Laurent Nottale

Planetary System around the Pulsar PSR 1257+12

Keywords: solutions

We apply the scale relativity framework, in which the formation of a planetary system is described in terms of a macroscopic Schrödinger equation, to the study of the three planets orbiting the pulsar PSR 1257+12. We describe in terms of wave packets the clouds from which the planets are formed, finally lying at their center of gravity. This leads us to predict that the semimajor axes of the planet orbits should be quantized as n 2 + n/2 (where n = 5, 7, 8 in this system, plus another possible planet at n = 2). We find prime integrals of the motion equation for the 3:2 near-resonant subsystem of two of the planets. This allows us to show that it has remained stable and that its evolution, since about one billion years, has not affected the structures present at the end of its formation. Our theoretical predictions can therefore be compared with its present state. We find that the observed ratios of planet semimajor axes differ from their theoretically expected values by only 4 × 10 -4 and 6 × 10 -5 , and that this remarkable agreement can be shown to be statistically highly significant.

The first extrasolar planetary system ever discovered has been the system of three planets found by Wolszczan around the pulsar PSR B1257+12 [16,17]. Even though this star is not solar-like, this system deserves a special study [10,11,[START_REF] Nottale | Workshop Pulsars, Theory and Observations[END_REF], because (i) the pulse timing measurements allow a very precise determination of the orbital elements of the planets and (ii) their small masses (four Earth mass for two of them and the Moon mass for the third one) allow both the celestial mechanics models of evolution and the scale relativity model of formation to become also very precise. This system therefore stands out as a kind of ideal gravitational laboratory for studying the formation and evolution of planetary systems and putting models to the test.

The planets probably result from an accretion disk formed around the very compact star after the supernova explosion. Even though this is a secondary process, there is general agreement that the formation process should be similar to the standard picture [START_REF] Currie | [END_REF].

We can therefore expect the purely gravitational formation process, described in the scale relativity approach by a macroscopic gravitational Schrödinger equation, to be still valid in this case. Moreover, the smallness of the planet masses implies very few perturbations between the subdisks from which the three planets have been formed and a negligible effect of self-gravitation, so that the theoretical predictions (based on conservation laws, in particular of the center of gravity of each subdisk) are expected to become very precise. The compacity of the star also suggests that planets be self-organized in terms of a smaller scale than the inner solar system (i.e., of a multiple of w 0 = 144 km/s).

On the one hand, the precision of pulse timing measurements has allowed to use the PSR B1257+12 system as a highly accurate probe of planetary dynamics. Indeed, it was soon pointed out that the near 3:2 resonance between the orbits of the two main planets should lead to precisely predictable and observable mutual gravitational perturbations [START_REF] Rasio | [END_REF][START_REF] Malhotra | [END_REF][START_REF] Rasio | Planets around pulsars[END_REF]9]. These non-Keplerian gravitational effects have been soon detected [17], and they have now been observed with high precision, yielding an irrefutable confirmation of the existence of planets around the pulsar and allowing a determination of the true masses and of the orbital inclinations of the planets [6].

On the other hand, the observed distances of the planets to the pulsar can be shown [10,11] to be in very good agreement with the scale relativity theoretical expectations. To the first approximation, we expect the semimajor axes to show peaks of probability scaling as n 2 , where n is integer. But, when perturbations to the simple one-object Kepler potential model are small and the trajectories are nearly circular (which is the case here, the present excentricities of the three planets being e A = 0, e B = 0.0186 ± 0.0002 and e C = 0.0252 ± 0.0002 [6]), one may use the center of gravity conservation law to refine the theoretical expectation and state that, after the formation process the planets should lie at a distance given by the center of mass of the planetesimal wave packet, which scales as n 2 + n/2, i.e. n 2 (1 + 2/n) [4,1]. This means a "correction" of 6 to 10 % with respect to the simple n 2 law for the pulsar planets. The agreement between the observed periods and the predicted ones is so precise that we have made three zooms by a factor of 10 in order to show the differences (less than 3 hours for periods of several months).

Already with the 1994 values of the orbital elements obtained by Wolszczan [17] from four years of observation, the agreement was excellent [10,11]. The three observed planets A, B and C are indeed found to correspond to ranks n = 5, 7 and 8 with a high relative precision. With the n 2 law the agreement is already at the level of 1 % and 0.3 %. One finds:

(P B /P C ) 1/3 = 0.878 to be compared with 7/8 = 0.875, (P A /P C ) 1/3 = 0.636 to be compared with 5/8 = 0.625.

With the n 2 + n/2 law, it becomes (see Fig. 1):

(P B /P C ) 1/3 = 0.8783 to be compared with (52.5/68) 1/2 = 0.8787, (P A /P C ) 1/3 = 0.6366 to be compared with (27.5/68) 1/2 = 0.6359.

A factor of about 10 has been gained with this conservative law. The relative agreement between observation and theoretical expectation has become respectively 4 × 10 -4 and 1.1 × 10 -3 for the B/C and A/C ratios.

Today the comparison between the data and the theoretical predictions can be considered again, since the determination of the orbital elements of the three planets has been greatly improved thanks to the increase to 13 years of the observing time and to the account of gravitational mutual effects [6]. This improvement of the data and of their fitting model has led to a change by 5 σ of some of the derived orbital elements. This could have degraded the agreement with the scale relativity model. On the contrary, it has yielded a new improvement by a factor of about 20 on the A/C ratio. One now finds, for P A = (25.262 ± 0.003) days, P B = (66.5419 ± 0.0001) days and P C = (98.2114 ± 0.0002) days [6]:

(P B /P C ) 1/3 = 0.8783 to be compared with (52.5/68) 1/2 = 0.8787, (P A /P C ) 1/3 = 0.63597 to be compared with (27.5/68) 1/2 = 0.63593.

The relative agreement is now at the level of respectively 4 × 10 -4 and 6 × 10 -5 for the B/C and A/C ratios. The probability to find such a close agreement, accounting for all other possible fractional ratios with n up to 10, can be estimated to be less than 10 -4 [11]. Moreover, using the standard pulsar mass M = (1.4±0.1) M ⊙ in the relation

P n = 2πGM w 3 (n 2 + n/2) 3/2 , (1)
one obtains for the coupling constant of this system w = (2.96 ± 0.07) × 144 km/s, which is consistent with an expected integer multiple of the inner solar system constant w 0 by a factor of 3 and with the value of the Keplerian velocity w ⊙ = 435 km/s at the Sun radius. The precision of these results suggests to attempt improving the model, both concerning the formation as the evolution of the system. Indeed, the presently observed orbital elements result from the formation era followed by about one billion years of evolution, while the theoretical expectations correspond to the end of the formation. The comparison between the observational and theoretical orbital elements should therefore take the system evolution into account.

Heliumoid model of the PSR1257+12 planetary system formation

Let us address the problem of the formation of two planets from two subdisks in the scale relativity approach. The question to be solved is whether the self-gravitation and the mutual interaction of the subdisks leads to non negligible corrections to the previous one-body model. Consider a star surrounded by a disk of planetesimals which can itself be separated in two parts, (B) and (C). The central star is of mass M and the individual planetesimals are assumed to have equal mass µ. The total Hamiltonian reads

Ĥ = B,C - ˜ 2 2µ ∆ - M µ r - B µ 2 r 12 - C µ 2 r 12 - BC µ 2 r 12 , (2)
where ˜ = 2µD, with D = GM/2w 0 . The three last terms are respectively the self potentials of the B and C rings and the potential of interaction between B and C, r 12 being the interdistance between two planetesimals. Assume that the total number of planetesimals in B and C is respectively N B and N C . The masses of the B and C rings, and therefore of the planets B and C which will finally be formed from these rings, are

m B = N B µ, m C = N C µ. (3)
The two first terms of the Hamiltonian then become

B,C - ˜ 2 2µ ∆ - M µ r = - ˜ 2 2µ 2 (m B ∆ B + m C ∆ C ) - M m B r B - M m C r C (4)
The distribution of planetesimals in the rings B and C are given by solutions of the gravitational Schrödinger equation

Ĥψ = Eψ, (5)
where Ĥ is given by Eq. ( 2). In the first approximation, since m B /M ≪ 1 and m C /M ≪ 1 (the ratios are ≈ 10 -5 for planets B and C of the PSR1257+12 system and ≈ 10 -7 for planet A), the potential is strongly dominated by the central body, and ψ B and ψ C are solutions of the equation

2D 2 ∆ψ + E µ + M r ψ = 0. (6)
In the second approximation, we can now use these solutions to derive the probability number density of planetesimals in the disks,

ρ B = |ψ B | 2 and ρ C = |ψ C | 2 .
The self potential of ring B can now be written as a sum of the planetesimal interaction gravitational energy over all couples of planetesimals in this ring, namely,

- i,j µ 2 r 12 = - i N B µ 2 B |ψ B | 2 r 12 dV B = -m 2 B I BB . (7)
This sum has therefore been reduced to the integral

I BB = B |ψ B1 | 2 |ψ B2 | 2 r 12 dV 1 dV 2 . (8)
Similar calculations can be made for the self-potential of C and for the BC interaction term. Namely, we define the integrals

I CC = C |ψ C1 | 2 |ψ C2 | 2 r 12 dV 1 dV 2 , (9)
I BC = BC |ψ B1 | 2 |ψ C2 | 2 r 12 dV 1 dV 2 . ( 10
)
Then the Hamiltonian now takes the form

Ĥ = - ˜ 2 2µ 2 (m B ∆ B +m C ∆ C )- M m B r B - M m C r C -m 2 B I BB -m 2 C I CC -m B m C I BC . (11)
The three last terms therefore also give the correction to the total energy of the system due to self-gravitation and to interactions of the two rings, which reads

E tot = - 1 2 w 2 0 m B n 2 B + m C n 2 C + 2m 2 B M I BB + 2m 2 C M I CC + 2m B m C M I BC , (12)
where n B and n C are the main quantum numbers of the B and C wave functions (namely, n B = 7 and n C = 8 for the PSR 1257+12 system), and where the I integrals are now dimensionless (namely, the lengths are expressed in gravitational "Bohr" units, r Bohr = GM/w 2 0 ). To be complete, one adds the third planet A to the description, whose mass is yet ≈ 200 times smaller than that of planets B and C, so that the self-potential correction to its energy is negligible, while the interactions given by the integrals I AB and I AC may be relevant.

Finally the relative correction on the energy of the three planets is found to be

∆E A E A = 2n 2 A n 2 B n 2 A + n 2 B m B M I AB + n 2 C n 2 A + n 2 C m C M I AC , (13)
∆E B E B = 2n 2 B m B M I BB + n 2 C n 2 B + n 2 C m C M I BC , (14)
∆E C E C = 2n 2 C m C M I CC + n 2 B n 2 B + n 2 C m B M I BC , (15)
where we have attributed the interaction contribution to each planet according to its energy, and where the self-gravity and of planet A and its effects on planets B and C have been neglected. The integrals I BB , I CC and I BC are exactly those which are encountered in the helium and heliumoide quantum problem (see, e.g. [START_REF] Landau | Quantum Mechanics[END_REF]). Their numerical calculation for n A = 5, n B = 7 and n C = 8, which are the values of the principal "graviquantum" number of the three planets of the PSR1257+12 system, and l = n -1 (quasi circularity), yields:

I 77 = 0.0173582, I 88 = 0.0134383, I 78 = 0.0221588, I 57 = 0.0377702, I 58 = 0.0307991.
Since the dimensionless energies are E n = -1/2n 2 , the perturbation can be expressed in terms of a correction on the expected main quantum numbers, δn/n = -(1/2)∆E/E. Using the measured values of the planet masses, m A = 0.020 ± 0.002, m B = 4.3 ± 0.2 and m C = 3.9 ± 0.2 Earth mass [6] and the standard pulsar mass of 1.4 M ⊙ , one finds very small corrections, δn A = -0.000052, δn B = -0.000090, δn C = -0.000102.

However, in the absence of a precise knowledge of the mass of the pulsar, only ratios of quantum numbers (or equivalently, of periods, semimajor axes, or energy) can be compared. The above corrections can then be expressed in another way. We fix as reference n C = 8, then the expected values of the two other planets become: (n A ) pred = 5.000012, while the observed value is (n A ) obs = 5.00029 ± 0.00020, and (n B ) pred = 7.000000, while the observed value is (n A ) obs = 6.996987 ± 0.000008.

As expected at the beginning of this study from the smallness of the planet masses, the corrections are negligible for this system, since they remain smaller than the observational errors despite the precision of the data. However, a new possible improvement of the determination of the orbital elements in the future may render them meaningful. Moreover, all this calculation is also relevant for multiple systems with larger planetary masses, in particular Jupiter-like planets, in which case the corrections are no longer negligible.

Problems and Exercises

Exercise 1 Apply the above heliumoid model of planetary system formation to the known multiple planetary systems with large planetary masses (several Jupiter masses), in order to estimate the self-gravity and interaction corrections to be applied to the values of the expected probability peaks. 0.3 Evolution of a 3:2 near resonant two-planet system 0.3.1 Three-body perturbation theory

The general three-body perturbation equations (see Brouwer & Clemence [START_REF] Brouwer | Methods of Celestial Mechanics[END_REF]) in the Lagrange-Laplace theory have been applied by Malhotra [9] to the PSR 1257+12 planetary system. The gravitational interactions of low mass planets such as the PSR 1257+12 planets, which are ≈ 10 5 times smaller than the pulsar mass, can be analysed in terms of osculating ellipses, i.e., of orbits that are instantaneously elliptical, but whose orbital parameters are time dependent. For a 3:2 near commensurability between two planets (1) and ( 2), identified respectively with planet B and C of the PSR 1257+12 planetary system, and assuming coplanar orbits, Malhotra found that the principal perturbation components in the interaction Hamiltonian are given by

H ′ = - Gm 1 m 2 a 2 [P(ψ, α) -α cos ψ] + 1 2 A 1 (α)(e 2 1 + e 2 2 ) -A 2 (α)e 1 e 2 cos(ω 1 -ω 2 ) +C 1 (α)e 1 cos(φ -ω 1 ) + C 2 (α)e 2 cos(φ -ω 2 ) , (16)
where the planet masses are m 1 ≪ M * and m 2 ≪ M * (M * being the central star mass), a 1 and a 2 are the osculating orbital semimajor axes, e 1 and e 2 the osculating eccentricities, n 1 = 2π/P 1 and n 2 = 2π/P 2 the mean orbital frequencies, and ω 1 and ω 2 the longitudes of periastron, and where (keeping Malhotra's notations)

α = a 1 /a 2 , ψ = λ 1 -λ 2 , φ = 3λ 2 -2λ 1 , P(ψ, α) = (1 -2α cos ψ + α 2 ) -1/2 , A 1 (α) = + 1 4 α b (1)
3/2 (α) = +2.50,

A 2 (α) = - 1 4 α b (2) 3/2 (α) = -2.19, C 1 (α) = - 1 2 6 + α d dα b (3) 1/2 (α) = -2.13, C 2 (α) = + 1 2 5 + α d dα b (2) 1/2 (α) = +2.59. ( 17
)
The λ j are the mean longitudes. The b(α) are Laplace coefficients. The numerical values of the A j and C j coefficients are those which corresponds to the value α = a 01 /a 02 = 0.771416(2) observed for the B and C planet ratio in the PSR 1257+12 system [6]. Three types of significant perturbations are apparent in this Hamiltonian: (1) those related to conjonctions of the planets, described by the term [P(ψ, α)α cos ψ],

(2) secular effects responsible for the slow precession of the absides, described by the terms with coefficients A 1 and A 2 , and (3) the effects of the 3:2 near-commensurability of the mean motions, described by the remaining terms.

Malhotra wrote the first time derivative of the orbital elements as:

ȧ1 a 1 = +2 m 2 M * n 1 α ∂ ∂ψ P(ψ, α) + α sin ψ + 2C 1 (α)e 1 sin(φ -ω 1 ) + 2C 2 (α)e 2 sin(φ -ω 2 ) (18) ȧ2 a 2 = -2 m 1 M * n 2 α ∂ ∂ψ P(ψ, α) + α sin ψ + 3C 1 (α)e 1 sin(φ -ω 1 ) + 3C 2 (α)e 2 sin(φ -ω 2 ) .
(19) Then, using the variables

h j = e j sin ω j , k j = e j cos ω j , (20)
for the two planets j = 1, 2 = B, C, the variations of the eccentricities and pericenters read

ḣ1 = A 11 k 1 + A 12 k 2 + B 1 cos φ, ḣ2 = A 21 k 1 + A 22 k 2 + B 2 cos φ, k1 = -A 11 h 1 -A 12 h 2 -B 1 sin φ, k2 = -A 21 h 1 -A 22 h 2 -B 2 sin φ. (21)
Setting

µ 1 = m 1 M * , µ 2 = m 2 M * , (22)
the coefficients of the matrix A ij and the vector coefficients B j read [9]

A 11 = µ 2 n 1 α A 1 (α), A 12 = µ 2 n 1 α A 2 (α), A 21 = µ 1 n 2 A 2 (α), A 22 = µ 1 n 2 A 1 (α), B 1 = µ 2 n 1 α C 1 (α), B 2 = µ 1 n 2 C 2 (α). ( 23
)

Analytic solution

A partial analytical solution for this system of equations had been found by Rasio et al. [START_REF] Rasio | Planets around pulsars[END_REF] in a simplified case. They have treated the inner planet as a massless "test particle" perturbed by the outer planet, which was assumed to move on an unperturbed Keplerian orbit. To this approximation and neglecting the conjunctions (close encounters), they found that the system can be integrated in terms of a period-eccentricity relation that reads, for small eccentricities,

n 1 = n * (1 + 3 e 2 1 ), ( 24
)
where n * is the value of the frequency for which e 1 = 0. However, this result holds only when m 2 ≫ m 1 , a condition which does not apply to the PSR 1257+12 system for which m 1 ≈ m 2 . In order to find a general analytical solution for the Malhotra system of equations, let us define a function that characterizes the conjunction effects on the semimajor axes variation,

Γ c = ∂ ∂ψ P(ψ, α) + α sin ψ, (25)
and a function that depends on eccentricities, namely,

Γ = C 1 (α)e 1 sin(φ -ω 1 ) + C 2 (α)e 2 sin(φ -ω 2 ). ( 26
)
In terms of these quantities, the semimajor axes equations now read

ȧ1 = +2 µ 2 n 1 a 1 α(Γ c -2Γ), ȧ2 = -2 µ 1 n 2 a 2 (Γ c -3Γ). ( 27
)
Therefore the two equations can be combined into a relation which no longer depends on the eccentricities,

ȧ1 4µ 2 n 1 a 1 α + ȧ2 6µ 1 n 2 a 2 = 1 6 Γ c . (28)
We can now use Kepler's third law for the osculating orbits to write

n 1 a 1 α = (GM * ) 1/2 a 1/2 1 a -1 2 , n 2 a 2 = (GM * ) 1/2 a -1/2 2 , (29)
so that the above relation becomes:

M 1/2 * 1 2m 2 a -1/2 1 ȧ1 + 1 3m 1 a -1/2 2 ȧ2 = Γ c 3a 2 . ( 30
)

Prime integrals of the equations of motion

Under the approximation where the conjunction term is neglected, it leads to a prime integral of the motion,

3m 1 a 1/2 1 + 2m 2 a 1/2 2 = Q a , (31)
where

Q a = 3m 1 a 1/2 01 + 2m 2 a
1/2 02 = cst. This analytical result shows that the main oscillations of the two planet orbits are strongly coupled and are always in opposition. This effect agrees with the numerical integration of Wolszczan [17, Fig. 2]. If we now take the conjunction term (label c) into account, we can neglect the variation of a 2 in the right-hand side of Eq. ( 30), and we obtain the integral 3µ 1 a

1/2 1 + 2µ 2 a 1/2 2 1 - 1 2 µ 1 2π P 2 Γ c (t)dt = Q a . (32)
For the system PSR1257+12, we have µ 1 ≈ 10 -5 in the correction term, 2π/P 2 = 23.37 yr -1 , while the integral of Γ c shows two components of amplitudes 0.05 and 0.25 yrs, so that the correction to the fluctuations of semimajor axes (which are themselves very small, of the order of 3 × 10 -4 ) remains smaller than 4 × 10 -5 in proportion. Therefore the prime integral of the motion given by Eq. ( 31) is valid to a very good approximation.

Let us now consider the eccentricity equations. From the relations e 2 1 = h 2 1 + k 2 1 and e 2 2 = h 2 2 + k 2 2 , we derive

e 1 ė1 = h 1 ḣ1 + k 1 k1 , e 2 ė2 = h 2 ḣ2 + k 2 k2 . ( 33
)
From Eqs. (21) we obtain the expressions

e 1 ė1 = A 12 (h 1 k 2 -k 1 h 2 ) + B 1 (h 1 cos φ -k 1 sin φ), e 2 ė2 = A 21 (h 2 k 1 -k 2 h 1 ) + B 2 (h 2 cos φ -k 2 sin φ). (34)
From the very definition of h j and k j , we find

h 1 k 2 -h 2 k 1 = e 1 e 2 sin(ω 1 -ω 2 ), (35)
so that the derivative of the eccentricities finally read ė1 = A 12 e 2 sin(ω

1 -ω 2 ) + B 1 sin(ω 1 -φ), ė2 = -A 21 e 1 sin(ω 1 -ω 2 ) + B 2 sin(ω 2 -φ). ( 36
)
We recognize in these expressions two contributions of respectively free and forced oscillations (see [9,5]). For the PSR 1257+12 system, the free oscillations correspond to long-term motion (periods ∼ 6200 yrs and ∼ 92000 yrs [9,5]), while the forced oscillation are on a shorter time scale (period 2π/(3n 2 -2n 1 ) = 5.586 yrs) and of smaller amplitude.

Let us consider only the long term motion characterized by the free oscillations. For them, the derivatives of the eccentricities read

ė1 = A 12 e 2 sin(ω 1 -ω 2 ), ė2 = -A 21 e 1 sin(ω 1 -ω 2 ), ( 37
)
and we therefore find the relation

A 21 e 1 ė1 + A 12 e 2 ė2 = 0. ( 38
)
It can be integrated in terms of a new conservative quantity (valid for long-term motion),

A 21 e 2 1 + A 12 e 2 2 = cst. ( 39
)
With

A 12 = µ 2 n 1 α A 2 , A 21 = µ 1 n 2 A 2 , it yields a prime integral of long-term motion µ 1 n 2 e 2 1 + µ 2 n 1 α e 2 2 = cst. ( 40
)
For the PSR 1257+12 planetary system, the relative variation of the semimajor axes is ≈ 2×10 -4 , while the eccentricities vary by ≈ ±50% on the long term. With α = a 1 /a 2 and (n 2 /n 1 ) 2 = (a 1 /a 2 ) 3 from Kepler's third law, the semimajor axes can therefore be considered as constant to the first approximation, as also shown by numerical integration [5], so that this conservative quantity can be written

m 1 a 1/2 1 e 2 1 + m 2 a 1/2
2 e 2 2 = cst, or equivalently

Q e = e 2 1 + m 2 m 1 α -1/2 e 2 2 = cst. ( 41
)
This result once again shows that the motion of planets B and C are tightly coupled and that their eccentricities oscillate on the long term in exact opposition (see Fig. 1c of Ref. [5]).

Expression for the eccentricity function Γ

Let us come back to the full motion, including the free and forced osciilations of the eccentricities. From equations (36), we derive expressions for sin(ω 1 -φ) and sin(ω 2 -φ), which we may now insert in the expression for the quantity Γ (Eq. 26). This quantity describes the eccentricity-dependent contribution to the variation of the semimajor axes. We find

Γ = C 1 B 1 e 1 ė1 + C 2 B 2 e 2 ė2 + C 2 A 21 B 2 - C 1 A 12 B 1 e 1 e 2 sin(ω 1 -ω 2 ). ( 42
)
When replacing the various coefficients by their expressions, one finds that

C 2 A 21 /B 2 = C 1 A 12 /B 1 = A 2 ,
so that the last term vanishes and Γ is finally given by

Γ = e 1 ė1 µ 2 n 1 α + e 2 ė2 µ 1 n 2 . ( 43
)
To the approximation (very good in this context) where n 1 , n 2 and α are constant, this function can now be easily integrated as

SΓ = Γ(t)dt = 1 2µ 2 n 1 α (e 2 1 + Ke 2 2 ), ( 44
)
where we have set

K = m 2 m 1 α -1/2 = µ 2 µ 1 α -1/2 , (45)
i.e., K = (m 2 /m 1 ) a 2 /a 1 . With the presently measured values of the masses of planets B and C, m B = (4.3 ± 0.2) M ⊕ and m C = (3.9 ± 0.2) M ⊕ [6], this parameter takes, for the PSR 1257+12 system, a value close to 1, namely,

K PSR = 1.03 ± 0.10. ( 46
)
We recognize in the term e 2 1 + Ke 2 2 the hereabove long-term motion prime integral (Eq. 41).

Relation between semimajor axes and eccentricities

These results allow us to finally integrate analytically the semimajor axes equations in function of the eccentricities and to provide an exact expression for the incomplete Rasio et al. [START_REF] Rasio | Planets around pulsars[END_REF] formula. Let us first neglect the conjunction contribution. The equation for the semimajor axis of planet (1) writes

ȧ1 a 1 = -4µ 2 n 1 α Γ(t). ( 47
)
Using the expression obtained for Γ, it writes

ȧ1 a 1 = -4(e 1 ė1 + Ke 2 ė2 ). ( 48
)
This equation is easily integrated as

ln a 1 a 1 * = -2(e 2 1 + Ke 2 2 ), (49)
which may be approximated for small values of the eccentricities by

a 1 a 1 * = 1 -2(e 2 1 + Ke 2 2 ), (50)
where a 1 * is the (virtual) value of the semimajor axis for which e 1 = e 2 = 0. This relation applies to the PSR1257+12 system, in which the eccentricities of planets B and C run between about 0.01 and 0.03. We may also express this result in terms of variations of the semimajor axis and eccentricities with respect to some initial conditions, a 10 , e 10 and e 20 ,

a 1 = a 10 {1 -2[(e 2 1 -e 2 10 ) + K(e 2 2 -e 2 20 )]}. (51)
This result allows to recover and to generalize the Rasio et al. [START_REF] Rasio | Planets around pulsars[END_REF] formula,

a 1 = a 1 * [1 -2e 2 1 ]
, in which e 2 was assumed to be constant (the term K(e 2 2 -e 2 20 ) vanishes in this case). However, it also shows that the correct result cannot be obtained as a mere "test particle" limit m 1 /m 2 → 0, since this increases the factor K instead of decreasing it. planetary system, plotted from its analytic expression. The fluctuations are mainly due to the conjunction effects by planets B and C. Except for this small oscillation, the period is stable on the long term (he motion of planet A is decoupled from the dynamics of planets B and C in the long-term scale [5]). planetary system, plotted from its analytic expression (see text). The fluctuations are due to the mutual effects between planets B and C. Except for this small oscillation, the period is stable on the long term. period derived from numerical integration of equations of motion by Wolszczan [17, Fig. 2].

Finally, the evolution of the semimajor axis of planet C ( i.e., planet 2 in the previous two-planet analysis) can be analytically integrated under the form: planetary system, plotted from its analytic expression (see text). The fluctuations are due to the mutual effects between planets B and C. Except for this small oscillation, the period is stable on the long term.

a C (t) = a C0 1 + 6 µ B 2π P C0 SΓ(t) - 1 3 SΓ c (t) . ( 60
The corresponding period evolution, which is given from Kepler's third law by a We can now compare to the expectation of the wave packet formation model (n 2 +n/2), not only the mean semimajor axis ratio of planets B and C as previously done, but the full variation with time of this ratio. It can be expressed in terms of the ratio of effective quantum numbers n B and n C (which is expected to be equal to 7/8). The time evolution of the ratio (n B /n C )/(7/8) is shown in Fig. 5.

An important point to notice is that this ratio, apart from very small fluctuations of about 3 × 10 -4 , is stable on long time scales. This definitively justifies the validity of the comparison of today's periods with those expected from the formation model, despite the evolution of the system over billion years. Moreover, we can see in this figure that, when taking into account the mutual effects between planets B and C, the minimal difference between the value expected at the end of the formation era and the observed value has been decreased again by a factor of ∼ 2, from 4.3 × 10 -4 to about 2.5 × 10 -4 (while it is 0.6 × 10 -4 for the A/C ratio).

This high precision suggests to push further the model and to attempt to obtain a theoretical estimate of the width of the probability peaks. In the framework of the macroscopic quantum-type scale relativity approach, this width is given by a generalized Heisenberg relation ∆ E∆t ≈ 2mD, with D = GM/2w. In this relation, the time fluctuation ∆t may be estimated from the characteristic revival time of localized wave packets. These wave packets are characterized by a revival time t rev = 2nP/3 where P ∝ n 3 is the classical period and a superrevival time t sr = 3n t rev /4 [1]. Inserting this last value in the macroscopic Heisenberg relation, one obtains ∆E/E = 1/(πn 3 ), i.e. ∆n/n = 1/(2πn 3 ). For n = 8 this relation gives ∆n/n ≈ 3 × 10 -4 . Though this is a very preliminary estimate (one should take the self-gravitation of the wave packet into account), the fact that it is of the same order of size as the observed difference seems very encouraging.

Eccentricities

Concerning eccentricities, the precession and near-resonance effects between planets B and C imply an important relative variation of nearly a factor 3. One finds, both in the numerical integration [5] and in the analytic solution [START_REF] Nottale | Scale Relativity and Fractal Space-Time: a New Approach to Unifying Relativity and Quantum Mechanics[END_REF], that the two planet eccentricities vary in opposition in the range 0.0125 to 0.0285 on a period of about 6200 years (plus a smaller component of period about 92000 years). It would therefore have no meaning to attempt obtaining a theoretical expectation of the individual eccentricities from the formation model. However, we have seen that the two eccentricities combine in terms of a conservative quantity (Eq. 41), Q e = e 2 B + Ke 2 C . With the observationnal values e B = 0.0186 ± 0.0002, e C = 0.0252 ± 0.0002, m B = (4.3 ± 0.2) M ⊕ and m C = (3.9 ± 0.2) M ⊕ [6] leading to K = 1.03 ± 0.10, one finds Q e (obs) = 0.00100 ± 0.00002(e) ± 0.00004(K).

Since this quantity is an invariant prime integral of the long term evolution of the system, it must have been fixed at the end of the formation era. One may therefore consider the possibility of deriving it as one among the possible quantized values in the macroscopic quantum-type model. In this aim, let us carry the model farther.

The planetesimal wave packets, being not only quantum-like wave packets but also gravitational structures, are expected to concentrate to form protoplanets, then the planets by final accretion. But during this concentration phase, the conditions under which the geodesic equation (i.e., the fundamental equation of dynamics) may be integrated under the form of a Schrödinger equation still apply. This results in the appearance of a scale factor f on the gravitational coupling constant, and therefore also on the main quantum numbers. This is another manifestation of the combination of the scale invariance of gravitation with a macroquantum Schrödinger description, which differs profoundly from the behavior of the microscopic atomic quantum regime whose scales are fixed by the constancy of the Planck constant . But the quantization of the solutions at each stage and the conservation of energy finally implies that the scale factor be integer (this process is quite similar to that yielding hierarchically embedded levels of organization in planetary systems, as verified in our Solar System, see [10] and Secs. 13.3.6 and 13.4 of [START_REF] Nottale | Scale Relativity and Fractal Space-Time: a New Approach to Unifying Relativity and Quantum Mechanics[END_REF]).

Applied to the PSR 1257+12 planet formation, the initial ranks n B = 7 and n C = 8 are successively transformed into n B = 7f and n C = 8f with f increasing during the wave packet concentration. The coupling constant w ⊙ becomes correspondingly f w ⊙ , then allowing the planet distances GM n 2 /w 2 to remain the same. The width of the orbitals, on the contrary, being given by σ n ∼ n 3/2 [START_REF] Landau | Quantum Mechanics[END_REF], relatively decreases as σ/a ∼ f -1/2 , as expected for such a concentration process.

Let us apply this process to the eccentricity quantization. We have seen that it is given by the amplitude of the Runge-Lenz vector [START_REF] Landau | Quantum Mechanics[END_REF], and is therefore expected to be quantized as e kn = k/n, with k = 0 to n -1. The first quantized value is e = 0. It yields a satisfying first approximation for the B and C planet eccentricities, < e >≈ 0.02. To a better level of precision, the first excited value is obtained for k = 1, i.e., e B = 1/7f and e C = 1/8f , with f integer. For f = 6, which corresponds to the beginning of the spatial separation of the orbitals, one obtains Q e = (1/42) 2 + K(1/48) 2 = 0.001014 (K = 1.03 fixed), in fair agreement with the observed value 0.001000 ± 0.000018 (the agreement being preserved for all possible values of K = 0.93 to 1.13). Reversely, this result can be put to the test in the future since, if correct, it provides a value of K:

K = -

Problems and Exercises

Open Problem 1 Use the above model of formation and evolution to predict the distances and periods, in terms of probability density peaks, of other possible planets in the PSR1257+12 planetary system.

Hints:

The n 2 + n/2 law yields in particular more probable short periods at 0.322 days (n = 1), 1.958 days (n = 2) and 5.96 days (n = 3). Wolszczan et al. [18] have obtained timing data for about 30 successive days. An analysis of the residuals of these data after substraction of the effects of the three known planets (the dispersion of these residual being still larger than the error bars) has yielded a marginal detection (at a significance level of 2.7 σ) of a planet with a period P = 2.2 days and a mass 0.035 M ⊕ , which is compatible with the n = 2 expectation [START_REF] Nottale | Workshop Pulsars, Theory and Observations[END_REF]. The absence of planet for n = 6 may be well understood from the numerical integrations and simulations of Gozdziewski et al. [5], which show this particular zone to be unstable (although there is an extended stable zone between planets A and B).

Figure 1 :

 1 Figure 1: Comparison between the observed periods of the three planets observed around the pulsar PSR B1257+12 and the scale relativity expectation (1994 data).The agreement between the observed periods and the predicted ones is so precise that we have made three zooms by a factor of 10 in order to show the differences (less than 3 hours for periods of several months).

Figure 2 :

 2 Figure 2: Evolution over 10 years of the period PA of planet A in the PSR 1257+12

Figure 3 :

 3 Figure 3: Evolution over 10 years of the period PB of planet B in the PSR 1257+12

Figure 4 :

 4 Figure 4: Evolution over 10 years of the period PC of planet C in the PSR 1257+12

3 / 2 C

 32 , is plotted in Fig.4. It also compares satisfactorily with the perturbation of orbital period derived from numerical integration of equations of motion by Wolszczan [17, Fig.2]. The coupling of the two B and C orbits, which leads to motion in opposition of the two planets, is clear on Figs.3 and 4. 0.5 Consequences for the formation model 0.5.1 Periods.

Figure 5 :

 5 Figure 5: Evolution over 10 years of the ratio [nB(t)/nC (t)]/[7/8] for the two planets B and C in the PSR 1257+12 planetary system. The minimal deviation from the expected value 1 reaches 2.5 × 10 -4 .

e 2 B

 2 -(1/42) 2 e 2 C -(1/48) 2 = 1.099 ± 0.065, (61) from which an estimate of the mass ratio can be obtained, m C m B = 0.965 ± 0.058, (62) more precise than the presently known value m C /m B = 0.91 ± 0.10 [6]. A possible future improvement of the observational values of the eccentricities would still improve this estimate.

  )

		98.23
		98.22
	Period, days	98.2 98.21
		98.19
		98.18
		0	2	4	6	8	10
				Time, years

Let us finally derive the expression for the variation of the semimajor axis of the outer planet. Its equation reads ȧ2

It is integrated, for small eccentricities, as

i.e., respectively to initial conditions,

We therefore find that the variation of the semimajor axes of the two planets (except for the small amplitude conjunction effects) depend only on the expression e 2 1 + Ke 2 2 which we have found to be a prime integral of the long-term motion. This allows us to conclude (within the approximations used in these analytical calculations) that there is no long-term variation of the semimajor axes of planets B and C. This result fully agrees with the Gozdziewski et al. [5] numerical integrations of their long-term motion, which also derived their stability.

Application to the PSR 1257+12 system evolution

Semimajor axes, analytic solutions

The evolution of the semimajor axis of the orbit of planet A of the PSR 1251+12 system is mainly determined by the conjunction effects by planets B and C. An analytic expression can be obtained for this evolution [START_REF] Nottale | Scale Relativity and Fractal Space-Time: a New Approach to Unifying Relativity and Quantum Mechanics[END_REF]: the corresponding period evolution, which is given from Kepler's third law by a 3/2 A , is plotted in Fig. 2.

The evolution of the semimajor axis of planet B ( i.e., planet 1 in the previous two-planet analysis) can be analytically integrated under the form:

where

and

The corresponding period evolution, which is given from Kepler's third law by a


Calculation of 6D Atomic Surfaces from a Given Approximant Crystalline Structure Using Approximate

Icosahedral Periodic Tilings

Abstract Approximant structures of quasicrystals such as α-A1MnSi, R-A1LiCu and c(Ti 2 Ni) are well described while atomic organization laws in quasicrystals of similar composition are still unknown. We suggest a method based on the projection method from R 6 to calculate, in a general way, the 6D atomic surfaces associated with a given approximant structure, of known atomic decoration, of any Bravais lattice, with respect to the geometrical features inside the unit cell and the distortions in Fourier space. From these atomic surfaces, we deduce three decorated 3D quasilattices of possibly different icosahedral Bravais lattice type (P, F, I) and lattice parameters, which in general present aperiodic density and stoichiometry fluctuations.


Microcrystalline structure and quasi-crystal formation in rapidly quenched (AI-Ti)-(Fe, Mn) alloys

A. R.

Microcrystalline structure and quasi-crystal formation in rapidly quenched (AI-Ti)-(Fe, Mn) alloys Abstract. We have performed an X-ray diffraction study of several microcrystalline binary and ternary alloys of the ( Al -Ti )-(Mn-Fe-Ni ) system prepared by melt-spinning. The results indicate that while Ti 2 Ni is fully amorphous, Ti 2 Fe forms metastable crystalline phase(s) with little or no i-phase fractions. The spectra of microcrystalline binary Ti-Mn alloys present similarities to that of the i-phase. Ternary (AlTi) 1-x Fe x (AlTi) 1-x Mn x are found to be isostructural near x = 30 at % but seem not to contain significant i-phase while the ternary Al-rich Al 72 Ti 10 Mn 18 seems to be partially quasi-crystalline (QC). Electron diffraction patterns obtained for binary Ti 56 Mn 44 whose X-ray spectrum resembles that of the i-phase show spots with five-fold sym-metry together with a periodic lattice and a lot of streaking.


A R Yavari

J L Verger-Gaugry

Phase fields in rapidly solidified MnTi and IVInAITi alloys studied by X-ray diffraction

While rapid solidification causes little modification of phase fields in MnxTi l_x alloys for x < 30at%, alloys richer in manganese present metastable phases. A structure intermediate between crystalline and quasi-crystalline phases is observed in the range 0.3 < x < 0.45. Aluminium addition is found to stabilize the C14 phase clown to manganese contents as low as 35at% and Mn35 (AITi)65 is found to be single-phased C14. The ternary alloy's C1 4 phase field emerges due to aluminium substitution for manganese at low manganese contents.

Introduction

On the titanium-rich side of the MnTi phase diagram (see Fig. 1), the high temperature bcc fl-Ti exhibits a maximum manganese solubility XMn < 30 at % while hexagonal /~-Ti at lower temperatures shows little manganese solubility [1]. Near XMn = 0.5, intermetallic MnTi (tetragonal, q~, together with the p phase of undetermined crystal structure [1]) with complicated crystalline structures appears and at higher manganese contents the C14-MgZn 2 type Mn2Ti hexagonal laves phase forms. We examine these phase fields in microcrystalline MnTi alloys obtained by rapid solidification. Waterstrat et al. [2] studied the Mn2Ti-C14 phase. Both the a and c lattice parameters of this hexagonal phase were found to expand with addition of the larger titanium atoms above 33 at % and the phase was found to be stable up to the composition Mn55Ti45. It was shown that this stability was due to partial occupation of manganese-type sites by titanium atoms. The stability of the C14 structure on the manganese-rich side is interrupted by the formation of Mn3Ti. We have also examined the effect of aluminium addition on the Mn2Ti Laves phase stability. To our knowledge, terniary MnA1Ti phase diagrams covering the composition range of the C14 phase have not yet been reported. It was found that addition of aluminium allows further substitution for manganese such that a single-phased hexagonal C14 structure with as low as 35at% Mn is obtained. Aluminium addition thus stabilizes the Laves phase.

Experimental procedure

MnTi and MnAITi alloys of various compositions were prepared by induction melting in a cold crucible and in alumina crucibles, respectively, and under argon gas. The ternary alloys were of compositions Mn• (Alsl Ti49)l x hereafter referred to as Mn.,.(A1Ti)~_.,.. The A1/Ti ratio was varied for a few more compositions to map out the C14 phase-field as indicated later in Fig. 6. The alloys were melt-spun on a copper substrate at speeds up to 80msec -1. Most of the ribbons were 30 to 50 #m thick. They were brittle and broke into small pieces upon collision with the quench-chamber walls. The entire chamber was evacuated to 10 .5 torr and then refilled with pure helium gas for the quenching experiments. X-ray diffraction patterns were obtained on the as-quenched samples and after annealing for 2h at 800~ (1073 K) in sealed quartz tubes under purified argon gas. A Siemens diffractometer was used with copper Kc~ radiation (2 = 0.154 nm).

Results

Fig. 1 shows the TiMn phase diagram. Pure titanium with hcp structure transforms the b cc structure at T< 900~ (1173K). However, /J-Ti containing manganese can be easily retained by quenching down to room temperature. Fig. 2 shows X-ray diffraction spectra for various rapidly solidified MnTi alloys. Pure titanium exhibits only h c p peaks (Fig. 2a) and MnTi with XMn = 15 at % predominantly h c p peaks (Fig. 2b) together with smaller//-Ti peaks. We note that the (1 10) peak of/~-Ti is superimposed on the (00 1) hcp peak. At 20at% Mn (Fig. 2c), hcp peaks have disappeared and only fl-Ti peaks are observed. This is consistent with the phase diagram of which shows that for XM, > 20 at %, the stable phases at T > 600~ are/?-Ti and MnTi (4) or p). It also shows that at temperatures below the/%Ti phase field, complicated intermetallic phases do not nucleate as easily as h c p e-Ti observed in the sample with xM. = 0.15at%. Figs 2d ande show that for xM, = 25 and 30at %, respectively, fl-Ti is again retained. With increasing manganese content, the /?-Ti peaks are displaced towards higher angles. The alloy with xMn = 0.30 should contain some intermetallic MnTi but no such peaks appear. This indicates that solid solubility of Mn in b c c titanium is enhanced by rapid solidification.

At the composition 30 ~< xMn ~< 50at %, X-ray spectra should normally show increasing predominance of peaks due to intermetallic MnTi (q~ and possibly p). However, in this composition range, diffraction continues to occur at the angles corresponding to /~-Ti peaks, but these peaks are drastically broadened (Figs 2f andg). Furthermore, a fourth and even broader peak appears near the (1 1 0)/~-Ti peak and a weak one at lower angles as can be seen in Fig. 2g. These diffraction peaks, which can be fitted with good approximation to an A1Mn-type icosahedral 4076 quasi-crystalline (qc) lattice [3] are, in fact, due to one of its approximate periodic structures [3,4].

The Bragg peaks of the rapidly solidified MnsoTis0 (Fig. 2h) can be indexed to the tetragonal 4) phases although they are significantly broadened. All but one of the major Bragg peaks of the Mn60Ti40 alloy (Fig. 2i) can be indexed to those of the MnzTi-C14 hexagonal Laves phase.

Fig. 3 shows the spectra for these samples before and after annealing for 2h at 800~ (1073K). As expected, annealed Mn60Ti40 shows only MnzTi-C14 peaks. The unidentified broad peak at 20 = 42.8 ~ and the shoulder at 20 = 41 ~ (see arrows in Fig. 3c) have disappeared after annealing thus indicating that they belong to a metastable phase which has not yet been identified although the peak positions are close to those of the major peaks of the qc approximate periodic structure of Fig. 3e. MnsoTis0 after annealing shows a much more developed ~b-phase pattern (Fig. 3h). All other compositions in the range 0.30 < xg,, < 0.50 show both q~-phase and/?-Ti peaks after annealing as expected from the phase diagram isotherms at 800 ~ C. Interestingly, the intermetallic phase peaks also begin to emerge in samples with 0.15 < xM. < 30at%

ZO (deg)

after annealing even though the annealing temperature places the composition range in the/3-Ti phasefield. Apparently the slow cooling upon removal from the furnace allows sufficient time for significant phase separation.

In a preliminary report on the structure of some microcrystalline MnAITi alloys [3], we reported an X-ray pattern for rapidly solidified Mn35(AITi)65. Subsequent identification revealed, somewhat unexpectedly, that the spectrum could be fully indexed by the C14 hexagonal symmetry of the Mn2Ti Laves phase. Because MnA1Ti ternary phase diagrams are unavailable, we have further examined the structures of rapidly quenched Mn55(A1Ti)45 and Mn20(AlTi)80 alloys before and after annealing at 800~ (1073 K). The X-ray diffraction patterns obtained for these samples are shown in Fig. 4. Of the three alloys only Mn_75 (A1Ti)65 is single-phased hexagonal C14 in its as-quenched and annealed states. The manganese-poor and manganese-rich alloys both present significant extra peaks that are modified after annealing. From the Bragg peak intensities of as-quenched Mn20(A1Ti)s0, it would appear that it is constituted of the C14 Laves phase up to a volume fraction more than 50%. The extra peaks due to an unidentified metastable phase give way after annealing to those of the tetragonal L10 AuCu-type ordered AITi (Figs 4d ande) while the C14 Laves peaks become less intense. Mn55 (A1Ti)45 also consists mostly of Mn2 Ti-C 14 phase with one major and a few minor unidentifed peaks indicated by arrows in Fig. 4f, which become more important after annealing but remain less intense than those of the C14 phase. The few available peaks of this minority phase have not yet allowed crystallographic identification. However, the same X-ray peaks appear in the slowly cooled ingot and a secondary electron micrograph of a section of this ingot shows large and small grains of C14 phase together with an important

20(deg)

Figure 4 X-ray diffraction patterns for rapidly solidified Mn~ (A1Ti)I ~ alloys in as-quenched (as-q) and annealed (ann) states. number of black areas (Fig. 5) which energy dispersive X-ray microanalysis indicated to be very rich in aluminium. Thus both Mn20(A1Ti)80 and Mnss(A1Ti)45 are constituted of more than 50vol % C14 while the Mn3~(A1Ti)65 alloy is fully hexagonal C14. Furthermore, Mn35(A1Ti)65 remained in the fully C14 phase after annealing. Further examination showed that even in ingot form, it presented a nearly identical X-ray spectrum that was fully indexed by C14 Bragg peaks. These results do not prove that the C14 phase is the equilibrium structure for Mn35(A1Ti)65 but indicate that if it is a metastable phase, it is a very stable one. In any case, it can be concluded that the MnzTi-type C14 Laves phase appears in a wide composition range of the ternary MnA1Ti alloys and a corresponding metastable phase diagram is traced in Fig. 6.

Discussion

Titanium-rich rapidly quenched Mn-Ti alloys exhibit phase fields and microstructures that are predictable from the phase diagram. The high-temperature b c c /3-Ti, which is stable up to XMn just below 30 at % at about 1400 K (see Fig. 1), is easily quenched down to room temperature. While in the rapidly quenched alloy with XMn = 0.3 only b c c Bragg peaks appear, at higher manganese contents these peaks are strongly broadened and new peaks begin to appear (Fig. 2).

Prior to this composition range and due to the size difference between titanium and manganese alloys, the lattice parameter of b c c titanium decreases rapidly with manganese addition. It can be seen in Fig. 7

which yields rM, = 0.1232 nm and rTi = 0.1402 nm in as-quenched /3-Ti c o m p a r e d with room-temperature values ru, = 0.135nm and rTi = 0.147nm for cubic manganese and h c p titanium, respectively. These differences imply a negative mixing volume of about 0.5 cm 3 mol 1 or 5.2% for the Mn30Tiv0 solid solution.

The phase diagram indicates that the alloys in the range 0.3 < XM, < 0.5 should exhibit a mixed structure of/LTi and intermetallic MnTi tetragonal 4) (or p) phases. However, the rapidly solidified alloys in the range 0.35 < XM, < 0.45at% show a strong broadening of the b c c peaks and the emergence of additional broad peaks which can together be indexed with an i-phase quasi-crystalline symmetry with the low-angle i-phase peaks being absent. Some electron diffraction patterns (EDP) show bright spots with fiveor ten-fold symmetry superimposed on a periodic network [3]. Dong et al. [4] have recently provided clear examples of such five-or ten-fold electron diffraction patterns superimposed on periodic networks in some EDPs of rapidly solidified FeTi 2 and Ni(TiV)2 whose corresponding Frank-Kasper type phases are predominantly built of icosahedral structural units. It appears that our alloys exhibit related structures [3] in the range between MnTi and MnTi2 and this is not surprising as icosahedral motifs are also predominant in the tetragonal MnTi and C14 Mn2Ti Laves phases. Fig. 3f shows how, upon annealing, this metastable phase transforms into the MnTi qS-phase together with residual /~-Ti. Such metastable phases may also be responsible for the unidentified Bragg peaks appear- ing in the spectrum for the rapidly solidified Mn60Ti40 (see arrows in Fig. 3i) which disappear after annealing. This point will be further investigated by TEM. The results of our investigation of the C14 phase field in ternary AIMnTi alloys revealed that compositions as poor in manganese as Mn35(A1Ti)65 form a single-phase C14 Laves phase upon solidification. This was somewhat unexpected as Waterstrat et al. [2] had previously shown that the stability of the Mn2Ti-C14 phase is limited to compositions with xMn ~> 55 at %. Fig. 8 shows a and c lattice parameters of the hexagonal C14 phase as a function of XMn as extracted from their results, and on which we have plotted our own values obtained from the diffraction patterns of Mn60Ti40 and Mn35(A1Ti)65 using a = 2d(1 10) and c = 4d(0 0 4) of Figs 2 and3. It can be seen that a and c and c/a for the ternary alloy fall nicely on the extrapolation of the lattice parameter of the binary MnTi alloys towards lower manganese content. Because Waterstrat et al. [2] have shown that the observed evolution of the lattice parameters is due to titanium substitution in manganese sites, it is concluded that in ternary Mn35(A1Ti)65, aluminium substitutes and compensates for manganese, thus rendering the C14 phase stable down to at least 35 at % Mn. The good linear fit of Fig. 8 is consistent with the nearly equal atomic diameters of titanium and aluminium atoms. While the C14 stable phase field has not been completely traced out in ternary MnAITi alloys, a metastable phase diagram is presented in Fig. 6.

Icosahedral qc phases have been observed in NiTi2 [5] and FeTi2 [6], and, as discussed earlier, MnTi alloys in the composition range between MnTi2 and MnTi show structures that are intermediate between quasicrystalline and crystalline phases [3]. These observations together with the ease of formation of qc structures in aluminium-rich A1Mn alloys could lead to the expectation that ternary alloys may form quasicrystals near the composition Mn35(AITi)65. It was seen, however, that this composition range massive substitution of manganese by aluminium allows easy nucleation and growth of the C14 phase whose formation must be suppressed to favour quasi-crystalline growth during solidification.

  Fig.1shows the TiMn phase diagram. Pure titanium with hcp structure transforms the b cc structure at T< 900~ (1173K). However, /J-Ti containing manganese can be easily retained by quenching down to room temperature. Fig.2shows X-ray diffraction spectra for various rapidly solidified MnTi alloys. Pure titanium exhibits only h c p peaks (Fig.2a) and MnTi with XMn = 15 at % predominantly h c p peaks (Fig.2b) together with smaller//-Ti peaks. We note that the (1 10) peak of/~-Ti is superimposed on the (00 1) hcp peak. At 20at% Mn (Fig.2c), hcp peaks have disappeared and only fl-Ti peaks are observed. This is consistent with the phase diagram of Fig.1

Figure 1

 1 Figure 1 Mn-Ti phase diagram as reconstructed by Murray [1]. 0022-2461/90 $03.00 + .12 9 1990 Chapman and Hall Ltd.4075

Figure 2 X

 2 Figure 2 X-ray diffraction patterns for rapidly solidified Mn~.Ti I ,. alloys in the range 0 < x < 60at. % (CuK~ radiation).

Figure 3 X

 3 Figure3X-ray diffraction patterns for rapidly solidified Mn,.Ti I , alloys in as-quenched (as-q) and annealed (ann) states.

Figure 5 Figure 6

 56 Figure 5 Secondary electron micrograph of Mnss(AITi)45 ingot showing aluminium-rich minority black phase.4 0 7 8

  that d(1 10) drops linearly with XMn up to 3 0 a t % and begins to increase in the range 0.3 < XMn < 0.35 which is consistent with the appearance of extra Bragg peaks belonging to a new phase in this range (Fig. 2). The lattice parameter a0 of pure b cc titanium at 900~ (1173K) is 0.3306nm. Using a coefficient of thermal contraction, ~ = 2 x 10-SK -l, one obtains for the hypothetical b c c titanium at room temperature, a0 = 0.3240nm and d(1 1 0) = no~21/2 = 0.2290nm. This value is plotted as an open circle on the vertical axis of Fig. 7 and it can be seen to fall on the linear fit of d (1 1 0) against XMn up to 30at %. This line corresponds to the simple relation 221/2 d ( l l 0 ) = 3 (XMnrMn + xxirvi)

Figure 7

 7 Figure 7 d(l 10) interplanar spacing plotted against manganese content in b c c (/3-Ti) Mn.,Ti~-x solidified alloys.