The modernity of the research of Romanian astronomer Constantin Popovici
We present the model of the photogravitational field proposed by the romanian mathematician Constantin Popovici, basic equations. We give the variation of orbital energy in the photogravitational two bodies problem and energy integral. In this case the orbital energy is not conserved.
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irk-123456789-1190942017-06-04T03:02:48Z The modernity of the research of Romanian astronomer Constantin Popovici Chiruta, C. We present the model of the photogravitational field proposed by the romanian mathematician Constantin Popovici, basic equations. We give the variation of orbital energy in the photogravitational two bodies problem and energy integral. In this case the orbital energy is not conserved. 2011 Article The modernity of the research of Romanian astronomer Constantin Popovici / C. Chiruta // Advances in Astronomy and Space Physics. — 2011. — Т. 1., вип. 1-2. — С. 106-109. — Бібліогр.: 15 назв. — англ. 987-966-439-367-3 http://dspace.nbuv.gov.ua/handle/123456789/119094 en Advances in Astronomy and Space Physics Advances in astronomy and space physics |
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We present the model of the photogravitational field proposed by the romanian mathematician Constantin Popovici, basic equations. We give the variation of orbital energy in the photogravitational two bodies problem and energy integral. In this case the orbital energy is not conserved. |
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Chiruta, C. |
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Chiruta, C. The modernity of the research of Romanian astronomer Constantin Popovici Advances in Astronomy and Space Physics |
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Chiruta, C. |
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Chiruta, C. |
| title |
The modernity of the research of Romanian astronomer Constantin Popovici |
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The modernity of the research of Romanian astronomer Constantin Popovici |
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The modernity of the research of Romanian astronomer Constantin Popovici |
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The modernity of the research of Romanian astronomer Constantin Popovici |
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The modernity of the research of Romanian astronomer Constantin Popovici |
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modernity of the research of romanian astronomer constantin popovici |
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Advances in astronomy and space physics |
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2011 |
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The modernity of the research of Romanian astronomer Constantin Popovici / C. Chiruta // Advances in Astronomy and Space Physics. — 2011. — Т. 1., вип. 1-2. — С. 106-109. — Бібліогр.: 15 назв. — англ. |
| series |
Advances in Astronomy and Space Physics |
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AT chirutac themodernityoftheresearchofromanianastronomerconstantinpopovici AT chirutac modernityoftheresearchofromanianastronomerconstantinpopovici |
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2025-07-08T15:13:00Z |
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2025-07-08T15:13:00Z |
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1837092129881456640 |
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The modernity of the research of Romanian astronomer
Constantin Popovici
C. Chiruta
The �Ion Ionescu de la Brad� University of Agricultural Sciences and Veterinary Medicine,
Aleea M. Sadoveanu, nr. 3, 700490, Iasi, Romania
kyru@uaiasi.ro
We present the model of the photogravitational �eld proposed by the romanian mathematician Constantin
Popovici, basic equations. We give the variation of orbital energy in the photogravitational two bodies
problem and energy integral. In this case the orbital energy is not conserved.
Introduction
The Constantin Popovici model was �rst proposed in 1923 (see e.g. [12]). In the following we try to
synthesize the remarkable results of Professor Constantin Popovici, as well as other contemporary researchers
in the photogravitational two bodies problem.
During the �rst part of the twentieth century, Romanian astronomer Constantin Popovici (1876-1956)
proposed an amendment to the law resulting from combining the Newtonian gravitation attraction and
radiation repulsion. He added a term which varies with the radial velocity [12], that leads to the law being
applied to the attractive and repulsive centrifugal forces.
It was also used by the Italian scientist Giuseppe Armellini in his work on the generalization of Newton's
law of universal gravitation (see e.g. [1]).
Born in 1878 in Iasi, Constantin Popovici achieved his elementary school, high school and higher education
in his hometown. He graduated from the Faculty of Science of the �Al. I. Cuza� University in 1900, having
a degree in mathematics. In 1905 he went to Paris, where he obtained an another license in mathematics,
then prepared and submitted a doctorate thesis in mathematics at the Sorbonne, in March 1908.
Back in his home country, he was appointed a professor of astronomy, geodesy and celestial mechanics at
the University of Iasi. In 1938 he moved to Bucharest, where he taught at the Department of Astronomy of
the Bucharest University until 1940 and led the Astronomical Observatory since 1937 till 1943.
In astronomy he studied the in�uence of light pressure on the movement of bodies within the solar system.
Constantin Popovici was an honorary member of the Romanian Academy (1948) and member (1949). He
died on November 26, 1956 and was buried in the �Belu� cemetery in Bucharest.
Constantin Popovici model has been reconsidered in the last decade by several Romanian scientists. E. g.,
M.-C. Anisiu [2, 3, 4] formulated the photogravitational problem of two body in Cartesian coordinates and
investigated the existence of equilibrium points; V. Mioc and C. Blaga [10, 11] using McGehee type variables
provided a qualitative study of the system of di�erential equations for the new variables; M. Barbosu and
T. Oproiu [5] determined the nature of equilibrium points for the two bodies photogravitational problem.
The photogravitational problem (for two, three or more bodies) has attracted more attention in recent
years [6, 8, 9, 15].
Constantin Popovici model. Basic equations
In Popovici C. model for the projection of the force onto the direction of the radius-vector one considers
the following relation [7]:
F = −A
r2
+
R
r2
−R
ṙ
c · r2
, (1)
where the �rst term corresponds to Newtonian attraction force (with A being the attraction of the luminous
body at unit distance r = 1), the second term represents the force caused by the light pressure of the central
106
Advances in Astronomy and Space Physics C. Chiruta
body (with R being the repulsion caused by light pressure at unit distance), and the last term introduced
by Popovici represents an increase of attraction force due to the �nite speed of light.
An expression similar to (1) was used by Giuseppe Armellini (1887− 1958) to generalize Newton's law of
universal gravitation. Armellini used the expression:
F = −Gmm′
r2
(1 + εṙ), (2)
where m and m′ are the masses of interacting bodies, G is the gravitational constant, r is the distance
between bodies and ε is the Armellini constant. The value of the constant was determined by comparison
between theory and observations. The law proposed by Armellini was �rst published in [1].
Starting from the relation (1) and using the notations k = A−R and ε = R
ck get (here ε is not the same
as in the relation introduced by Armellini):
F = −A−R
r2
− Rṙ
cr2
= − k
r2
(1 + εṙ). (3)
The following form was used by romanian researchers using the notation l = R/c [3] :
F = − k
r2
− Rṙ
cr2
= − k
r2
(1 + lṙ). (4)
The variation of orbital energy
in the photogravitational two bodies problem
The orbital energy in the Popovici model is not conserved. Starting from the relation (3):
m~̈r = − k
r2
(1 + εṙ) · ~r
r
, (5)
one can derive:
d
dt
(m~̇r2) = −2
d
dt
(
k
r
)
− 2εk
(
ṙ
r
)2
(6)
In the manuscript [12] one can �nd the following Popovici's theorem:
�L'énergie ne se conserve plus. Cette quantité
E =
v2
2
− k
r
(7)
que l'on appelle dans la Mécanique newtonienne l'énergie, varie dans le même sens avec le temps. La relation
suivante, qui est en même temps l'équation du mouvement:
dE
dt
= αk
(
r′
r
)2
, r′ =
dr
dt
, (8)
dE
dt > 0 attraction; dE
dt < 0 répulsion, nous fait voir comment l'énergie est dépensée par le mécanisme de la
propagation�.
Theorem 1. The variation of the energy of an in�nitesimal particle (during relative motion) has the
following property:
Ė = −εk
(
r′
r
)2
, (9)
where ε = R
ck . In this case if the energy does not conserve, then there is no prime integral.
Remark: If Ė = 0 one can �nd a �rst integral of energy to generalize the classical one, where energy
varies with time [7, 13] .
107
Advances in Astronomy and Space Physics C. Chiruta
Theorem 2. In the photogravitational two bodies problem, the C. Popovici model, the energy integral
takes its generalization form:
√
I − 2ρα
√
I − ρ2 · eS = H, where S =
α√
1− α2
arctan
√
I − ρ2 − ρα
ρ
√
1− α2
, (10)
where I = 2E + k2
C2 , ρ = C
r − k
C , α = εk
2C and H is a constant of integration.
Proof:
Starting from the relation which can be obtained from theorem 1, dE
dt = dE
dr · dr
dt = −εk
(
ṙ
r
)2, and knowing
that the theorem of the areas remains valid in this case, and the force is central, one can obtain:
(
dr
dt
)2
+
C2
r2
− 2k
r
= 2E,
and get the equality:
dE
dr
= −εk
r2
√
2E +
k2
C2
−
(
C
r
− k
C
)2
. (11)
Using the notations I = 2E + k2
C2 , ρ = C
r − k
C one obtains:
dE
dr
= −εk
r2
√
I − ρ2. (12)
Then di�erentiating relations one gets:
dI
dρ
= 2
dE
dρ
=
2εk
C
√
I − ρ2. (13)
Use the notation α = εk
2C a simpli�ed relation can be obtained:
dI
dρ
= 4α
√
I − ρ2. (14)
Changing the variable I → u as I = ρ2(1 + u2) one derives:
dI
dρ
= 2ρ(1 + u2) + 2ρ2u
du
dρ
, (15)
or:
2ρ2u
du
dρ
= 4αρu− 2ρ(1 + u2), (16)
from which one can obtain the following di�erential equation:
− udu
u2 − 2αu + 1
=
dρ
ρ
, (17)
that can be reorganized as:
dρ
ρ
= − (u− α)du
(u− α)2 + 1− α2
− αdu
(u− α)2 + 1− α2
. (18)
Integrating this relation one can obtain:
ln ρ = −1
2
ln(u2 − 2αu + 1)− α√
1− α2
arctan
u− α√
1− α2
+ H1, (19)
108
Advances in Astronomy and Space Physics C. Chiruta
or
H = ρ
√
1− 2αu + u2 · eS1 , S1 =
α√
1− α2
arctan
u− α√
1− α2
. (20)
And coming back to the �rst notation one can obtain:
√
I − 2ρα
√
I − ρ2 · eS = H, where S =
α√
1− α2
arctan
√
I − ρ2 − ρα
ρ
√
1− α2
, q.e.d. (21)
Remark: ε = 0 means that the following equalities take place: S = 0, eS = 1 and I = 2E + k2
C2 = H2,
thus E = 1
2
(
H2 − k2
C2
)
= h is constant. It means that the energy integral from the keplerian motion exists.
Conclusions
We discuss the photogravitational model proposed by the Romanian astronomer Constantin Popovici.
The term introduced by Constantin Popovici in order to consider the fact that the light pressure propagates
with a �nite velocity unlike the attraction force which propagates instantaneously is also important. This
has a breaking e�ect, similar to the motion in a resisting medium that had been marked out by Radziewsky
since 1950 [14].
Acknowledgement
The author is grateful to Dr. T. Oproiu for valuable comments which helped in improving the text.
References
[1] Armellini G. Rendiconti., Accad. Naz. Lincei, V. 26, p. 209 (1937)
[2] Anisiu M. C. Analele stiinti�ce ale Univ. �Al. I. Cuza�, V. 16, pp. 115-119 (1995)
[3] Anisiu M. C. Rom. Astron. J., V. 5, no. 1, pp. 49-54 (1995)
[4] Anisiu M. C. Rom. Astron. J., V. 13, no. 2, pp. 171-177 (2003)
[5] Barbosu M., Oproiu T. General Mathematics, V. 12, no. 2, University �Lucian Blaga�, Sibiu, pp. 19-26 (2004)
[6] Chorny G. F. Celestial Mech. Dyn. Astr., V. 97, pp. 229-248 (2007)
[7] Chiruta C., Oproiu T. PADEU, Astron. Dept. of the Eotvos Univ., V. 19, pp. 117-128 (2007)
[8] Kunitsyn A. L., Tureshbaev A. Celestial Mechanics, V. 35, pp. 105-112 (1985)
[9] Kunitsyn A., L., Polyakhova E. N. Astronomical and Astrophysical Transactions, V. 6, pp. 283-293 (1995)
[10] Mioc V., Blaga C. Rom. Astron. J., V. 11, pp. 45-51 (2001)
[11] Mioc V., Blaga C. Serb Astron. J., V. 165, pp. 9-16 (2002)
[12] Popovici C. Bul. Astron., V. 3, pp. 257-261 (1923)
[13] Robertson H. P. Mon. Not. Roy. Astron. Soc., V. 97, pp. 423-438 (1937)
[14] Radzievsky V. V. Astron Zh. 27, V. 4, pp. 250-256 (1950)
[15] Zimovshchikov A. S., Tkhai V. N. Solar System Research, V. 38, no. 2, pp. 155-164 (2004)
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