TIME SCALE OF OUR UNIVERSE — OPIK 219 



correction, evidently, must be added to that found by him. The total 

 correction amounts thus to 1.7 + 1.3 = 3.0 mag, or an increase in the 

 distances of nebulae (except the nearest, \Yhich are based on cepheids) 

 in a ratio of 4 to 1. The constant of recession, or the rate of increase of 

 velocity with distance as based on observed red shifts, now becomes 

 145 km./sec. per magaparsec (3.25 million light-years), only one- 

 quarter of the formerly assumed value. The expansionistic time scales 

 are increased fourfold, and even the shortest will yield more than the 

 lower limit — the age of the earth. 



The retention of the cosmological constant by Eddington and Le- 

 niaitre was justified by the need to extend the time scale ; the slow phase 

 of expansion, when gravitational attraction and cosmic repulsion 

 nearly balanced each other, allowed this to be done almost indefinitely. 

 Now, with the increased distances, cosmic repulsion becomes a super- 

 structure of a purely esthetic nature, serving no practical purpose. 

 Besides, Einstein, the originator of the concept, has disavowed the 

 cosmological constant ever since, in spite of the then favorable numeri- 

 cal aspect of the problem. 



Without the cosmological constant, the Friedmann-Einstein cos- 

 mological models (54) furnish a working hypothesis best suited to 

 deal with the expanding universe. These models are very similar to 

 an ordinary gravitating sphere in uniform expansion. Gravitation, 

 working against expansion, is slowing it down. Wlien the velocity of 

 expansion is below a certain limit, the expansion will be ultimately 

 stopped by gravitation, and contraction will start ; when the velocity 

 of expansion equals or exceeds the limit (velocity of escape), gravita- 

 tion will be unable to stop it and the sphere will disperse into space, 

 expansion never ceasing. According to the general theory of relativ- 

 ity, and without cosmological repulsion, a similar state of affairs in the 

 expanding universe prevails. The first case, when expansion is ulti- 

 mately stopped by gravitation, would correspond to positive curvature 

 of space, or to closed space and a relapse of the universe, after maxi- 

 mum expansion, into the original state of high density (atom or 

 nebula) . The second case would correspond to zero or negative curva- 

 ture, to open and infinite space, and to a one-way development of the 

 universe by perpetual expansion. 



For an expansion constant of 145 km./sec. per megaparsec the line 

 between the two cases is set by a certain limiting value of the average 

 density of matter in space (i. e., if all the matter of the universe were 

 spread unifonnly over its entire volume, instead of being concentrated 

 into galaxies, stars, and atoms), equal to 3.9 XIQ-^^ gm./cm.^ The 

 volume of the earth filled with matter of so low a density would contain 

 only a mass of 42 milligrams. 



The probable value of the average density of matter in space can be 

 estimated in the following way. There are in the universe, on the 



