How Long Would It Take For An Advanced Alien Civilization To Reach Earth Using 'Conventional' Propulsion?

 

                                 Location of our solar system in Milky Way galaxy

In their paper,  ‘Galactic Civilizations: Population Dynamics and Interstellar Diffusion’  (Icarus, Vol. 46, June 1981, page 293) authors William Newman and Carl Sagan considered a more of less conventional passage to Earth of of an advanced alien civilization.  The mode considered was conventional spacecraft travel - treated as "diffusion" through cosmic space, i.e. in terms of applying known diffusion front equations to simulate the transfer of a large alien armada to Earth over possible centuries.

 The authors began with a standard diffusion equation, treating the spread of any colonizing civilization similarly to any medium that diffuses – for example, viruses, or general infections, or even human populations (say in the early colonization of the New World).. The basic diffusion equation used was (p. 301, eqn.12):

d(r)/dt = DIV (D(x,t,r) · grad r(x,t)

where, r(x,t) describes the population density at time t, and position x, and D is the diffusion coefficient in terms of x, t and r. The preceding equation is then tweaked and used as the basis for future refinements. Rather than weary the casual reader with the dozens and dozens of equations leading to the Results section (page 314), I will simply commence at that section and then go from there.

The authors' first major computation is of N’, the steady state number of extant advanced civilizations in the Milky Way. This is essential to obtain because it is one of the key variables used to compute the mean distance between advanced civilizations in the Milky Way:

L _M = (2.5 x 10 ¹¹  /N’) ^1/3

Where the numerator refers to the number of  stars estimated in the galaxy, and N the estimated civilization-bearing stars. The result is in parsecs, assuming the mean separation between stars in the galaxy is 1pc = 3.26 light years. (Bear in mind while our region near the outer rim is sparse with stars, the interior third of the Milky Way is teeming with them, very densely packed)

Based on a star –planet formation factor, f * ≈  1, and a mean lifetime for an advanced civilization of 10⁶ years, the authors obtain: N’ = f(10⁶) = 10⁶ , or one million advanced civilizations in the Milky Way alone.

My inclination is actually to increase f  to  ≈ 2, based on the discovery (as of this writing) of 5,425 actual extra-solar planets in 4,001 planetary systems. All of which were unknown at the time Newman and Sagan published their paper. This would yield double the number of advanced civilizations, but we will retain the more conservative estimate.

Then, the mean distance between advanced civilizations in the Galaxy is:

L_M = (2.5 x 10 ¹¹  /10⁶ ) ^1/3  = 63 pc = 205 Light years

Readers may well not appreciate this, but this is literally “next door neighbors” in terms of the galaxy!

The authors’ next task is to obtain the velocity of the colonization wavefront which they give as (Eqn. (79), page 316):

V = (v  _j )(D g) ^1/2

Here, (v _j ) is a dimensionless constant of order unity(1), and g ≈  0.1 (based on the rate of migration of human populations today (Newman and Sagan estimated 0.01 /yr, but that was nearly 30 years ago before the age of globalization). The diffusion coefficient, D, they (very) conservatively estimate at: D ≈ (2 x 10 ^-8 pc ² /yr).

Thus, the colonization wave velocity would be:

V = (1)[ (2 x 10 ^-8 pc ² /yr) (0.1 /yr)] ^1/2 = 6.3 x  10 ^-9 pc/ yr

But the authors adjust this value (with "bias") i.e. toward younger civilizations - using historic Earth colonization rates-   to: 3 x  10 ^- 4 pc/ yr.  I removed some of that bias using a statistical algorithm, leading to the slower  colonization wave rate of : 

V =   4.4 x  10 ^-5 pc/ yr

 

Which would imply 1.4 x  10⁶   or 1.4 million years  (instead of 210,000 yrs.) before the colonization wave reached Earth, assuming a 63 pc distance to the planet of the nearest advanced colonizers.  (The lifetime of these advanced colonizer civilizations is assumed to be greater than 30 million years. In other words, they all would have had to survive their critical nuclear energy development phase.)

Now, before anyone gets too ecstatic, bear in mind:

1) Sagan and Newman based their diffusion coefficient on relatively low travel speeds (v much less than c) since anything near v ~ c would be enormously expensive in terms of shielding, propulsion (page 312). They opted then for speeds far below relativistic (e.g. ≈ 40,000 km/h).

2)They deliberately assumed a “random walk” diffusion with directional bias “away from population centers".

I personally suspect the first is way too conservative and ignores the sort of ingenuity and enterprise that may well apply to a truly advanced civilization which is also space faring. And again, just because we can’t imagine humans attaining relativistic speeds, doesn’t mean advanced aliens couldn’t. So, just a shift (reduction) of the base travel time to about one ten thousandth of what the authors use enhances the diffusion wave speed, V to 0.004 pc/yr.

This reduces the time to encounter to 1.57 x 10 ⁴ yrs. or just over 15,700 years. A blink of an eye in geological time.

Thus, if the alien colonizers commenced a journey in our general direction from about 3ky before the Holocene geological era, then they’d be roughly 700 years away from finding us. (Give or take 1000 years in terms of uncertainties). This means the colonizers may well be “right around the corner”. Or may even have already entered our terrestrial environment.

The conclusion here is that any of the strange craft we've seen in the released Pentagon files could well be craft from an advanced civilization, based on fine tuning the Sagan- Newman diffusion wave equations.  This doesn't prove the unusual craft seen in the released Pentagon video, e.g.  https://youtu.be/Ce6ZevfbIK0

is an actual alien craft, only that this hypothesis can't be excluded. Especially in light of the Sagan-Newman paper, especially with factors tweaked after the emergence of the exoplanet discovery era.