Why do Star Clusters dissipate?
Collisions
Virial Theorem:
Kinetic Energy:
Potential Energy:
yielding:
2𝐸𝑘𝑖𝑛 = −Ω
2𝐸𝑘𝑖𝑛 = 𝑛 ∙ 𝑚𝑖 ∙ 𝑣
2
= 𝑀 ∙ 𝑣
2
𝑣 … mean velocity of the members
relative to the cluster centre
Ω = −
1
2
∙
𝐺 ∙ 𝑀2
𝑅
2
𝑣
2
=
𝐺 ∙ 𝑀
2𝑅
2
Escape Velocity:
Collisions:
Density r and cross section s:
Example of a typical Open Cluster:
𝑡 𝑐𝑜𝑙𝑙= 1025 s => Collisions play no role
𝑣∞
2
= 4 ∙ 𝑣
2
𝑡 𝑐𝑜𝑙𝑙 ≈
1
𝜌 ∙ 𝜎 ∙ Δ𝑣
𝜌 =
𝑁
𝑅
3 𝜎 = 4𝜋 ∙ 𝑅∗
2
⇒ 𝑡 𝑐𝑜𝑙𝑙 =
𝑅
3
4𝜋∙𝑁∙𝑅∗
2∙Δ𝑣
𝑁 = 1000, Δ𝑣 = 10 km/s, 𝑅∗ = 2.5R⨀, 𝑅 = 5 pc
Even in the most inner core parts, collisions are
highly improper, but could occur
Conclusions:
1.Binary and Multiple systems are not results
of collisions in later stages but form already
at the very beginning
2.Members do, in general, not escape due to
collisions (swing-by effect), but their
peculiar velocity component is part of the
cluster formation or due to SNs
Why do Star Clusters dissipate?
Crossing time - escaping
Crossing Time:
Members can escape from a Star
Cluster on a relatively short time scale
Reason: Velocity dispersion caused by
the cluster formation and SN events
𝑡 𝑐𝑟𝑜𝑠𝑠 =
ത𝑅
Δ𝑣
Δ𝑣 = 10 km/s, 𝑅 = 5 pc ⇒ 𝑡 𝑐𝑟𝑜𝑠𝑠 = 4.9 ∙ 108
yr
Why do Star Clusters dissipate?
Differential Galactic Rotation
Total Mass of the Milky Way:
Gravitational acceleration of the complete star
cluster 𝑔 𝑂𝐶𝐿 and the individual member 𝑔∗ :
The difference of these two values, is the
force, of which “the Milky Way” tries to pull away
a star from the cluster
𝑀 𝑀𝑊 = 2 ∙ 1011M⨀
𝑔∗ =
𝐺 ∙ 𝑀 𝑂𝐶𝐿
𝑅 𝐺𝐶 − 𝑟 2
𝑅 𝐺𝐶 … Distance of the
star cluster’s centre to the
Galactic centre
𝑟 … Distance from star to
the star cluster’s centre
𝑔 𝑂𝐶𝐿 =
𝐺 ∙ 𝑀 𝑀𝑊
𝑅 𝐺𝐶
2
On the other side we have the gravitational force of
the star cluster. The stability radius 𝑟𝑠 is defined as:
For 1000 M⨀ => Diameter 20 pc
𝑔 𝑀𝑊,∗ =
2 ∙ 𝐺 ∙ 𝑀 𝑀𝑊 ∙ 𝑟
𝑅 𝐺𝐶
3 for 𝑟 ≪ 𝑅 𝐺𝐶
2 ∙ 𝐺 ∙ 𝑀 𝑀𝑊 ∙ 𝑟
𝑅 𝐺𝐶
3 =
𝐺 ∙ 𝑀 𝑂𝐶𝐿
𝑟𝑠
2 ⇒ 𝑟𝑠 = 𝑅 𝐺𝐶 ∙
𝑀 𝑂𝐶𝐿
2𝑀 𝑀𝑊
ൗ1
3
𝑟𝑠 = 10.9 ∙
𝑀 𝑂𝐶𝐿
1000
ൗ1
3
for 𝑅 𝐺𝐶 = 8 kpc in units of M⨀, pc
Summary
• Star Cluster dissipate because of
1. Differential Galactic Rotation
2. Internal Velocity Dispersion
3. Collisions in the first few Myrs
4. SN Explosions and corresponding Shock
Waves
5. ( Collisions with “Field Stars”)
• Explanations of the existence of Globular
Clusters?
• Valid for all Spiral Galaxies