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The '''Leonard–Merritt mass estimator''' is a formula for estimating the mass of a spherical [[stellar system]] using the apparent (angular) positions and [[proper motions]] of its component [[star]]s. The distance to the stellar system must also be known.
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Like the [[virial theorem]], the Leonard–Merritt estimator yields correct results regardless of the degree of velocity anisotropy. Its statistical properties are superior to those of the virial theorem. However, it requires that two components of the velocity be known for every star, rather than just one for the virial theorem.<ref>{{Cite journal
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  | doi = 10.1086/167287
  | last1 = Leonard  | first1 = Peter
  | last2 = Merritt | first2 = David
  | author2-link = David Merritt
  | title = The Mass of the Open Star Cluster M35 as Derived from Proper Motions
  | journal = The Astrophysical Journal
  | volume = 339
  | issue =
  | pages = 195–208
  | date = 10 April 1989
  | bibcode = 1989ApJ...339..195L
  | issn =}}
</ref>
 
The estimator has the general form
 
<math>
 
\langle M(r)\rangle = {16\over 3\pi G}
\langle R\left(2V_R^2 + V_T^2\right)\rangle.
 
</math>
 
The angle brackets denote averages over the ensemble of observed stars. <math> M(r) </math> is the mass contained within a distance <math>r</math> from the center of the stellar system; <math>R</math> is the projected distance of a star from the apparent center; <math>V_R</math> and <math>V_T</math> are the components of a star's velocity parallel to, and perpendicular to, the apparent radius vector; and <math>G</math> is the [[gravitational constant]].
 
Like all estimators based on moments of the [[Jeans equations]], the Leonard–Merritt estimator requires an assumption about the relative distribution of mass and light. As a result, it is most useful when applied to stellar systems that have one of two properties:
# All or almost all of the mass resides in a central object, or,
# the mass is distributed in the same way as the observed stars.
Case (1) applies to the nucleus of a galaxy containing a [[supermassive black hole]]. Case (2) applies to a stellar system composed entirely of luminous stars (i.e. no [[dark matter]] or [[black holes]]).
 
In a cluster with constant mass-to-light ratio and total mass <math>M_T</math>, the Leonard–Merritt estimator becomes:
 
<math>M_T = {32\over 3\pi G}
\langle R \left(2V_R^2 + V_T^2\right)\rangle.</math>
 
On the other hand, if all the mass is located in a central point of mass <math>M_0</math>, then:
 
<math>M_0 = {16\over 3\pi G}
\langle R\left(2V_R^2 + V_T^2\right)\rangle.</math>
 
In its second form, the Leonard–Merritt estimator has been successfully used to measure the mass of the [[supermassive black hole]] at the center of the [[Milky Way]] [[galaxy]]<ref>{{Cite journal
  | last1 = Schödel | first1 = R
  | last2 = Ott | first2 = T | first3 = R
  | last4 = Eckart | first4 = A
  | last5 = Mouawad | first5 = N
  | last3 = Alexander | first6 = T
  | title = Stellar Dynamics in the Central Arcsecond of Our Galaxy
  | journal = The Astrophysical Journal
  | volume = 596
  | issue = 2
  | pages = 1015–1034
  | date = 1 October 2003
  | bibcode = 2003ApJ...596.1015S
  | issn =|arxiv = astro-ph/0306214 |doi = 10.1086/378122 }}
</ref>
.<ref>{{Cite journal
  | doi = 10.1051/0004-6361/200810922
  | last1 = Schödel  | first1 = Rainer
  | last2 = Merritt | first2 = David
  | author2-link = David Merritt
  | last3 = Eckart | first3 = Andreas
  | title = The nuclear star cluster of the Milky Way: proper motions and mass
  | journal = Astronomy and Astrophysics
  | volume = 502
  | issue = 1
  | pages = 91–111
  | date = 1 July 2009
  | bibcode = 2009A&A...502...91S
  | issn =|arxiv = 0902.3892 }}
</ref>
 
==See also==
* [[Virial theorem]]
* [[Proper motion]]
* [[Globular cluster]]
 
==References==
{{Reflist}}
 
{{Use dmy dates|date=September 2010}}
 
{{DEFAULTSORT:Leonard-Merritt Mass Estimator}}
[[Category:Astrometry]]
[[Category:Celestial mechanics]]
[[Category:Stellar astronomy]]
[[Category:Supermassive black holes]]

Latest revision as of 17:13, 8 May 2014

If an existing Word - Press code is found vulnerable, Word - Press will immediately issue an update for that. It is thus, on these grounds that compel various web service provider companies to integrate the same in their packages too. Should you go with simple HTML or use a platform like Wordpress. 2- Ask for the designs and graphics that will be provided along with the Word - Press theme. The number of options offered here is overwhelming, but once I took the time to begin to review the video training, I was amazed at how easy it was to create a squeeze page and a membership site.

Generally, for my private income-making market websites, I will thoroughly research and discover the leading 10 most worthwhile niches to venture into. You will have to invest some money into tuning up your own blog but, if done wisely, your investment will pay off in the long run. This is the reason for the increased risk of Down Syndrome babies in women over age 35. So, if you are looking for some option to build a giant e-commerce website, then e-shopping preferable CMS tools will be helpful for you. If you adored this article and also you would like to obtain more info about wordpress backup plugin kindly visit the page. As soon as you start developing your Word - Press MLM website you'll see how straightforward and simple it is to create an online presence for you and the products and services you offer.

It is very easy to install Word - Press blog or website. The only problem with most is that they only offer a monthly plan, you never own the software and you can’t even install the software on your site, you must go to another website to manage your list and edit your autoresponder. I hope this short Plugin Dynamo Review will assist you to differentiate whether Plugin Dynamo is Scam or a Genuine. To turn the Word - Press Plugin on, click Activate on the far right side of the list. So, if you are looking online to hire dedicated Wordpress developers, India PHP Expert can give a hand you in each and every best possible way.

Numerous bloggers are utilizing Word - Press and with good reason. And, make no mistake,India's Fertility Clinics and IVF specialists are amongst the best in the world,and have been for some time. Websites that do rank highly, do so becaue they use keyword-heavy post titles. A whole lot worse, your site will likely be useless as well as your merchandise won't sell if no one has the endurance to wait for the web pages to load. If your site does well you can get paid professional designer to create a unique Word - Press theme.

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