Possible origin of Larson’s lows
A.W. Zaharow
Ufa state petrotechnical university, Ufa, Russia∗
(Dated: April 10, 2007)
It was found that approximately constant column densities of giant molecular
clouds (Larson’s low) can be explained as cloud existence condition in external (galac-
tic) gravitational field. This condition can be also applied to objects (clumps and
cores) embedded into the cloud and its gravitational field. Derived existence condi-
tion do not rely on any internal dynamic of a cloud and embedded objects.
1. Introduction
Giant molecular clouds (GMCs) play a crucial role in the star formation process
([1],[2],[3],[4],[5]). GMCs are complex objects with masses ∼ 105−6M⊙, diameters ∼ 50pc
and average densities nH2 ∼ 10
2cm−3 (e.g. [6]). GMCs are generally gravitationally bound
and may contain several sites of star formations. Equilibrium of self-gravitating gas was the-
oretically investigated in many works (e.g. [4],[9] and references in these papers). Internal
structure of GMS is usually very complicated. The inhomogeneous structure of cloud could
be described as a set of descrete clumps ([7]). These clumps themselves contain dense cores
with densities n ∼ 103−5cm−3. The one point of view at this time is that this cloud structure
is due to supersonic turbulence. Remarkably enough, the properties of cloud complexes are
rather simply interrelated. Total masses, mean densities and average velosity dispersions
vary with sizes (effective radii) roughly as M ∝ R2,ρ ∝ R−1, σ ∝ R1/2 (Larson’s lows) ([8]).
In this paper I propose simple hypothesis to explain relation between masses and sizes of
clumps and cores, embedded into clouds.
∗Electronic address: zaharow@gmail.com
http://arxiv.org/abs/0704.1214v1
mailto:zaharow@gmail.com
2. Larson’s lows
Let us briefly consider some of the most salient characteristics of GMCs summarized by
Larson ([8]). See alsow ([2]). The first relation is the line width-size relation: molecular
clouds are supersonically turbulent with line widths ∆v that increase as a power of size,
∆v ∝ Rp. Larson himself estimated that p ≃ 0.38. Subsequent work has distinguished
between the relation valid for a collection of GMCs and that valid within individual GMC.
Within low-mass cores, Caselli & Myers ([10]) found that the nonthermal velocity dispersion
σnt ≃ 0.55R
pc km s
−1 , (2.1)
which is near to relation
σ ∝ R1/2 . (2.2)
Larson’s second result was that GMCs and clumps within them are gravitationally bound.
It implies that
σ2 ∝ GM/R . (2.3)
This relation is result of virial equilibrium.
His third conclusion was that all GMCs have about the same column density
N ≃ const . (2.4)
Column density is defined as
N ∝ M/R2 ∝ nR . (2.5)
As Larson pointed out, only two of these conclusions are independent: any one of them
can be derived from other two. Opposite to the second conclusion the first and the second
Larson’s lows have no evident explanation. I think this explanation may be as follows.
Embedded clumps and cores moves through the cloud in its gravitational field and are
subject of turbulent motion and various accelerations. If their own gravitational field is not
enough to hold on these objects they must rapidly decay. It is easy to derive necessary
condition for confinement gaseous body with mass M and radius R in external gravitational
field of cloud with mass Mcl and radius Rcl. The variation of self gravitational potential ∆φ
must be greater then that of the cloud on size R. Bearing in mind that
∆φ ≃ GM/R (2.6)
∆φcl ≃ R
GM/Rcl ≃ GMR/R
cl (2.7)
we get
GM/R ≥ GMclR/R
cl (2.8)
or equivalently
M/R2 ≥ Mcl/R
cl . (2.9)
Inequality (2.9) is the main result of this work. It is important for undestanding the third
Larson’s low (2.4). Inequality (2.9) is strong only for very massive and compact objects
within the cloud which are hard to generate in turbulent motion. So inequality can not
be strong for almost all objects. Further, in expression (2.7) we did not take into account
placement of object within the cloud. Summarizing all we can generalize third Larson’s low
in the form
Mcl(r)
, (2.10)
where r is the dinstance of object from the center of cloud, Mcl(r) is the mass of cloud within
r, C is the non-dimensional constant of order one. As far as clouds itself are concerned they
move in galactic gravitational field through the interstellar gaseous media (ISM) and we can
get for them analogous condition
Mgal(r)
, (2.11)
where r is the dinstance of cloud from the center of galactic, Mgal(r) is the mass of galactic
within r.
3. Summary
We have seen that it is possible to understand the third Larson’s low and some proper-
ties of molecular clouds and objects within them in terms of their existence conditions in
external gravitational field. In particular, we have seen possible reason why the clouds have
approximately constant column densities. I have to stress now that conditions (2.10) and
(2.11) do not rely on any internal dynamic of cloud. It will be also interesting to examine
relations (2.10) and (2.11) with astronomical data sets.
[1] McLaughlin Dean E. , Pudritz Ralph E. A model for the internal structure of molecular cloud
cores. arXiv:astro-ph/9605018.
[2] McKee Christopher F. The dynamical structure and evolution of giant molecular clouds.
arXiv:astro-ph/9901370.
[3] Williams Jonathan P., Blitz Leo, McKee Christopher F. The structure and evolution of molec-
ular clouds: From clumps to cores to the IMF. arXiv:astro-ph/9902246.
[4] McKee Christopher F. , Holliman John H.,II Multi-pressure polytropes as models for the struc-
ture and stability of molecular clouds. I. Theory. arXiv:astro-ph/9903213.
[5] Padoan Paolo, Nordlund Ake, Kritsuk Alexei G., Norman Michael L., Shing Pak Li Two
regimes of turbulent fragmentation and the stellar IMF from primordial to present day star
formation arXiv:astro-ph/0701795.
[6] Blitz L. 1993. Giant molecular clouds. In Protostars and Planets III, eds E.H. Levy and
J.I. Lunine, (Tucson: Univ. of Arizona Press), pp 125-161.
[7] Blitz L., Stark A. A. 1986. Detection of clump and interclump gas in the Rosette molecular
cloud complex. Astrophys. J. Lett. 300:L89-L93
[8] Larson R. B., 1981, MNRAS, 194, 809.
[9] Zaharow A. W. Gravitational stability of finite massive bodies. arXiv:astro-ph/0610450.
[10] Caselli P., Myers P. C. 1995, ApJ, 446, 665.
http://arxiv.org/abs/astro-ph/9605018
http://arxiv.org/abs/astro-ph/9901370
http://arxiv.org/abs/astro-ph/9902246
http://arxiv.org/abs/astro-ph/9903213
http://arxiv.org/abs/astro-ph/0701795
http://arxiv.org/abs/astro-ph/0610450
	Introduction
	larson's lows
	summary
	References

  It was found that approximately constant column densities of giant molecular
clouds (Larson's low) can be explained as cloud existence condition in external
(galactic) gravitational field. This condition can be also applied to objects
(clumps and cores) embedded into the cloud and its gravitational field. Derived
existence condition do not rely on any internal dynamic of a cloud and embedded
objects.

Introduction
Giant molecular clouds (GMCs) play a crucial role in the star formation process
([1],[2],[3],[4],[5]). GMCs are complex objects with masses ∼ 105−6M⊙, diameters ∼ 50pc
and average densities nH2 ∼ 10
2cm−3 (e.g. [6]). GMCs are generally gravitationally bound
and may contain several sites of star formations. Equilibrium of self-gravitating gas was the-
oretically investigated in many works (e.g. [4],[9] and references in these papers). Internal
structure of GMS is usually very complicated. The inhomogeneous structure of cloud could
be described as a set of descrete clumps ([7]). These clumps themselves contain dense cores
with densities n ∼ 103−5cm−3. The one point of view at this time is that this cloud structure
is due to supersonic turbulence. Remarkably enough, the properties of cloud complexes are
rather simply interrelated. Total masses, mean densities and average velosity dispersions
vary with sizes (effective radii) roughly as M ∝ R2,ρ ∝ R−1, σ ∝ R1/2 (Larson’s lows) ([8]).
In this paper I propose simple hypothesis to explain relation between masses and sizes of
clumps and cores, embedded into clouds.
∗Electronic address: zaharow@gmail.com
http://arxiv.org/abs/0704.1214v1
mailto:zaharow@gmail.com
2. Larson’s lows
Let us briefly consider some of the most salient characteristics of GMCs summarized by
Larson ([8]). See alsow ([2]). The first relation is the line width-size relation: molecular
clouds are supersonically turbulent with line widths ∆v that increase as a power of size,
∆v ∝ Rp. Larson himself estimated that p ≃ 0.38. Subsequent work has distinguished
between the relation valid for a collection of GMCs and that valid within individual GMC.
Within low-mass cores, Caselli & Myers ([10]) found that the nonthermal velocity dispersion
σnt ≃ 0.55R
pc km s
−1 , (2.1)
which is near to relation
σ ∝ R1/2 . (2.2)
Larson’s second result was that GMCs and clumps within them are gravitationally bound.
It implies that
σ2 ∝ GM/R . (2.3)
This relation is result of virial equilibrium.
His third conclusion was that all GMCs have about the same column density
N ≃ const . (2.4)
Column density is defined as
N ∝ M/R2 ∝ nR . (2.5)
As Larson pointed out, only two of these conclusions are independent: any one of them
can be derived from other two. Opposite to the second conclusion the first and the second
Larson’s lows have no evident explanation. I think this explanation may be as follows.
Embedded clumps and cores moves through the cloud in its gravitational field and are
subject of turbulent motion and various accelerations. If their own gravitational field is not
enough to hold on these objects they must rapidly decay. It is easy to derive necessary
condition for confinement gaseous body with mass M and radius R in external gravitational
field of cloud with mass Mcl and radius Rcl. The variation of self gravitational potential ∆φ
must be greater then that of the cloud on size R. Bearing in mind that
∆φ ≃ GM/R (2.6)
∆φcl ≃ R
GM/Rcl ≃ GMR/R
cl (2.7)
we get
GM/R ≥ GMclR/R
cl (2.8)
or equivalently
M/R2 ≥ Mcl/R
cl . (2.9)
Inequality (2.9) is the main result of this work. It is important for undestanding the third
Larson’s low (2.4). Inequality (2.9) is strong only for very massive and compact objects
within the cloud which are hard to generate in turbulent motion. So inequality can not
be strong for almost all objects. Further, in expression (2.7) we did not take into account
placement of object within the cloud. Summarizing all we can generalize third Larson’s low
in the form
Mcl(r)
, (2.10)
where r is the dinstance of object from the center of cloud, Mcl(r) is the mass of cloud within
r, C is the non-dimensional constant of order one. As far as clouds itself are concerned they
move in galactic gravitational field through the interstellar gaseous media (ISM) and we can
get for them analogous condition
Mgal(r)
, (2.11)
where r is the dinstance of cloud from the center of galactic, Mgal(r) is the mass of galactic
within r.
3. Summary
We have seen that it is possible to understand the third Larson’s low and some proper-
ties of molecular clouds and objects within them in terms of their existence conditions in
external gravitational field. In particular, we have seen possible reason why the clouds have
approximately constant column densities. I have to stress now that conditions (2.10) and
(2.11) do not rely on any internal dynamic of cloud. It will be also interesting to examine
relations (2.10) and (2.11) with astronomical data sets.
[1] McLaughlin Dean E. , Pudritz Ralph E. A model for the internal structure of molecular cloud
cores. arXiv:astro-ph/9605018.
[2] McKee Christopher F. The dynamical structure and evolution of giant molecular clouds.
arXiv:astro-ph/9901370.
[3] Williams Jonathan P., Blitz Leo, McKee Christopher F. The structure and evolution of molec-
ular clouds: From clumps to cores to the IMF. arXiv:astro-ph/9902246.
[4] McKee Christopher F. , Holliman John H.,II Multi-pressure polytropes as models for the struc-
ture and stability of molecular clouds. I. Theory. arXiv:astro-ph/9903213.
[5] Padoan Paolo, Nordlund Ake, Kritsuk Alexei G., Norman Michael L., Shing Pak Li Two
regimes of turbulent fragmentation and the stellar IMF from primordial to present day star
formation arXiv:astro-ph/0701795.
[6] Blitz L. 1993. Giant molecular clouds. In Protostars and Planets III, eds E.H. Levy and
J.I. Lunine, (Tucson: Univ. of Arizona Press), pp 125-161.
[7] Blitz L., Stark A. A. 1986. Detection of clump and interclump gas in the Rosette molecular
cloud complex. Astrophys. J. Lett. 300:L89-L93
[8] Larson R. B., 1981, MNRAS, 194, 809.
[9] Zaharow A. W. Gravitational stability of finite massive bodies. arXiv:astro-ph/0610450.
[10] Caselli P., Myers P. C. 1995, ApJ, 446, 665.
http://arxiv.org/abs/astro-ph/9605018
http://arxiv.org/abs/astro-ph/9901370
http://arxiv.org/abs/astro-ph/9902246
http://arxiv.org/abs/astro-ph/9903213
http://arxiv.org/abs/astro-ph/0701795
http://arxiv.org/abs/astro-ph/0610450
	Introduction
	larson's lows
	summary
	References
