Exploring Dark Sky Ahead: Calculating Escape Velocity of Stars & Galaxies

[Buzz Bloom]

From time to time I see discussions about the expansion of the universe that describe a future state where the view of the sky from Earth (assuming Earth were to still be in existence) will be completely dark. There are two possible explanations of this prediction.

1. The universe will eventually exhaust its supply of hydrogen, the fuel that stars use to make light (and to make other things as well).
2. The expansion of the universe will cause shining stars and galaxies to eventually move outside of the Earth's observable universe.

It is explanation (2) I would like some help with.

Consider a system of currently shining stars and galaxies of total mass M (including dark stuff) which fits into a sphere of radius R. If the distribution of the mass is approximately radially symmetric, we can calculate the approximate escape velocity V_esc of a test particle at the edge of the sphere.

V_esc = Sqrt (2 G M / R)

V_esc can then be compared with the velocity V with which a test particle at the periphery of a system will be moving away from the system center due the universe expansion.

V = R h_inf,
where hinf is the calculated limit of the Hubble constant as time approaches infinity. If V < V_esc, then the system would be gravitationally sufficiently bound to avoid disruption by universe expansion (assuming spherical symmetry).
Otherwise, not.

My questions are:
1. In general, what is the approximate effect of a non-radially symmetric, non-spherical shape on this calculation of V_esc?
2. In particular, if we assume the mass distribution is two-dimensional and circularly symmetric, how would this effect the calculation of V_esc?

Below are some examples. Here are the values for G and hinf used for these examples.

G = 6.69 x 10¹¹ m³ s^-2 kg^-1
h_inf = 1/17.3 Gy^-1 = 1/5.45 x 10¹⁷ s^-1
https://www.physicsforums.com/threads/the-hypersine-cosmic-model.819954/page-3#post-5197970Example 1. Laniakea super cluster
Laniakea is very much NOT a spherically nor circularly symmetric example.
https://en.wikipedia.org/wiki/Laniakea_Supercluster
M = 10¹⁷ solar masses = 2 x 10⁴⁷ kg
R = 260 million light years = 2.46 x 10²⁵ m
V_esc = 1.14 x 10⁶ m/s
V = 4.51 x 10⁷ m/s

Since V >> V_esc, I would guess that the shape of the Laniakea supercluster would likely have no effect on the system's not being gravitationally bound with respect to the expanding universe.

Example 2. Local Group
This example is much closer to being symmetrical than Laniakea.
http://mnras.oxfordjournals.org/content/384/4/1459.full.pdf+html
M = 5.27x 10¹² solar masses = 1.05 x 10⁴³ kg
https://en.wikipedia.org/wiki/Local_Group
R = 5 million light years = 4.73 x 10²³ m
V_esc = 5.46 x 10⁴ m/s
V = 8.68 x 10⁵ m/s

Since V > V_esc, the Local Group is also not gravitationally bound with respect to the expanding universe.

Example 3. Milky Way
This is approximately a circularly symmetric example.
https://en.wikipedia.org/wiki/Milky_Way
M = 7 x 10¹¹ solar masses = 1.4 x 10⁴² kg
R = 160 thousand light years = 1.51 x 10²² m
V_esc = 1.11 x 10⁵ m/s
V = 2.78 x 10⁴ m/s

Since V < V_esc, the Milky Way IS gravitationally bound with respect to the expanding universe, assuming spherically symmetry. Would it still be calculated as bound if only two dimensional and circularly symmetric?

Also, I note it as unexpected that the Milky Way has a greater value for V_esc than does the much large Local Group.

 

[mfb]

Also, I note it as unexpected that the Milky Way has a greater value for Vesc than does the much large Local Group.

So you can escape from the Local Group without leaving the Milky Way?
That is not possible, so some assumption has to be wrong. Here it is the assumption of a spherical mass distribution.

The galaxies are not all moving away from each other with the speed the Hubble expansion would predict: gravity pulled them together in the past as well. This is significant for all gravitationally bound structures.

2. In particular, if we assume the mass distribution is two-dimensional and circularly symmetric, how would this effect the calculation of V_esc?

Should increase it a bit, but that is not a typical distribution. The dominant dark matter has a roughly spherical distribution within galaxies, but within larger structures you get several spheres.

2. The expansion of the universe will cause shining stars and galaxies to eventually move outside of the Earth's observable universe.

There is a more subtle effect: even in gravitationally bound systems, star orbits are chaotic over longer timescales. Stars or their remnants will eventually get enough energy to leave the galaxy or galaxy cluster, or fall into the central black hole. If they leave the system, expansion of space will eventually shift them out of the observable universe.

 

[Buzz Bloom]

Hi @mfb:

Thank you for your prompt and useful post.

So you can escape from the Local Group without leaving the Milky Way?
That is not possible, so some assumption has to be wrong. Here it is the assumption of a spherical mass distribution.

The value of V_esc is calculated for something at the perihery of of the sysyem. So, expansion would remove something from the local group at its periphery, but not something from the Milky Way at its periphery. If the Milky Way were at the periphery of the local group, then this would be a contradiction regarding something at the periphery of both. I think the odd ordering of the two V_esc values probably just implies that the Milky Way must be well inside the Local Group.

Stars or their remnants will eventually get enough energy to leave the galaxy or galaxy cluster, or fall into the central black hole. If they leave the system, expansion of space will eventually shift them out of the observable universe.

Thank you especially for pointing this out. I missed this third mechanism entirely.

Regards,
Buzz

 

[mfb]

I think the odd ordering of the two V_esc values probably just implies that the Milky Way must be well inside the Local Group.

It does not. It implies escape velocities are not that simple.
To add another complication: there is mass outside the structure you are looking at as well, which will lower the escape velocity.

 

[Buzz Bloom]

Hi mfb:

Thanks again.

It does not. It implies escape velocities are not that simple.

That illustrates why I like this forum so much. The opportunities to learn never end.

Regards,
Buzz

 

[Buzz Bloom]

Should increase it a bit, but that is not a typical distribution. The dominant dark matter has a roughly spherical distribution within galaxies, but within larger structures you get several spheres.

Hi @mfb:

Is there a physical theoretical explanation of why the stars in most galaxies are seen in their photographs to have a generally flat 2D distribution while the much greater quantity of accompanying dark matter has been determined to have a more spherical distribution?

Regards,
Buzz

 

[D H]

Is there a physical theoretical explanation of why the stars in most galaxies are seen in their photographs to have a generally flat 2D distribution while the much greater quantity of accompanying dark matter has been determined to have a more spherical distribution?

TL;DR response: The second law of thermodynamics.First off, some galaxies aren't flat. A good portion of galaxies are elliptical rather than spiral. Some astronomers and cosmologists are of the opinion that most galaxies are elliptical. We see mostly spiral galaxies because of the current state of the evolution of the universe and because we only have local information. Extremely remote galaxies are only a few pixels across, even in the best images of those galaxies.

Stars, like dark matter, interact almost exclusively by gravitation. (Stars occasionally do collide, but this is so extremely rare that this can be discounted.) Except in the case extreme conditions such as a pair of neutron stars, gravitation is more or less a conservative force. A system that conserves energy and angular momentum is going to more or less maintain its shape. Something besides gravitation is needed to explain the fact that our galaxy is spiral, as are many of the other galaxies whose shape can be ascertained.

That "something else" is electromagnetism. The dust and gas that eventually form stars interact electromagnetically as well as by gravitation. Electromagnetism dominates over gravitation, by many orders of magnitude, in the the near collision of a pair of dust or gas particles. Particles that interact via electromagnetism emit photons. These photons drain energy from the clouds of gas and dust that eventually form stars, which means that those clouds are dissipative systems.

This is where the second law of thermodynamics comes into play. A system that is closed but is not isolated will migrate toward a state that minimizes the total energy of the system. Those gas and dust clouds will (and do) evolve toward a state that minimizes the energy of the cloud, which is a more or less flat configuration. This is very important both on the stellar level and on the galactic level.

 

[Buzz Bloom]

Hi @D H:

Thank you very much for you prompt post with its excellent answer to my question. I had a thought that EM must play some role, but I couldn't think of way for it to work.

Electromagnetism donates over gravitation

Just to nit about what I think is a typo: "donates" should be "dominates".

Regards,
Buzz

BTW: I recently started a conversation with you which you may not have noticed. It includes some questions about a closed thread. I would very much appreciate any information you could give me about that topic.

 

[D H]

Hi @D H:
Just to nit about what I think is a typo: "donates" should be "dominates".

DYAC! Your nit is correct.