Dynamical parallax: Difference between revisions
Content deleted Content added
m removed a subtitle |
m Dating maintenance tags: {{Clarify}} |
||
| (29 intermediate revisions by 15 users not shown) | |||
| Line 1: | Line 1: | ||
{{Short description|Iterative technique to find the properties of the stars in a binary pair}} |
|||
{{Underconstruction}} |
|||
In [[astronomy]], the [[distance]] to a [[visual binary|visual binary star]] may be estimated from the [[mass]]es of its two components, the size of their [[orbit]], and the period of their |
In [[astronomy]], the [[distance]] to a [[visual binary|visual binary star]] may be estimated from the [[mass]]es of its two components, the [[angular size]] of their [[orbit]], and the period of their orbit about one another.<ref>{{cite book |
||
| author = Patrick Moore |
| author = Patrick Moore |
||
| title = Philip's Astronomy Encyclopedia |
| title = Philip's Astronomy Encyclopedia |
||
| url = https://archive.org/details/philipsastronomy00ehrl_838 |
|||
| edition = revised and expanded ed. |
|||
| url-access = limited |
|||
| edition = revised and expanded |
|||
| year = 2002 |
| year = 2002 |
||
| publisher = Philip's |
| publisher = Philip's |
||
| location = London |
| location = London |
||
| isbn = 0-540-07863-8 |
| isbn = 0-540-07863-8 |
||
| page = [https://archive.org/details/philipsastronomy00ehrl_838/page/n128 120] |
|||
| page = 120 |
|||
}}</ref> A '''dynamical parallax''' is an (annual) [[parallax]] which is computed from such an estimated distance. |
}}</ref> A '''dynamical parallax''' is an (annual){{clarify|reason=How is this annual?|date=November 2024}} [[parallax]] which is computed from such an estimated distance. |
||
To calculate a dynamical parallax, the angular [[semi-major axis]] of the orbit of the stars |
To calculate a dynamical parallax, the angular [[semi-major axis]] of the orbit of the stars is observed, as is their [[apparent brightness]]. By using [[Isaac Newton|Newton's]] generalisation of [[Johannes Kepler|Kepler's]] [[Kepler's third law|Third Law]], which states that the total mass of a binary system multiplied by the [[square (algebra)|square]] of its [[orbital period]] is proportional to the [[cube (arithmetic)|cube]] of its [[semi-major axis]],<ref>{{cite web |
||
| |
|url = http://www.site.uottawa.ca:4321/astronomy/index.html#dynamicalparallax |
||
| |
|title = dynamical parallax |
||
| |
|accessdate = 2006-07-18 |
||
| |
|work = Astronomy Knowledge Base |
||
|url-status = dead |
|||
}}</ref> together with the [[mass-luminosity relation]], the distance to the binary star can then be determined.<ref>[http://csep10.phys.utk.edu/astr162/lect/binaries/masslum.html The Mass-Luminosity Relation], University of Tennessee, Astronomy 162: Stars, Galaxies, and Cosmology, lecture notes. Accessed July 18, 2006.</ref> |
|||
|archiveurl = https://web.archive.org/web/20060705214728/http://www.site.uottawa.ca:4321/astronomy/index.html#dynamicalparallax |
|||
|archivedate = 2006-07-05 |
|||
}}</ref> together with the [[mass–luminosity relation]], the distance to the binary star can then be determined.<ref>[http://csep10.phys.utk.edu/astr162/lect/binaries/masslum.html The Mass–Luminosity Relation], University of Tennessee, Astronomy 162: Stars, Galaxies, and Cosmology, lecture notes. Accessed July 18, 2006.</ref> |
|||
With this technique, the masses of the two stars in a binary system are estimated, usually as the mass of the Sun. Then, using [[Kepler's laws]] of [[celestial mechanics]], the distance between the stars is calculated. Once this distance is found, their distance from the observer can be found via the arc subtended in the sky, giving a preliminary distance measurement. From this measurement and the [[apparent magnitude]]s of both stars, the luminosities can be found, and from the mass–luminosity relationship, the masses of each star. These masses are used to re-calculate the separation distance, and the process is repeated. The process is iterated many times, and accuracies within 5% can be achieved.<ref name="double">{{cite book|last=Mullaney|first=James|title=Double and multiple stars and how to observe them|publisher=Springer|year=2005|isbn=1-85233-751-6|url=https://archive.org/details/doublemultiplest0000mull|url-access=registration|page=[https://archive.org/details/doublemultiplest0000mull/page/27 27]}}</ref> |
|||
== |
==See also== |
||
* [[Parallax in astronomy]] |
|||
* [[Photometric parallax method]] |
|||
There is a simple formula relating distance to parallax. Firstly, two measurements are made at both sides of the Earth's orbit. For a nearby star, it will appear displaced relative to background stars. The distance to this nearby star is given by |
|||
* [[Spectroscopic parallax]] |
|||
:<math> d = 1AU/tan(p) </math> |
|||
Where p is the parallax, defined as half the apparent angular displacement, and AU refers to the astronomical unit. (1AU = 1.49*10^8km)<ref name="parallax">{{cite book |
|||
| author = Martin Harwit |
|||
| title = Astrophysical concepts |
|||
| isbn = 0-387-94943-7 |
|||
| publisher = springer |
|||
}}</ref> |
|||
==Visual Binary== |
|||
A visual binary is a gravitationally bound system that can be resolved into two stars with the aid of a telescope. These stars are estimated, via Kepler's 3rd law, to have periods ranging from a number of years to thousands of years. A visual binary consists of two stars, usually of a different brightness. Because of this, the brightest star is called the primary and the fainter one is called the companion. If the primary is too bright, relative to the companion, this can cause a glare making it difficult to resolve the two components. <ref>''The Binary Stars'', [[Robert Grant Aitken]], New York: Dover, 1964, p. 41.</ref> |
|||
For a visual binary system, measurements taken need to specify, in arc-seconds, the apparent angular separation on the sky and the position angle- which is the angle measured eastward from North in degrees- of the companion star relative to the primary star. Taken over a period of time, the apparent relative orbit of the visual binary system will appear on the celestial sphere.<ref name="Binary">{{cite book |
|||
| author = Michael Zeilik, Stephan A. Gregory, and Elske V. P. Smith |
|||
| title = Introductory Astronomy and Astrophysics |
|||
| isbn = 978-0030062285 |
|||
| publisher = Brooks/Cole |
|||
}}</ref> |
|||
==Kepler's Laws== |
|||
The two stars orbiting each other, as well as there centre of mass, must obey Kepler's laws. |
|||
This means that the orbit is an ellipse with the centre of mass at one of the two focii (Kepler's 1st law) and the orbital motion satisfies the fact that a line joining the star to the centre of mass sweeps out equal areas over equal time intervals (Kepler's 2nd law). The orbital motion must also satisfy Kepler's 3rd law.<ref name=Theory /> |
|||
Keplar's 3rd Law can be stated as follows: "The square of the orbital period of a planet is directly proportional to the cube of its semi-major axis." Mathmatically, this translates as |
|||
:<math> T^2 \propto a^3 </math> |
|||
where <math>T</math> is the orbital period of the planet and <math>a</math> is the semi-major axis of the orbit.<ref name="Theory">{{cite book |
|||
| author = Leanard Susskind and Hrabovsky |
|||
| title = The Theoretical Minimum: What You Need To Know To Start Doing Physics |
|||
| year = 2013 |
|||
| isbn = 978-1846147982 |
|||
| publisher = the Penguin Group |
|||
}}</ref> |
|||
===Newton's Generalisation=== |
|||
Consider a binary star system. This consists of two objects, of mass <math>m_1</math> and <math>m_2</math>, orbiting around their centre of mass. <math>m_1</math> has position vector <math>r_1</math> and orbital velocity <math>v_1</math>, and <math>m_2</math> has position vector <math>r_2</math> and orbital velocity <math>v_2</math> relative to the centre of mass. The separation between the two stars is denoted <math>a</math>, and is assumed to be constant. Since the gravitational force acts along a line joining the centers of both stars, we can assume the stars have an equivalent time period around their center of mass, and therefore a constant separation between each other.<ref>{{cite web |
|||
| url = http://www.egglescliffe.org.uk/physics/gravitation/binary/binary.html |
|||
| title = The Physics of Binary Stars |
|||
| accessdate = 15/10/2013 |
|||
}}</ref> |
|||
To arrive at Newton's version of Kepler's 3rd law we can start by considering Newton's 2nd law which states: "The net force acting on an object is proportional to the objects mass and resultant acceleration." |
|||
:<math> F_{net} = \sum \, F_{i} = ma </math> |
|||
where <math>F_{net}</math> is the net force acting on the object of mass <math>m</math>, and <math>a</math> is the acceleration of the object.<ref name="Astro">{{cite book |
|||
| author = Bradley W. Carroll and Dale A. Ostlie |
|||
| title = An Introduction to Modern Astrophysics |
|||
| year = 2013 |
|||
| isbn = 978-1292022932 |
|||
| publisher = Pearson |
|||
}}</ref> |
|||
Applying the definition of centripetal acceleration to Newton's second law gives a force of |
|||
:<math> F = mv^2/r </math> <ref name="phy">{{cite book |
|||
| author = Hugh D. Young |
|||
| title = University Physics |
|||
| year = 2010 |
|||
| isbn = 0321501306 |
|||
| publisher = Bertrams |
|||
}}</ref> |
|||
Then using the fact that the orbital velocity is given as |
|||
:<math> v = 2\pi r/T </math> <ref name=phy /> |
|||
we can state the force on each star as |
|||
:<math> F_{1} = 4\pi^2 m_{1}r_{1}/T^2 </math> and <math> F_{2} = 4\pi^2 m_{2}r_{2}/T^2 </math> |
|||
If we apply Newton's 3rd law- "For every action there is an equal and opposite reaction" |
|||
:<math> F_{12} = -F_{21} </math> <ref name=Astro /> |
|||
We can set the force on each star equal to each other. |
|||
:<math> 4\pi^2 m_{1}r_{1}/T^2 = 4\pi^2 m_{2}r_{2}/T^2 </math> |
|||
This reduces to |
|||
:<math> r_{1}m_{1} = r_{2}m_{2} </math> |
|||
If we assume that the masses are not equal, then this equation tells us that the smaller mass remains farther from the centre of mass than does the larger mass. |
|||
The separation <math>a</math> of the two objects is |
|||
:<math> a = r_{1}+r_{2} </math> |
|||
Since <math>r_1</math> and <math>r_2</math> would form a line starting from opposite directions and joining at the centre of mass. |
|||
Now we can substitute this expression into equation 4 and rearrange for <math>r_1</math> to find an expression relating the position of one star to the masses of both and the separation between them. Equally, this could have been solved for <math>r_2</math>. We find that |
|||
:<math> r_{1} = m_{2}a/(m_{1}+m_{2}) </math> |
|||
Substituting this equation into the equation for the force on one of the stars, setting it equal to Newton's Universal Law of Gravitation (namely, <math>F=Gm_{1}m_{2}/a^2</math>)<ref name=Astro />, and solving for the period squared yields the required result. |
|||
:<math> T^2 = 4\pi^2 a^3/G(m_{1}+m_{2}) </math><ref name=Astro /> |
|||
This is Newton's version of Kepler's 3rd Law. It requires that the mass is measured in solar masses, the orbital period is measured in years, and the orbital semi-major axis is measured in astronomical units. |
|||
==Determining Stellar Masses of a Binary System == |
|||
==The Mass-Luminosity Relationship== |
|||
==References== |
==References== |
||
{{Reflist}} |
{{Reflist}} |
||
==See also== |
|||
*[[Photometric parallax method]] |
|||
*[[Spectroscopic parallax]] |
|||
{{DEFAULTSORT:Dynamical Parallax}} |
{{DEFAULTSORT:Dynamical Parallax}} |
||
[[Category:Binary stars|-]] |
[[Category:Binary stars|-]] |
||
[[Category:Astrometry]] |
[[Category:Astrometry]] |
||
{{Astronomy-stub}} |
|||
Latest revision as of 18:54, 22 November 2024
In astronomy, the distance to a visual binary star may be estimated from the masses of its two components, the angular size of their orbit, and the period of their orbit about one another.[1] A dynamical parallax is an (annual)[clarification needed] parallax which is computed from such an estimated distance.
To calculate a dynamical parallax, the angular semi-major axis of the orbit of the stars is observed, as is their apparent brightness. By using Newton's generalisation of Kepler's Third Law, which states that the total mass of a binary system multiplied by the square of its orbital period is proportional to the cube of its semi-major axis,[2] together with the mass–luminosity relation, the distance to the binary star can then be determined.[3]
With this technique, the masses of the two stars in a binary system are estimated, usually as the mass of the Sun. Then, using Kepler's laws of celestial mechanics, the distance between the stars is calculated. Once this distance is found, their distance from the observer can be found via the arc subtended in the sky, giving a preliminary distance measurement. From this measurement and the apparent magnitudes of both stars, the luminosities can be found, and from the mass–luminosity relationship, the masses of each star. These masses are used to re-calculate the separation distance, and the process is repeated. The process is iterated many times, and accuracies within 5% can be achieved.[4]
See also
[edit]References
[edit]- ^ Patrick Moore (2002). Philip's Astronomy Encyclopedia (revised and expanded ed.). London: Philip's. p. 120. ISBN 0-540-07863-8.
- ^ "dynamical parallax". Astronomy Knowledge Base. Archived from the original on 2006-07-05. Retrieved 2006-07-18.
- ^ The Mass–Luminosity Relation, University of Tennessee, Astronomy 162: Stars, Galaxies, and Cosmology, lecture notes. Accessed July 18, 2006.
- ^ Mullaney, James (2005). Double and multiple stars and how to observe them. Springer. p. 27. ISBN 1-85233-751-6.