a celestial body toward or away from an observer.
RADIAN: A unit of angle equal to about 57 degrees. The length along the arc of a circle covering by one radian is equal to the radius of the circle. The complete angle around the circle (360 degrees) is equal to 2 pi radians.

Current Exoplanet search strategies have not yet found Earth-mass planets, for a number of reasons:

(Doppler Wobble) Method Most sensitive to massive planets close to their parent stars Required sensitivity to find Earths in the Habitable Zone is the ability to measure speeds of a few .

Periodic expansion and contraction of a star that may be merely an optical effect of recession. [A84]

Method - Another name for the Wobble Method and Doppler Spectroscopy.
Radiation - A catch-all term for any fast-moving particles that are emitted from an object. Most radiation is in the form of light (electromagnetic radiation) or subatomic particles (nuclear radiation).

- the direction toward or away a star is moving from us - is 24 km/sec for Albireo A and 18.8 km/sec for Albireo B. Both are heading in our direction, but at different speeds.

(These are not true B-V colors, but show only relative change.) The bottom graph shows the "

" of the star, that is, how fast the star appears to be moving along the line of sight as determined from the Doppler shift (positive values indicating recession).

This evidence comes from the gravitational perturbations exerted on the star by the unseen companion planet that can be exposed by very accurate measurement of the

of the star (see the related discussion of detecting unseen companions in binary star systems).

We can decompose kinetic energy (½mv2) into two pieces by using the Pythagorean theorem to decompose v2 into the sum of two terms:

squared (vr2) and the component of velocity perpendicular to r ("transverse" velocity) squared.

Using the world-renowned HARPS (High Accuracy

Planet Searcher) spectrograph on the 3.6-metre telescope at the European Southern Observatory in Chile, a team led by Garik Israelian of the Institutos de Astrofisica de Canarias in Tenerife sampled 500 stars with ages between six and nine billion years.

The age of the Universe is determined from its expansion rate: the Hubble constant, which is the ratio of the

is easy to measure, but the distances are not.

The innovation, called the 6dF instrument, is being used by a multinational consortium, the

Experiment (RAVE), to measure the radial velocities of more than half a million stars. It is mounted on the Australian National University's UK Schmidt Telescope at Siding Spring in New South Wales.

Both discoveries were made using the "

" technique, in which a planet's gravitational tug is detected by the wobble it produces in the parent star. Butler, Marcy and collaborators, including Dr. Deborah Fischer of San Francisco State University and Dr.

In fact, the large possibility that a planet is orbiting Barnard's Star is largely due only to the extreme accuracy of the HARPS (High Accuracy

Planet Searcher) spectrograph. The instrument is able to measure deviations in a star's radial motion that are as small as 3.

The spectrometer allows precise measurements of a star's

The first is the differing calculated velocities for cluster and nebula (the

of NGC 2438 is about 77 km/sec recession, which is 43 km/sec different from the cluster's value). The second is the advanced age of planetary nebulae.

A planet was tentatively discovered orbiting 51 Pegasi by a wobble in the star's

, monitored by looking at 5000 lines in the visible spectrum. The wobble could be caused by a 4.2-day period, an orbital radius 1/6 of Mercury, and a mass at least half that of Jupiter.

between the source and observer.
c is the velocity of light, 300,000 km/s.
&lambdao is the wavelength observed from the star and &lambdae is the wavelength in the lab.

imaging and spectroscopy, the physical properties of sources discovered at nonvisible wavelengths determining the interior structures of stars through long-term programs of spectroscopic monitoring of stellar oscillations searching for planetary systems and subsolar mass objects by means of long-term

Radial motion is motion towards or away from the observer.

can be determined using the doppler shift. Motion towards the observer shifts spectral lines towards the blue motion away from the observer shifts spectral lines towards the red.
.

WR 124 is one of the fastest known runaway stars in the Milky Way, with a

of about 200 km/s. The star was discovered by the American astronomer and spectroscopy pioneer Paul W. Merrill in 1938. It is classified as an eruptive variable with a range of 0.08 magnitudes.

A great deal of information can be extracted by detailed examination of spectral lines, such as redshift,

, expansion velocity, temperature, and much more. This section will grow as more authors cover particular spectral methods that appear in the literature.

Vesto M. Slipher
1875-1969
American
first to measure the

The star, designated C2306265-085103 by the

Experiment (RAVE) survey of 33 A.T., is a member of the so-called "Aquarius Stream" of stars that marks the demise of a dwarf galaxy whose core has been identified as the Omega Centauri globular cluster.

In analogy to a stop sign increasing in angular size as you approach it on the road, the angular size of the cluster will increase or decrease as well this, combined with the

of the cluster, can be used to estimate its distance.

The star has a surface gravity of around 3.8 cgs and it is also hotter than our sun, having surface temperatures of around 15.540 K. This is around 2.6 times hotter than our Sun. The

of Alkaid has been estimated to be at around -10.9 km / -6.7 mi per second.

In summary, we have learned how astronomers measure the brightness of stars, and how they determine their surface temperatures using both color indices and spectral class. The doppler shift in the spectral lines gives a star's

Relativistic Doppler shift Wavelength shift from the

of a source as calculated in special relativity, so that very large red shifts do not imply that the source moves faster than light. Relativity Two theories proposed by A.

## Motion Affects Waves

In 1842, Christian Doppler first measured the effect of motion on waves by hiring a group of musicians to play on an open railroad car as it was moving along the track. He then applied what he learned to all waves, including light, and pointed out that if a light source is approaching or receding from the observer, the light waves will be, respectively, crowded more closely together or spread out. The general principle, now known as the Doppler effect, is illustrated in Figure 1.

Figure 1: Doppler Effect. (a) A source, S, makes waves whose numbered crests (1, 2, 3, and 4) wash over a stationary observer. (b) The source S now moves toward observer A and away from observer C. Wave crest 1 was emitted when the source was at position S4, crest 2 at position S2, and so forth. Observer A sees waves compressed by this motion and sees a blueshift (if the waves are light). Observer C sees the waves stretched out by the motion and sees a redshift. Observer B, whose line of sight is perpendicular to the source’s motion, sees no change in the waves (and feels left out).

In Figure 1a, the light source (S) is at rest with respect to the observer. The source gives off a series of waves, whose crests we have labeled 1, 2, 3, and 4. The light waves spread out evenly in all directions, like the ripples from a splash in a pond. The crests are separated by a distance, λ, where λ is the wavelength. The observer, who happens to be located in the direction of the bottom of the image, sees the light waves coming nice and evenly, one wavelength apart. Observers located anywhere else would see the same thing.

On the other hand, if the source of light is moving with respect to the observer, as seen in Figure 2b, the situation is more complicated. Between the time one crest is emitted and the next one is ready to come out, the source has moved a bit, toward the bottom of the page. From the point of view of observer A, this motion of the source has decreased the distance between crests—it’s squeezing the crests together, this observer might say.

In Figure 2b, we show the situation from the perspective of three observers. The source is seen in four positions, S1, S2, S3, and S4, each corresponding to the emission of one wave crest. To observer A, the waves seem to follow one another more closely, at a decreased wavelength and thus increased frequency. (Remember, all light waves travel at the speed of light through empty space, no matter what. This means that motion cannot affect the speed, but only the wavelength and the frequency. As the wavelength decreases, the frequency must increase. If the waves are shorter, more will be able to move by during each second.)

The situation is not the same for other observers. Let’s look at the situation from the point of view of observer C, located opposite observer A in the figure. For her, the source is moving away from her location. As a result, the waves are not squeezed together but instead are spread out by the motion of the source. The crests arrive with an increased wavelength and decreased frequency. To observer B, in a direction at right angles to the motion of the source, no effect is observed. The wavelength and frequency remain the same as they were in part (a) of the figure.

We can see from this illustration that the Doppler effect is produced only by a motion toward or away from the observer, a motion called radial velocity. Sideways motion does not produce such an effect. Observers between A and B would observe some shortening of the light waves for that part of the motion of the source that is along their line of sight. Observers between B and C would observe lengthening of the light waves that are along their line of sight.

You may have heard the Doppler effect with sound waves. When a train whistle or police siren approaches you and then moves away, you will notice a decrease in the pitch (which is how human senses interpret sound wave frequency) of the sound waves. Compared to the waves at rest, they have changed from slightly more frequent when coming toward you, to slightly less frequent when moving away from you.

A nice example of this change in the sound of a train whistle can be heard at the end of the classic Beach Boys song “Caroline, No” on their album Pet Sounds. To hear this sound, watch this video of the song. The sound of the train begins at approximately 2:20.

### The Basic of Star’s Spectroscopy

Spectroscopy is a branch study in astronomy that focus on astronomical objects’ spectrum. From the spectrum, we can get informations, such as its temperatures, chemical compositions, movement speed, etc. That’s why spectroscopy can be considered as one of the fundamental field in astronomy. The spectrum of a star (or any other astronomical object) is acquired by using an instrument called spectrograph.

One of the fundamental law in spectroscopy is Kirchoff Law (1859) which stated that:

1. If a liquid or high pressure gas is ignited, they will emit energy in all wavelength which will produce a continuous spectrum .
2. If a low temperature gas is ignited, it will only emit energy in certain range wavelength and produce spectrum which have a dark background and some bright lines. That kind of spectrum is called the emission spectrum. The wavelength of each bright lines are the precise indicator of what gas that produce them. So, the same gas will produce bright lines in certain exact wavelength.
3. If a white light (which is a equal mixture of all colors) is passed through a cool low temperature gas, the gas will absorb energy at certain wavelength. The result spectrum will be continuous spectrum as the background with some dark lines in certain exact wavelength. The dark lines called absorption lines and that kind of spectrum is called the absorption spectrum. The wavelength of each dark lines are the precise indicator of what gas that produce them. So, the same gas will produce dark lines in certain exact wavelength.

Balmer Series
Switzerland scientist, Balmer, state a series equation to predict the wavelength of the absorption lines of hydrogen gas. The equation is widely known as Balmer series equation.

with : λ : the wavelength of the absorption lines [cm]
RH : Rydberg constant (= 109678 )

Planck postulates that light is radiated in the form of small discrete package called quantum. This theory is the foundation of the birth of a new field in physics called quantum physics.

Planck state that energy of each photon

h : Planck’s constant (h = 6,63 x 10^-34 J.s)
f : frequency of the photon [Hz]
c = speed of light (= 3.10^5 km/s)
λ = photon’s wavelength

Star’s spectrum
Star’s spectrum pattern is wide in variety. In 1863, an astronomer called Angelo Secchi classified star’s spectrum in 4 groups based on the similarities of its’ absorption lines.

Miss A. Maury from Harvard Observatory establish another way to classify star’s spectrum and it was revised by Miss Annie J. Cannon. Miss Cannon’s classification is the most widely adopted today.

Table 1 : Resume of the classification of star’s spectrum (to remember it use the donkey bridge : O h B e A F ine G irl (or G uy), K iss M e ). (you can click the figure to get bigger and clearer version of the table above .

Sub-classification of star’s spectrum
Star’s spectrum classification O, B, A, F, G, K, M is divided again to several sub-classes :
B0, B1, B2, B3, . . . . . . . . ., B9
A0, A1, A2, A3, . . . . . . . . ., A9
F0, F1, F2, F3, . . . . . . . . . ., F9

Bigger number represent lower temperature! The use of this sub-class is to narrow the specification’s range and become more precise.
(for further information, check this site.)

M-K Classification (Star’s Luminosity Class)

Stars with same certain spectrum’s class is found to have different luminosities . In 1913, Adam dan Kohlscutter from Mount Wilson Observatory showed that the width of spectrum’s lines can be used to estimate star’s luminosity.
Based on these facts. in 1943 Morgan and Keenan from Yerkes Observatory divided stars to several luminosity class as shown in the table below.

Class 1b

Class II

Class III

Class IV

Class V

Morgan Keenan’s Luminosity Class (M-K class) is sketched in a Hertzprung-Russell diagram (H-R diagram) below.

Now, star’s classifications use the combination of spectrum class and luminosity class. For example : A G2 V star is a main sequence star that belongs to spectrum class G2

Star’s motion
Contrary to widely beliefs that star isn’t moving in space, star DO move in space. However, the movement of stars is hard to track. Beause of its immense distance, the movement of star only produce extremely small apparent movement in sky. We have to wait several years (or decades!) to track star’s movement in sky. Warning : the star’s movement that is discussed above is not the apparent daily motion of the star !

The star’s angular motion of a star is called proper motion ( μ ). Proper motion is usually measured in arc-second per year. Star with biggest proper motion is Barnard Star with μ = 10”,25 per year (In 180 years, this star will (only) move in extent as full Moon’s disk).

Relationship between tangential velocity (Vt) and the proper motion (μ):

Vt = tangential speed of the star [km/s]

μ = proper motion of the star [“/ year]

the above equation also can be stated as :

with p is the parallax of the star (in arc second).

The proper motion is measured by two quantities: the position angle and the proper motion itself. The first quantity indicates the direction of the proper motion on the celestial sphere (with 0 degrees meaning the motion due north, 90 degrees due east, and so on), and the second quantity gives the motion’s magnitude, in seconds of arc per year.

The equations used to find the quantity of star’s proper motion are :

with :
μα = proper motion in right ascension
μδ = proper motion in declination
μα and μδ is measurable –> μ and θ can be determined.

Beside proper motion, information about star’s motion can be obtained from its radial motion , which is the component of star’s motion that lies parallel to our line of sight.
Radial velocity (Vr) can be measured by its spectrum lines that shift ( Doppler shift ). For star which radial velocity (Vr) is significant compared to the speed of light:

For Vr being much smaller compared to the speed of light (c), the equation can be simplified to:

with :
Δλ = the difference between static wavelength (λo) and observed wavelength (λ). [Å or nm]
λo = static wavelength. [Å or nm]
c = speed of light (300.000 km/s )

Now, we are able to calculate Vt and Vr as discussed above and we will be able to calculate star’s true motion ( linear motion ):

## Technology for the Precision Radial Velocity Measurement Technique

Astronomical spectrographs have proven to be powerful tools for exoplanet searches. When a star experiences periodic motion due to the gravitational pull of an orbiting planet, its spectrum is Doppler-modulated in time. This is the basis for the Precision Radial Velocity (PRV) method, one of the first and most efficient techniques for detecting and characterizing exoplanets. Since spectrographs have their own drifts which must be separated from the periodic Doppler shift, a stable reference is always needed for calibration. Optical Frequency Combs (OFCs) and line-referenced etalons are capable of providing the instrument precision needed for detecting and characterizing Earth-like planets in the Habitable Zone of their Sun-like host stars. While “stellar jitter” (a star’s photospheric velocity contribution to the RV signal) is unavoidable, the contribution to the error budget from Earth’s atmosphere would be eliminated in future space missions. Thus, there is a need to develop robust spectral references with Size, Weight and Power (SWaP) suitable for space qualified operation to calibrate the next generation of high-resolution spectrographs with precision corresponding to <

1 cm/s over multiple years of observations.

This subtopic solicits proposals to develop cost effective component and subsystem technology for low SWaP, long-lived, robust implementation of radial velocity measurement instruments both on the ground and in space. Research areas of interest include but are not limited to:

• Integrated photonic spectrographs
• PRV spectrograph calibration sources
• High efficiency photonic lanterns
• Advanced fiber scrambling techniques for modal noise reduction
• Software for advanced statistical techniques to mitigate effects of telluric absorption and stellar jitter on RV precision and accuracy

• Fischer et al. (2016) State of the Field: Extreme Precision Radial Velocities http://adsabs.harvard.edu/abs/2016PASP..128f6001
• Plavchan et al. (2015) Radial Velocity Prospects Current and Future: A White Paper Report prepared by the Study Analysis Group 8 for the Exoplanet Program Analysis Group (ExoPAG) http://adsabs.harvard.edu/abs/2015arXiv150301770P
• Plavchan et al. (2019) EarthFinder Probe Mission Concept Study (Final Report): https://smd-prod.s3.amazonaws.com/science-red/s3fs-public/atoms/files/Ea.
• Gris-Sanchez et al. (2018) Multicore fibre photonic lanterns for precision radial velocity Science: https://academic.oup.com/mnras/article/475/3/3065/4769655
• Jvanovic, N. et al. (2012). Integrated photonic building blocks for next-generation astronomical instrumentation I: the multimode waveguide. Optics Express, 20:17029.
• Yi, X., et al. (2016) Demonstration of a near-IR line-referenced electro-optical laser frequency comb for precision radial velocity measurements in astronomy. Nature Communications, 7:10436.
• Halverson, S., et al, (2014) "The habitable-zone planet finder calibration system", Proc. SPIE 9147, Ground-based and Airborne Instrumentation for Astronomy V, 91477Z: https://doi.org/10.1117/12.2054967
• Suh, M.-G., et al. (2019) Searching for exoplanets using a microresonator astrocomb. Nature Photonics, 13(1):25–30.
• Obrzud, E., et al. (2019) A Microphotonic Astrocomb. Nature Photonics, 13 (1):31–35.
• Chang, L., et al. (2018) Heterogeneously integrated GaAs waveguides on insulator for efficient frequency conversion, Laser Photonics Reviews, 12, 1800149: https://doi.org/10.1002/lpor.201800149
• Halir, R., et al. (2012) Ultrabroadband supercontinuum generation in a CMOS-compatible platform, Optics letters, 37, 1685: https://di.org/10.1364/OL.37.001685

Expected TRL or TRL range at completion of the project: 3 to 5

Desired Deliverables of Phase II

Desired Deliverables Description

This subtopic solicits proposals to develop cost effective component and subsystem technology for low SWaP, long-lived, robust implementation of radial velocity measurement instruments both on the ground and in space. Research areas of interest include but are not limited to:

• Integrated photonic spectrographs that meet PRV specifications (e.g. wavelength coverage, resolution, throughput, and polarization). These devices should be able to accept multiple fibers - at least two for the science light and simultaneous calibration light source. Ideally, they should be able to include on-chip cross-dispersion to eliminate bulk optics.
• PRV spectrograph calibration sources, particularly optical frequency combs (a.k.a. “astrocombs”) from the UV through the NIR (

• Spectral flattening to provide uniform power across the spectral band covered by the instrument
• Spectral broadening to obtain wide spectral coverage, preferably octave-spanning to enable self-referencing
• Integrated photonic solutions including nonlinear waveguides, microresonators or other comb generators, pump lasers, and f-2f beat-note generation
• Low phase-noise solutions
• Tunability of comb lines to scan spectrograph detectors for pixel characterization

Proposals should show an understanding of the science needs, as well as present a feasible plan to fully develop the relevant subsystem technologies and to transition into future NASA program(s).

Phase I will emphasize research aspects for technical feasibility, infusion potential into ground or space operations, clear and achievable benefits (e.g., reduction in SWaP and/or cost, improved RV precision), and show a path towards a Phase II proposal. Phase I Deliverables include feasibility and concept of operations of the research topic, simulations and measurements, validation of the proposed approach to develop a given product (TRL 3-4), and a plan for further development of the specific capabilities or products to be performed in Phase II. Early development and delivery of prototype hardware/software is encouraged.

Phase II will emphasize hardware/software development with delivery of specific hardware or software products for NASA targeting demonstration operations at a ground-based telescope in coordination with the lead NASA center. Phase II deliverables include a working prototype or engineering model of the proposed product/platform or software, along with documentation of development, capabilities, and measurements (showing specific improvement metrics), documents and tools as necessary. Proposed prototypes shall demonstrate a path towards a flight-capable platform. Opportunities and plans should also be identified and summarized for potential commercialization or NASA infusion.

State of the Art and Critical Gaps

The classical bulk optic spectrographs that are traditionally used for PRV science impose architectural constraints due to their large mass and limited optical flexibility. The spectrograph is the single element that if replaced with a photonic alternative could dramatically alter the course of astronomical instrumentation. Integrated Photonic Spectrographs (IPS) are wafer thin devices that could reduce instrument volume by up to three orders of magnitude. Furthermore, high resolving power spectrographs (R

150,000) with simultaneous UV, visible, and NIR coverage and exquisite long-term stability are required for PRV studies. Spectrometers that are fiber-fed with high illumination stability, excellent wavelength calibration, and precise temperature and pressure control represent the immediate future of precision RV measurements.

As spectrograph stability imposes limits on how precisely the Radial Velocity (RV) can be measured, spectral references play a critical role in characterizing and ensuring this precision. Only Laser Frequency Combs (LFCs) and line-referenced Fabry-Pérot etalons are capable of providing the broad spectral coverage and long term (years) stability needed for extreme PRV detection of exoplanets. While both frequency combs and etalons can deliver high precision spectrograph calibration, the former requires relatively complex and sophisticated hardware in the visible portion of the spectrum. Visible band frequency combs for astronomy (a.k.a. astrocombs) were initially based on mode-locked laser comb technology. However, the intrinsic free spectral range of these instruments, 100s of MHz to 1 GHz, is too fine to be resolved by astronomical spectrographs of R

150,000 or less. Thus, mode filtering of comb lines to create a more spectrally sparse calibration grid is necessary. The filtering step introduces complexity and additional sources of instability to the calibration process, as well as instrument assemblies too large in mass and volume for flight.

Commercial fiber laser astrocombs covering 450 - 1400 nm at 25 GHz line spacing and <3 dB intensity variations over the entire bandwidth are available for ground-based astronomical spectrographs and have been developed for HARPS-S and ESPRESSO RV instruments. However, the cost for these systems is often so prohibitive that recent RV spectrograph projects such as CARMENES and Keck Planet Finder either do not use a frequency comb or include it only as a future upgrade, owing to the cost impact on the project.

Alternatively, frequency combs produced by Electro-Optic Modulation (EOM) of a laser source have been demonstrated at observatories for PRV studies in the near-IR. EOM combs produce modes spaced at a RF modulation frequency, typically 10-30 GHz, and are inherently suitable as ground-based astrocombs. Significantly, EOM combs avoid the line filtering step of commercial mode-locked fiber laser combs. Comb frequency stabilization can be accomplished in a variety of ways, including referencing the laser pump source to a molecular absorption feature or another frequency comb. Where octave spanning EOM combs are available, f-2f self-referencing provides the greatest stability.

EOM combs must be spectrally broadened to provide the octave bandwidth necessary for f-2f stabilization for stability traceable to the Standard International (SI) second. This is accomplished through pulse amplification followed by injection into Highly Non-Linear Fiber (HNLF) or nonlinear optical waveguides, but the broadening process is accompanied by multiplication of the optical phase noise from the EOM comb modulation signal and must be optically filtered. Also, at these challenging microwave pulse repetition rates, the pulse duty-cycle requires pulse amplification to 4-5 Watts of average optical power in order to generate the high enough peak intensity needed for nonlinear broadening. This necessitates use of high power, non-telecom amplifiers that are more prone to lifetime issues, making EOM combs not optimal for flight either. It is important to note that very little comb light is actually required on the spectrograph detectors for calibration. In fact, most of the generated comb light must be deliberately attenuated to avoid detector saturation.

Power consumption of the frequency comb calibration system will be a significant driver of mission cost for space-based PRV systems, and motivates the development of a comb system that operates with less than 20 Watts of spacecraft power. Thus, for flight applications, it is highly desirable to develop frequency comb technology with low power consumption,

10 GHz mode spacing, compact size, broad (octave spanning) spectral grasp across both the visible and NIR, phase noise insensitivity, stability traceable to the definition of the SI second, and very importantly, long life.

Relevance / Science Traceability

The NASA Strategic Plan (2018) and Space Mission Directorate Science Plan (2014) both call for discovery and characterization of habitable Earth analogs and the search for biosignatures on those worlds. These goals were endorsed and amplified upon in the recent National Academy of Science (NAS) Exoplanet Report which emphasized that a knowledge of the orbits and masses is essential to the complete and correct characterization of potentially habitable worlds. PRV measurements are needed to follow up on the transiting worlds discovered by Kepler, K2, and Transiting Exoplanet Survey Satellite (TESS). The interpretation of the transit spectra which James Webb Space Telescope (JWST) will obtain will depend on knowledge of a planet’s surface gravity which comes from its radius (from the transit data) and its mass (from PRV measurements or in some cases Transit Timing Variations). Without knowledge of a planet's mass, the interpretation of its spectrum is subject to many ambiguities.

These ambiguities will only be exacerbated for the direct imaging missions such as the proposed Habitable Exoplanet Observatory (HabEx) and Large Ultraviolet Optical Infrared Surveyor (LUVOIR) flagships which will obtain spectra of Earth analogs around a few tens to hundreds of stars. Even if a radius can be inferred from the planet's brightness and an estimate of its albedo, the lack of a dynamical mass precludes any knowledge of the planet's density, bulk composition, and surface gravity which are needed to determine, for example, absolute gas column densities. Moreover, a fully characterized orbit is challenging to determine from just a few direct images and may even be confused in the presence of multiple planets. Is a planet in a highly eccentric orbit habitable or not? Only dynamical (PRV) measurements can provide such information. Thus, highly precise and highly stable PRV measurements are absolutely critical to the complete characterization of habitable worlds.

The NAS report also noted that measurements from space might be a final option if the problem of telluric contamination cannot be solved. The Earth’s atmosphere will limit precise radial velocity measurements to

10 cm/s at wavelengths longer than

700 nm and greater than 30 cm/s at >900 nm, making it challenging to mitigate the effects of stellar activity without a measurement of the color dependence due to stellar activity in the PRV time series. A space-based PRV mission, such as has been suggested in the NASA EarthFinder mission concept study, may be necessary. If so, the low SWaP technologies developed under this SBIR program could help enable space-based implementations of the PRV method.

### Assorted References

For objects beyond the immediate neighbourhood of the Sun, initially it is necessary to choose a standard of rest (the reference frame) from which the solar motion is to be calculated. This is usually done by selecting a particular kind of star or…

…extrasolar planets has been the radial velocity method, which measures the motion of host stars in response to gravitational tugs by their planets. Swiss astronomers Michel Mayor and Didier Queloz discovered the first planet using this technique, 51 Pegasi b, in 1995. (Mayor and Queloz won the 2019 Nobel

Radial velocities, measured along the line of sight spectroscopically using the Doppler effect, are known for nearly all of the recognized stars near the Sun. Of the 54 systems within 17 light-years, most have well-determined radial velocities. The radial velocities of the rest…

, toward the observer), called radial velocity, is obtained directly from spectroscopic observations. If λ is the wavelength of a characteristic spectral line of some atom or ion present in the star and λL is the wavelength of the same line measured in the laboratory, then the difference Δλ, or…

### Work of

…his spectrographic determinations of the radial velocities of stars—i.e., their motions toward the Earth or away from it. In addition, he discovered many spectroscopic binary stars, and in 1924 he published a catalog listing more than 1,000 of them.

…accurate measurements of a star’s radial velocity (that is, its velocity toward or away from the observer). When a planet orbits a star, the planet and the star orbit around their common centre of mass, and the star’s motion around the centre of mass can be seen as a shift…

## Creating a 6D map of the Galaxy

When we combine the measurement of the proper motion with that of the parallax, we can get a measure of the real space velocity perpendicular to the line-of-sight. Combine this with the radial velocity and we have a measurement of the space velocity. Similarly, the position on the sky and the parallax combined give the position in space of the object. This way, Gaia will produce a 6-dimensional map of the sky (3 dimensions of the position in space and 3 dimensions of space velocity).

Velocities in space are determined by the distribution of mass in space. For the Earth the most important mass is that of the Sun, around which it describes an orbit. For stars it is the mass of the Galaxy, around the centre of which those stars describe each their own orbit. Combining the 3-dimensional positions with the space velocities will provide us with a unique source of information on topics such as the distribution of mass in the Galaxy.