Observed Properties of Molecular Clouds

Question 1

cs From the Equation for Cloud Core Supported by Thermal Pressure Alone

Answer 1

1/2 m v² = 3/2 kb T
-this is for velocity in all three dimensions, so considering only the velocity along the line of sight
1/2 m vx² = 1/2 kb T
=>
Δv ~ cs = √[kb*T/μ*mh]

Question 2

Significance of Speed of Sound

Answer 2

-speed of sound in the medium sets the speed at which information/disturbances, e.g. shock waves, will pass through the cloud

Question 3

Virial Equation for a Cloud Supported Only by Thermal Pressure

Answer 3

3VcPs = 2U + Ω

Question 4

cs From the Virial Equation for a Cloud Supported Only by Thermal Pressure

Answer 4

3*Vc*Ps = 2U + Ω
-the external (surface) pressure is negligible, thus:
2U = -Ω
=>
cs = √[G*Mc/5*Rc]

Question 5

What is cs for a typical molecular cloud?

Answer 5

0.2 km/s

Question 6

What do the projected (line of sight) velocities look like?

Answer 6

• we observe the projected line of sight velocities, and due to Doppler shifting, we see emission over a range of velocities
• the emission line is Gaussian shaped with a dispersion of order 0.2km/s
• the full-width half maximum is about 2.3 times the dispersion

Question 7

FWHM

Answer 7

FWHM = Δv = √[8ln2] σ

Question 8

Larson’s Study of Molecular Clouds

Answer 8

-log of thermal velocity dispersion, ln σ, is proportional to log of cloud size, lnL

Question 9

Thermal and Non-Thermal Velocity Widths

Answer 9

Δv² = Δvth² + Δvnt²

-where v is total, vth is thermal and vnt is non-thermal

Question 10

Are thermal or non-thermal components of velocity width dominant?

Answer 10

• non-thermal velocities are observed to be dominant over the thermal component
• if we consider successively smaller clouds, the velocity approaches the ambient thermal veloctiy

Question 11

Does the presence or absence of a protostar effect the relationship between cloud size and velocity width?

Answer 11

• a protostar heats the cloud surrounding it
• but the same proportional relationship between logR and logΔv is still found
• this is further evidence of a non-thermal dominating component

Question 12

Larson’s Empirical Law

Answer 12

-from his compilation of the available data, Larson derived an empirical relationship between line width and cloud (core) size:
σ (km/s) = 1.1 * [L(pc)]^(0.38)
-where 0.1pc≤L≤100pc

Question 13

Crossing Time

Answer 13

-the timescale associated with internal motions:
τ ~ L/σ
-during this time, appreciable dissipation of turbulent motions will occur, gravitational collapse and star formation will probably also occur, at least in some parts of them molecular cloud
-within a crossing time, the cloud can then be partially or completely dispersed or restructured by the effects of stellar winds, HII regions etc.

Question 14

Relationship Between Crossing Time and Free-Fall Time

Answer 14

τ ~ 2*tff

Question 15

Crossing Time for a Typical Molecular Cloud

Answer 15

τ ~ 210^5 yr for L~0.1pc
τ ~ 1.710^7 yr for L~100pc
-thus even the largest molecular cloud complexes must be rather transient and will be completely restructured if not completely dispersed after only a few time 10^7yr

Question 16

Expected vs Observed Star Formation Rate in the Milky Way

Answer 16

• the Milky Way contains 1-3*10^9M☉ of molecular gas
• combine the Jeans mass and free-fall time together and one concludes that molecular clouds within our galaxy should be highly unstable to gravitational collapse
• we should be observing a star formation rate that converts 200-400M☉ per year into stars
• but we calculate an actual rate of only ~3M☉ per year
• SO molecular clouds cannot be being supported by thermal pressure alone

Question 17

Possible Other Sources of Cloud Support

Answer 17

• the obvious conclusion from the difference in predicted and observed star formation rates is that molecular clouds cannot be being supported by thermal pressure alone
• this further implies that cloud collapse times cannot be gauges by the free-fall time scale since this based on a cloud supported only by thermal pressure
• we still continue to use this value as a useful lower limit to the cloud collapse time
• cloud lifetimes are estimated to be ≥10Myr
• candidates for other sources of cloud support are rotation, magnetic fields and turbulence

Question 18

Is rotation a source of support for molecular clouds?

Answer 18

-clouds exhibit velocity gradients ~ 1km/s and Ω~10^(-14)rad/s
Δv ~ RΩ
-input comes from Galactic rotation or cloud-cloud collisions
-for a typical molecular cloud:
Δvrot = 0.03km/s
-compared with the thermal component:
Δvth = 0.2km/s
-rotational energies are generally small compared to gravitational energies

Question 19

Radius of a Typical Molecular Cloud

Answer 19

R=0.1pc, M=5M☉

R=1pc, M=10M☉

Question 20

Are magnetic fields a source of support for molecular clouds?

Answer 20

-perturbations in a molecular cloud can give rise to magnetohydrodynamic (MHD) waves called Alfven waves
-they propagate with Alfven speed, vA:
vA = B / √[4πμmh]
-expression for non-thermal velocity dispersion in terms of B:
σnt = Δvnt/√[8ln2] ~ vA/√[3] = B/√[12πμ*mh]
=>magnetic fields can support clouds if |B_| is sufficiently high
-so a cloud with a weak B field would need another mechanism of cloud support

Question 21

Is turbulence a source of support for molecular clouds?

Answer 21

• the supersonic line widths are interpreted as evidence for supersonic turbulence
• initially thought to be a mechanism of supporting clouds against gravity
• now considered to be a fundamental part of determining cloud properties such as lifetime, morphology and star formation rate
• turbulence is a multiscale phenomenon in which kinetic energy cascades from large scales to small scales
• the issue with turbulence is that it decays very quickly (in a crossing time) which has implications for the formation and lifetimes of GMCs

Question 22

What are molecular clouds most likely supported by?

Answer 22

-magnetic fields through the propagation of Alfven waves (if the B_ field is sufficiently strong) and turbulent motions