Heat expansion of star-like cosmic objects induced by the cosmological expansion

7 Pages·0.11 MB·English

by  E. Schmutzer

Heat expansion of star-like cosmic objects induced
by the cosmological expansion∗
E. Schmutzer, Jena, Germany
Friedrich Schiller University
Received: 2001
2
0
0 Abstract
2
The heat expansion of a star-like cosmic object, induced by the cosmo-
n
logical bremsheat production within a moving body, that was predicted by
a
the Projective Unified Field Theory of the author, is approximately treated.
J
The difference to planet-like bodies investigated previously arises from an-
2
othermaterialconstitution.Theexponential-likeexpansionlawisappliedtoa

model with numerical valuesof theSun. Theresults are not in contradiction
1
v toempirical facts.
7
0
0 1 Review
1
0
These days it is generally recognized in science that, as the Sun burns through its
2
hydrogen on the main sequence, it steadily grows hotter and therefore more lumi-
0
/ nous. Itis alsoacceptedthat throughthis fact the Suncontinuoously growslarger.
h
Recently an interesting paper on this subject has been published (Korycansky et
p
- al. 2001).
o Thispaperdevotedtoasimilarsubjectfollowsthetheoreticallineofourprevious
r
t publication on the bremsheat expansion of a moving spherical planet-like body,
s
induced by the expansion of the cosmos (Schmutzer 2000a). This hypothetical
a
: bremsheateffecthasbeenderivedfromtheProjectiveUnifiedFieldTheory(PUFT)
v
oftheauthor(Schmutzer1995). Themovingstar-likebodyconsideredheremaybe
i
X orbitingaroundagalacticcentralbody. Asusualinastrophysics,theGausssystem
r of units is used. Let us first repeat the general theoretical results.
a
Theradialexpansionvelocityofthesurfaceofthesphere(rradiusofthesphere)
reads
dr dσ
3
v = =4E rσ . (1)
S
dt dt
The quantitiesoccurringmean(inourspecific terminology): σ is the scalaricworld
function determined by the scalaric cosmologicalfield equation, whereas
2
Σ γ M
S N c
E = (2)
S
4 (cid:18) σF0 (cid:19)
is the scalaric cosmic expansion factor with following meaning of the symbols ap-
−8 −1 3 −2
pearing: γ = 6.68·10 g cm s is the Newtonian gravitational constant and
N
M the mass of the central body. In this context one should take account that the
c
∗DedicatedtoProf. Dr. R.Kippenhahnontheoccasionofhis75thbirthday
1
2 Astron. Nachr.
relation M = M σ holds, where M is the scalmass (invariant constant mass) of
c c c
the central body. Further
f α
S c
Σ = (3)
S
3c
Q
is the scalaric material factor (α cubic thermal coefficient of dilation, c specific
c Q
heat, f heat consumptionfactor being anindividual quantity ofthe order ofmag-
S
nitude 1), and F0 = σ|R×V| is the modified arial velocity of the orbiting body
(constant of integration) appearing in its angular momentum (V orbital velocity,
|R| distance between the moving body and the galactic central body).
For further mathematical treatment of the equation (1) we assume constancy
of the scalaric cosmic expansion factor (2). Then we are able to integrate with the
result
4 4
r=r exp[E (σ −σ )], (4)
b S b
where
a) r =r(t=t ) and b) σ =σ(t=t ) (5)
b b b b
mean the value of the radius of the sphere and the value of the scalaric world
function at the time of the birth of the cosmic object considered (index b refers to
birth).
2 Application of the theory to a star-like cosmic
object (Sun as model)
For rough application of the theory presented above we refer to approximate nu-
mericalvaluesoftheSunasanumericalmodel(furthershort: sun)orbitingaround
the center of the Galaxy (index p refers to present, year = y):
a) R = 2.6·1022cm (radius of the orbit),
b) V = 2.2·107cms−1 (orbital velocity of the sun),
c) M = 1.8·1044g (acting mass of the Galaxy),
c (6)
d) r = 6.96·1010cm (present radius of the sun),
p
e) t = 4.66·109y (age of the sun).
sun
Let us mention that the value of the central mass acting at the place of the sun
resulted from our theory of the rotation curve (Schmutzer 2001). Obviously this
value is different from the usually accepted value of the mass of the whole Galaxy
which is somewhat greater.
Apartfromthese individualvalues (6)we further needthe presentcosmological
values
a) σ = 65.19,
p
dσ −9 −1 (7)
b) = 1.14·10 y .
(cid:18)dt(cid:19)
p
E.Schmutzer: Heat expansion of star-like cosmic objects ... 3
These values result from the solution of the system of the cosmological differential
equations of the closed isotropic homogeneous cosmological model investigated in
detail (Schmutzer 2000b).
By means of (6a,b) and (7a) we find the value F0 =3.73·1031cm2s−1.
Themainproblemarisingnowistofindawaytogetinformationonthescalaric
expansion factor E of the sun (2). For comparing it is appropriate to remember
S
the results obtained for planet-like cosmic bodies (Schmutzer 2000). We used the
following values applied in geophysics for the earth:
a) α = 6.84·10−5K−1,
c
(8)
b) c = 1.5·107cm2s−2K−1.
Q
For f =0.5 we obtaind the values
S
a) Σ = 7.6·10−13cm−2s2,
S
(9)
b) E = 9.35·10−8.
S
The situation with respect to the sun is insofar qualitatively different from the
situation just mentioned, since the interior of the sun consists of gas (mainly H,
partly atomic burning to He) with a very high temperature.
For simplicity we approximately treat this problem by referring to an perfect
gas for which the relation
ε= m0 u2 = 3kT (10)
2 2
is valid (m0 mass of a gas particle, u velocity of the gas particles subjected to
averaging, k Boltzmann constant, T kinetic temperature). As it is well known, for
a perfect gas the formulas
1 3k
a) α = = ,
c T m0u2
(11)
3k
b) c =
Q
2m0
hold. Hence the relations (3) and (2) take the form
2fSm0
Σ = (12)
S
3kT
and
2
fSm0 γNMc
E = . (13)
S
6kT (cid:18) σF0 (cid:19)
Using the numerical values (6) and (7) for the sun, we find
2
γ M
N c 7 2 −2
=2.45·10 cm s . (14)
(cid:18) σF0 (cid:19)
As mentioned above, it is our aim to treat cosmic bodies globally. Therefore it
seems to be acceptable to take the value T = 5·106K as an appropriate mean
value of the kinetic temperature of the sun. Let us further remember the value of
−16 2 −2 −1
the Boltzmann constant k=1.38·10 gcm s K and the value of the proton
mass m =1.68·10−24g (being used as the mass of the H-atom constituent of the
p
gas considered). Hence we obtain from (13) the numerical result (f =1)
S
−9
E =9.92·10 . (15)
S
4 Astron. Nachr.
3 Numerical evaluation and plotting for the Sun
(as a model)
3.1 Numerical formulas for radius and expansion velocity
Inserting the values (15), (6d) and (7) into formula (1), we find for the present
expansion velocity of the surface of the sun
−1
v =0.87cmy . (16)
p
Further we get information from formula (4) on the temporal behavior of the
radius of the sun. Inserting (16) into this equation we arrive at
−9 4 4
r =r exp[9.92·10 (σ −σ )]. (17)
p b p b
Nowfrom(6d)weknowthequantityr andfrom(7a)thequantityσ .Asmentioned
p p
above, there is a good consensus among astrophysicists that the age of the sun is
about4.66billionyears. Thereforeweneedthevalueofσ atthetimeofbirthofthe
sun. In order to find this value a small excursus to the results of our cosmological
model quoted above is necessary.
We arrived at an age of this cosmos of 18·109y, i.e. we need the cosmological
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values at the time t = 13.34·10 y. Checking our list of numerical data for the
b
wholecosmosafterthebigstart(Urstart),wefindthevaluesσ =σ(t=t )=55.44
b b
and σ4 =9.45·106.
b
Now we are prepared to calculate the radius of the sun at the birth with the
result
10
r =6.39·10 cm. (18)
b
3.2 Plotting of the temporal course of the radius and the
velocity
Theabscissaofthefigurespresentedinthefollowingreferstotherescaledtimeη =
0.945·10−9ty−1. Thispracticallymeans thatthe unit ∆η =1roughlycorresponds
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to10 y. Referringtothefiguresoftheradius,theunitoftheradiusattheordinate
is givenincm; referringto the figuresofthe velocity,the unit ofthe velocityat the
ordinate is cm/y.
The temporal course of the radius of the spherical body is presented in Fig. 1
for the whole time scale since the big start.
Thetemporalcourseofthe radialvelocityofthesurfaceofthe sphericalbodyis
presented in Fig. 2 which shows the course of the velocity for the whole time scale
since the big start.
3.3 Annotation to the mass of the Galaxy
IninvestigatingtheorbitalmotionofthesunaroundthecentralbodyoftheGalaxy,
for physical reasons it is necessary to use the gravitational force just at the place
of the moving sun. Because of the matter (particularly dark matter) outside the
sun it is obvious that the mass causing the gravitational force mentioned is not
the total mass of the Galaxy. We recently treated the problem of the radial mass
distributioninthe Galaxywithinthe frameworkofourtheoryofthe rotationcurve
of stars (Schmutzer 2001). In that paper we came to the numerical value (6c) of
the gravitationally acting mass at the place of the sun. All numerical calculations
performed above are based on this value.
E.Schmutzer: Heat expansion of star-like cosmic objects ... 5
rHhL
7·1010
6.8·1010
6.6·1010
6.4·1010
6.2·1010
h
2.5 5 7.5 10 12.5 15 17.5
5.8·1010
Figure 1: Temporal course of the radius for the whole time scale from the big start
to 18·109y
vHhL
2
1.5
1
0.5
h
2.5 5 7.5 10 12.5 15 17.5
Figure 2: Temporal course of the radial velocity of the surface for the whole time
scale from the big start to 18·109y
Fromformula(2)welearnthatthescalariccosmicexpansionfactorE contains
S
M quadratically. Thatmeansunderthecircumstancesoftheexponentialcharacter
c
of the expansionlaw (4)an extraordinarysensitivity of the radius onthe mass M .
c
With the aim of illustration we also performed the calculations on the basis of
45
the total mass of the Galaxy M = 2·10 g (to be found in astrophysical
c/total
literature), arriving at the rescaled temporal curves for the radius R(η) and the
radial expansion velocity V(η), both presented in Fig. 3 and Fig. 4. Hence the
present value of the radial expansion velocity V =55cm/y results.
p
The following figures show the temporal course of R(η) and V(η).
Let us conclude this paper with the general remark that in Fig. 1 and Fig. 2
the curves show a quasi-continuous course through the whole time scale from the
big start to the presence, i.e. there is no hint at the time of birth of the cosmic
object considered. In contrast to this behavior in Fig. 3 and Fig. 4 the birth is
significantlymarkedby asteeprise. All inallitseems thatthis theorycorresponds
to a longtime evolution of the star-like cosmic objects from “small to large” and
not according to the accretion concept from “large to small”.
6 Astron. Nachr.
RHhL
1·109
8·108
6·108
4·108
2·108
h
2.5 5 7.5 10 12.5 15 17.5
Figure 3: Temporal course of the radius for the whole time scale from the big start
to 18·109y (different numerical variant)
VHhL
0.5
0.4
0.3
0.2
0.1
h
2.5 5 7.5 10 12.5 15 17.5
Figure 4: Temporal course of the radial velocity of the surface for the whole time
scale from the big start to 18·109y (different numerical variant)
I would like to thank Prof. Dr. R.Kippenhahn (Goettingen) for interesting in-
formation on the physics of the sun and Prof. Dr. A.Gorbatsievich (Minsk) for
advice and help.
References
Korycansky,D.G.,G.LaughlinandAdams,F.C.: 2001,arXiv:astro-ph/0102126
(7 Feb), Astrophysics and Space Science (in press)
Schmutzer, E.: 1995, Fortschritte der Physik 43, 613
Schmutzer, E.: 2000a, Astron. Nachr. 321, 227
Schmutzer, E.: 2000b, Astron. Nachr. 321, 209
Schmutzer, E.: 2001, Astron. Nachr. 93, 103
E.Schmutzer: Heat expansion of star-like cosmic objects ... 7
Address of the author:
Ernst Schmutzer Cospedaer Grund 57, D-07743 Jena Germany