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18.2 Measuring stellar masses  (Page 3/10)

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Motions of two stars orbiting each other and what the spectrum shows.

Motions of Two Stars Orbiting Each Other and What the Spectrum Shows. This figure has four binary star spectra, each with blue wavelengths on the left and red wavelengths on the right. Above each spectrum is a diagram showing the orbit of the two binary stars. Spectrum 1 has two spectral lines, one from each star. The lines for star B is roughly in the center of the spectrum, and the line for star A is a little to the left. The orbit shows the stars at opposite sides horizontally, with an arrow pointing down from star A and an arrow pointing up from star B, indicating that the stars are moving horizontally to our line of sight. In spectrum 2, both lines merge into one and the line is labeled “A + B”. The orbit shows the stars at opposite sides horizontally, with an arrow pointing right from star A and an arrow pointing left from star B, indicating that the stars are moving perpendicularly to our line of sight. In spectrum 3 the line for star B is near the center and that of star A is on the right. The orbit shows the stars at opposite sides horizontally, with an arrow pointing up from star A and an arrow pointing down from star B, indicating that the stars are moving horizontally to our line of sight. Finally, in spectrum 4, the lines have again merged near the center and the line is labeled “B + A”. The orbit shows the stars at opposite sides horizontally, with an arrow pointing left from star A and an arrow pointing right from star B, indicating that the stars are moving perpendicularly to our line of sight.
We see changes in velocity because when one star is moving toward Earth, the other is moving away; half a cycle later, the situation is reversed. Doppler shifts cause the spectral lines to move back and forth. In diagrams 1 and 3, lines from both stars can be seen well separated from each other. When the two stars are moving perpendicular to our line of sight (that is, they are not moving either toward or away from us), the two lines are exactly superimposed, and so in diagrams 2 and 4, we see only a single spectral line. Note that in the diagrams, the orbit of the star pair is tipped slightly with respect to the viewer (or if the viewer were looking at it in the sky, the orbit would be tilted with respect to the viewer’s line of sight). If the orbit were exactly in the plane of the page or screen (or the sky), then it would look nearly circular, but we would see no change in radial velocity (no part of the motion would be toward us or away from us.) If the orbit were perpendicular to the plane of the page or screen, then the stars would appear to move back and forth in a straight line, and we would see the largest-possible radial velocity variations.

A plot showing how the velocities of the stars change with time is called a radial velocity curve ; the curve for the binary system in [link] is shown in [link] .

Radial velocities in a spectroscopic binary system.

Radial Velocities in a Spectroscopic Binary System. The upper portion of this figure has four binary star spectra, each with blue wavelengths on the left and red wavelengths on the right. Spectrum 1 at left has two spectral lines, one from each star. The line for star A is near the center and that for star B is toward the right. In spectrum 2, both lines merge into one and is labeled “A + B”. In spectrum 3 the line for star B is near the center and that of star A is on the right. Finally, in spectrum 4, the lines have again merged near the center and labeled “B + A”. The bottom portion shows a graph of measured radial velocities vs. time. The vertical axis is labeled “Radial Velocity (km/s)”, in 40 km/s increments. The horizontal axis is labeled “Time (days)”, in 2 day increments. Curves are plotted corresponding to the motion of stars A and B that are shown in the spectra above the plot. Both curves begin at day zero on the left at +40 km/s. At day 4, corresponding to spectrum 1, star A has a velocity of +15 km/s and B +110 km/s. At day 9 for spectrum 2, both stars are at +40 km/s. At day 13 (spectrum 3) star A is near +65 km/s and star B near -30 km/s. Finally, near day 17 (spectrum 4), both stars are again at +40 km/s.
These curves plot the radial velocities of two stars in a spectroscopic binary system, showing how the stars alternately approach and recede from Earth. Note that positive velocity means the star is moving away from us relative to the center of mass of the system, which in this case is 40 kilometers per second. Negative velocity means the star is moving toward us relative to the center of mass. The positions on the curve corresponding to the illustrations in [link] are marked with the diagram number (1–4).

This animation lets you follow the orbits of a binary star system in various combinations of the masses of the two stars.

Masses from the orbits of binary stars

We can estimate the masses of binary star systems using Newton’s reformulation of Kepler’s third law (discussed in Newton’s Universal Law of Gravitation ). Kepler found that the time a planet takes to go around the Sun is related by a specific mathematical formula to its distance from the Sun. In our binary star situation, if two objects are in mutual revolution, then the period ( P ) with which they go around each other is related to the semimajor axis ( D ) of the orbit of one with respect to the other, according to this equation

D 3 = ( M 1 + M 2 ) P 2

where D is in astronomical units, P is measured in years, and M 1 + M 2 is the sum of the masses of the two stars in units of the Sun’s mass. This is a very useful formula for astronomers; it says that if we can observe the size of the orbit and the period of mutual revolution of the stars in a binary system, we can calculate the sum of their masses.

Most spectroscopic binaries have periods ranging from a few days to a few months, with separations of usually less than 1 AU between their member stars. Recall that an AU is the distance from Earth to the Sun, so this is a small separation and very hard to see at the distances of stars. This is why many of these systems are known to be double only through careful study of their spectra.

Questions & Answers

what does the ideal gas law states
Joy Reply
Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.
Kate Reply
To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.
Muhammed
What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?
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Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.
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A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate
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Sharon
8m/s²
Aishat
What is Thermodynamics
Muordit
velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.
Mehmet
A boat travels due east at a speed of 40meter per seconds across a river flowing due south at 30meter per seconds. what is the resultant speed of the boat
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50 m/s due south east
Someone
which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer
Ramon Reply
I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.
Someone
Scratch that
Someone
temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....
Someone
about the amount of heat stored in the system then in that case since the mass of water in the kettle is greater so more energy is required to raise the temperature b/c more molecules of water are present in the kettle
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field is a region of space under the influence of some physical properties
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Esrael
Two bodies attract each other electrically. Do they both have to be charged? Answer the same question if the bodies repel one another.
JALLAH Reply
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Dlovan
Are you really asking if two bodies have to be charged to be influenced by Coulombs Law?
Robert
like charges repel while unlike charges atttact
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Source:  OpenStax, Astronomy. OpenStax CNX. Apr 12, 2017 Download for free at http://cnx.org/content/col11992/1.13
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