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There are at least two reasons why one would study the gravitational field of a disk. The first is that many astronomical objects, such as spiral galaxies like the Milky Way, are disk-like. The second is that the field of a disk is interesting, particularly when compared to that of a spherical, or near-spherical, object, which is much easier to analyze because of its high degree of symmetry. It is hoped that this study will augment previous work on this subject. The aspects presented in this paper are as follows: 1) both the radial and vertical gravitational fields of a thin disk within the plane of the disk and above it; 2) a comparison of some of the field results obtained by Lass and Blitzer (1983) involving elliptic integrals to those obtained by a standard numerical integration, now available online, and separately through the use of Legendre polynomials; 3) the logarithmic divergence of the radial field at the edge of a thin disk; 4) the fields in the plane of a disk containing a central hole, particularly within the hole, such as the rings of Saturn; 5) circular orbits within the plane of a single disk and half way between two disks, and their stability; 6) the escape velocity at a point within the Milky Way, particularly at the position of the solar system and without any added, or subtracted, orbital effects around the galactic center; and 7) the radial field at the circular edge of a disk of finite thickness.

Although many astronomical objects, such as stars, planets, and certain galaxies, are spherical, or nearly spherical, in shape, many others are disk-like. Examples of the latter are bar spiral galaxies (such as the Milky Way), the rings of Saturn, viewed as disks with a large central hole, and protoplanetary disks, consisting of dense gas and dust surrounding a newly formed star that add mass to the star [

Certain Previous Work (Lass and Blitzer)

Two examples of previous work on the gravitational field of an infinitely thin disk are by Lass and Blitzer [

With reference to

V ( r , z ) = 2 σ G [ π | z | − z 2 + ( a + r ) 2 E ( k ) − [ ( a 2 − r 2 ) / z 2 + ( a + r ) 2 ] K ( k ) − [ ( ( a − r ) / ( a + r ) ) z 2 / z 2 + ( a + r ) 2 ] Π ( n , k ) ] . (1)

For r > a, the first term, π | z | , is eliminated because there is no surface mass beyond r = a. Thus, the z-component of the field at z = 0 is zero there, while it is not zero for z < a, where there is surface mass. Regarding the meaning of the symbols not noted in ^{2} = n. As to the complete elliptic integrals, they have the following integral representations, as stated in [

K ( k ) = ∫ 0 1 d t ( 1 − t 2 ) ( 1 − k 2 t 2 ) ; E ( K ) = ∫ 0 1 d t 1 − k 2 t 1 − t 2 ; Π ( n , k ) = ∫ 0 1 d t ( 1 − n t 2 ) ( 1 − t 2 ) ( 1 − k 2 t 2 ) ; (2)

Although it is of no consequence, we should be aware of a slight difference in notation between [^{2} in [^{2} as n.

To obtain the force in the radial direction, F_{r}, we calculate ?∂V(r, z)/∂r, which involves the sums of the derivatives of the three products of functions in Equation (1). When that is done, the force within the plane of the disk (where z = 0) can be expressed as:

F r ( z = 0 ) / 2 G σ = − [ 1 + r a 2 r a ( 1 + r a ) ] K ( k ) + [ ( 1 + r a ) ( 2 + r a ) 2 r a ] E ( k ) − [ ( 1 − r a ) 2 2 ( 1 + r a ) ] Π ( n , k ) (3)

In this equation, r_{a} ≡ r/a, is a relative coordinate. If z ≠ 0, then z_{a} ≡ z/a, would also be used in the equation for F_{r}(z ≠ 0)/2Gσ. Since the product rule for derivatives is used in deriving Equation (3), one might wonder why no derivatives of the elliptic integrals appear in it. This apparent anomaly exists because the derivatives of all three elliptic integrals can be expressed in terms of the elliptic integrals themselves, as shown in [

The scaled radial field in the plane of the disk is plotted in _{a} ≤ 1. The elliptic integrals were evaluated numerically with the help of “Keisan Online Calculator” [_{a} for r_{a} ≤ 1 and is finite for r_{a} = 1. That is not true for the field of the disk, a two-dimensional object, for which the field behavior is curved for r_{a} < 1 and diverges to negative infinity at r_{a} = 1, as suggested by the arrow pointing straight down at that point in _{a} = 1 is logarithmic, and is caused by K(k) in that region. The logarithmic behavior of K(k) for 0.8 ≤ r_{a} ≤ 1.2 is shown in _{a} = 1 is reasonable when one considers the divergence at a point mass (of zero dimension) to be (1/r^{2}) and at a line mass of any length to be a more gradual (1/r) (one-dimensional). In this case, we are dealing with the edge of a two-dimensional mass distribution and expect an even more gradual divergence, which the logarithmic divergence is.

In addition, a view of the complete elliptic integral of the third kind in [

obtained in the same way as that for K(k), demonstrates that Π(n, k) diverges as 1/(1 − r_{a})^{2}. However, its coefficient in Equation (3) contains (1 − r_{a})^{2}. Thus, the final term in Equation (3) is finite at r_{a} = 1, as is the second term because of the normal behavior of E(k) over the entire range of k. Therefore, the behavior of the field at r_{a} = 1 is due do solely to K(k) there.

Certain Current Work

In this section, we examine two other methods of calculating the same radial field as above. These are the use of “standard” numerical integration and Legendre Polynomials.

The gravitational field, V(r_{o}), will be first calculated and then F_{r}(r_{o}) will be calculated from −∂V/∂r_{o}, as above. Now,

V ( r o ) = − 2 G σ [ ∫ 0 a d y ∫ 0 a 2 − y 2 d x ( r o − y ) 2 + x 2 + ∫ 0 a d y ∫ 0 a 2 − y 2 d x ( r o + y ) 2 + x 2 ] (4)

The first term represents the contribution from the right half of the disk in _{a} and y_{a}, the expression for the radial component of the force, F(r_{a}):

F ( r a < 1 ) = 2 G σ [ ∫ 0 1 d y a ( r a − y a ) ( 1 − y a 2 + 1 + r a 2 − 2 y a r a ) 1 + r a 2 − 2 y a r a + ∫ 0 1 d y a ( r a + y a ) ( 1 − y a 2 + 1 + r a 2 + 2 y a r a ) 1 + r a 2 + 2 y a r a + ln ( ( 1 − r a ) / ( 1 + r a ) ) ] (5)

For r_{a} > 1, the expression for the force is the same, except that (1 − r_{a}) within the logarithm is replaced by (r_{a} − 1). We note that the logarithmic divergence appears explicitly in the expression for the force at r_{a} = 1, as opposed to the expression in Equation (3). The “standard” numerical integration is used to calculate numerical value of the two integrals in Equation (5) over the range of r_{a}. This procedure is available online in [_{a} = 1, which we shall now examine.

_{l}(cosθ). The gravitational potential, V(r, θ), is expressed, for this geometry, as:

V ( r , θ ) = ∑ 0 ∞ A l r l P l ( cos θ ) ; r < a (6a)

V ( r , θ ) = ∑ 0 ∞ B l r − ( l + 1 ) P l ( cos θ ) ; r > a (6b)

In the case of other azimuthally symmetric geometries, such as the annular region between a disk and a ring, the potential could be expressed as the sum of these two equations. These expressions are standard for solutions of Laplace’s Equation with azimuthal symmetry and can be found in any textbook on the subject.

Since P_{l}(1) = 1 for all l, we first determine V(r, θ) on-axis, where θ = 0, as in _{l}(cosθ) and sum over the chosen number of products in the resultant series. In our case, where the potential is in the plane of the disk, θ = π/2, and all terms involve even l = 2n. As stated in [_{2n+1}(0), are zero. The on-axis potential, V(r, 0) is:

V ( r , 0 ) = − ( 2 G σ ) π [ r 2 + a 2 − r ] (7)

For r < a, one factors out the a-term under the square root, which allows for the expansion of the potential in powers of (r/a). Factoring out the r-term allows for the expansion in powers of (a/r). After performing the expansions out to a respectable six terms (the chosen number), making use of the expressions for P_{2n}(0), and calculating the derivative −∂V/∂r, we obtain the following for F_{r}:(using the relative coordinate r_{a})

F r = − ( 2 G σ ) π [ 0.5 r a + 0.1875 r a 3 + ( 0.1171 ⋯ ) r a 5 + ( 0.0854 ⋯ ) r a 7 + ( 0.06729 ⋯ ) r a 9 + ( 0.0555 ⋯ ) r a 11 ] ; r a < 1 (8a)

F r = − ( 2 G σ ) π [ 0.5 r a 2 + 0.1875 r a 4 + ( 0.1171 ⋯ ) r a 6 + ( 0.0854 ⋯ ) r a 8 + ( 0.06729 ⋯ ) r a 10 + ( 0.0593 ⋯ ) r a 12 ] ; r a > 1 (8b)

Although it is difficult to resolve details, _{a} = 1. An expanded view of this region is shown in _{a} < 1.2. As the curve of F_{r} vs r_{a} grows steeper in its approach, from either direction, to the logarithmic divergence at r_{a} = 1, more terms in the Legendre expansion are needed to accurately describe it. An even more respectable eight terms in the expansion would shrink the range of disagreement to 0.9 < r_{a} < 1.1. As a motivation for their paper, Lass and Blitzer [

So far, we have only looked at the radial field in the plane of the disk, but what about the radial field at different values of z_{a} and the vertical field at different values of r_{a}? Both of these fields can be readily calculated using the “standard” numerical integration, as in Equation (4), but with the inclusion of z_{o} in the expression for distance, as in Equation (9). The potential gradient, −∂V/∂r, is then calculated.

V ( r o , z o ) = − 2 G σ [ ∫ 0 a d y ∫ 0 a 2 − y 2 d x ( r o − y ) 2 + z o 2 + x 2 + ∫ 0 a d y ∫ 0 a 2 − y 2 d x ( r o + y ) 2 + z o 2 + x 2 ] (9)

_{a} = 0, 0.2, and 0.4. The basic shapes are the same for all three curves, but there is no logarithmic divergence for values of z_{a} > 0 because there is no edge to the mass distribution at those levels. As with the first curve, these two curves do peak at r_{a} = 1 because at that point, all of the mass exerts a force in the same direction; there is no cancellation in the radial direction. Although that is also true for r_{a} > 1, the distance to r_{a} has increased to all of the mass points on the disk.

_{z} (=−∂V/∂z), for various values of r_{a}, both less than and greater than 1. For all r_{a} < 1, F_{z} = −2πσG at z = 0. This

result follows from the Gaussian formulation of the law of gravitation, ÑF = −4πGρ, where F is the gravitational field and ρ is the mass density at each point. Using a Gaussian pillbox, as in electrostatics, that straddles the surface mass density of the disk, along with the integral form of Gauss’ law, leads to the result. All mass points on the disk, other than the one in question, only contribute to the field parallel to the surface of the disk at z = 0.

We note that F_{z} = 0 for all r_{a} > 1and z = 0, where there is no surface mass (σ = 0). We could also note that by planar anti-symmetry, the field above the disk points in the negative z-direction, while that below points in the positive z-direction. Since there is no discontinuity in the field for r_{a} > 1 and z_{a} = 0, what else could the field be for those values of r_{a} and z_{a}? Apropos of these comments, _{z} in the plane of a disk with a hole in it. Within the entire area of hole and the region beyond the disk, F_{z} = 0 in the plane of the disk.

This is definitely not true for the radial field, as we now discuss in the case of a solid disk containing a hole of half of its area and centered within it. The radial field can readily be obtained from linear superposition by subtracting the field of a smaller disk, identical in size and position of the hole, from that of the solid disk. _{a} ≈ 0.81. At that point, the field becomes negative and stays that way for all r_{a} > 0.81, eventually approaching zero for large r_{a}.

By comparison, _{s}, of a sphere of mass density ρ, radius a, and a central hole of half the volume of the sphere. Because of its spherical symmetry, this is a simple problem, also solved with linear superposition, whose solution is:

F s = ( 4 π 3 ) G a ρ [ r a − 1 2 r a 2 ] ; 0.5 1 / 3 ≤ r a ≤ 1 (10a)

F s = ( 2π 3 ) [ G a ρ ] r a 2 ; r a ≥ 1 (10b)

F_{s} = 0 within the hole, where 0 ≤ r_{a} ≤ 0.5^{1/3}.

Until now, we have concentrated on a disk of zero thickness. In this section, we consider the radial gravitational field of a uniform disk of non-zero thickness T equal to a maximum of one tenth of its fixed radius “a”. Two different situations are considered, as shown in _{a} = 0.1. The field at X is determined as a function of the relative thickness, Z_{a} (≤T_{a}). In situation B, the relative thickness is a constant T_{a}, but the field point X moves from the lower corner along the outer surface of the disk to the top corner. The field at X is determined as a function of its relative position, z_{a}. There are two contributions to the field in situation B: the section of the disk above z_{a} and the one below. For a disk of constant radius with respect to height, these contributions will have already been determined in situation A.

These contributions are determined by breaking up the disk into thin slices and considering each slice to be a disk of zero thickness, having a vertical separation from point X. The contribution from a given slice is calculated as in Equation (9) and taking its radial gradient at r_{a} = 1. The total contribution is obtained by integrating the individual contributions over the relative thickness, Z_{a}.

_{a}. The function is obviously linear in ln(z_{a}) over the range of consideration. The curve would deviate significantly from linearity for much larger z_{a} because the

contribution must asymptotically approach zero for very thick disks. Calling the function F(z_{a}),

F ( z a ) = − ln ( z a ) + 0.0869 (11)

To obtain the total contribution at X for situation A, these contributions must be summed from 0 to Z_{a}. Thus, the total contribution over that thickness is:

∫ 0 Z a F ( Z a ) d Z a = − Z a ln ( Z a ) + 1.0869 Z a (12)

Aside from a scale factor, this equation is plotted in _{a} = 0.05 is obvious, as are values at the end points, both of which receive contributions from the entire disk.

The properties of these circular orbits considered are the orbital period as a function of their radius measured from the center of the disk and the mass of the disk. The disk is assumed to be an approximate representation of the Milky Way with the same mass and diameter. The estimated diameter of the Milky Way is 10^{5} ly [^{42} kg [^{3} ly [

Finding the period of the orbit is a simple problem in orbital mechanics. Assuming the radial field, F(R), beyond the disk is as in

F ( R ) = V 2 R , (13)

where V and R are the orbital speed and radius, respectively. The period T is

T = 2 π R V = 2 π R / F (14)

^{5} ly. Galaxies that are less than this distance from the Milky Way are listed in [^{3} ly, or 1.5 galactic radii, which suggests a rapid march toward the asymptotic limit (but not as rapid as in the case of a sphere!). The result for a disk was arrived at by noting that the period of the orbit of radius R around a point mass is, coincidentally, proportional to R^{1.5}. Thus, if

Regarding orbital stability perpendicular to the plane of the disk, we can see in

(i.e., a vertical oscillation about the plane of the disk). One could imagine a horizontal orbit half way between two identical disks, but such an orbit would be vertically unstable, since a perturbation toward either disk would only create a growing force imbalance toward that disk. This is particularly true for r_{a} < 1, where the magnitude of the negative force is monotonically decreasing with z_{a}. The orbiting object would eventually collide with the disk. This trend would not necessarily hold for r_{a} > 1, such as r_{a} = 1.1, where the magnitude of the negative force increases with z_{a} for 0 ≤ z_{a} ≤ 0.3.

Radial stability of a circular orbit is discussed in [

3 ρ + g ′ ( ρ ) g ( ρ ) > 0 (15)

where ρ is the radius of the unperturbed circle, g(ρ) is the negative of the gravitational force, and g'(ρ) is its derivative at ρ. _{a} ≳ 1.15. If the field behaved according to an inverse-square law, as outside a sphere, stability would occur for all values of r_{a}.

In this section, we shall calculate the escape velocity from the disk-approximation of the Milky Way based solely on the gravitational attraction to an escaping object as a function of its location within the galaxy. The object’s orbital motion, or

other motion, within the galaxy, which influences the escape velocity, is not considered. As is well known, the escape velocity or, more properly, the escape speed, v_{esc}, is computed by equating the object’s initial kinetic energy to the negative of its initial potential energy. Thus,

v e s c = 2 V ( r a ) (16)

where V(r_{a}) is the potential energy, as in Equation(4), using a relative coordinate. This expression gives the escape velocity as a function of position within the disk. _{a} for it ≈ 0.5, where its computed escape velocity from Equation (16) is 684 km/sec. A reported value is ≈550 km/sec [

In this paper, we have presented certain aspects of the gravitational characteristics of the disk geometry, including an ideally thin disk, a disk of finite thickness, and a thin disk in the form of a ring, though not thin walled. This geometry has been less studied than that of a sphere, which, because of its higher degree of symmetry, is simpler to analyze. Aside from its interest simply as a physics problem, certain astronomical objects, such as the Milky Way and protoplanetary disks, are disk-like in shape. In that regard, we have also presented calculations of the circular orbits in the plane of the Milky Way, approximated as a disk, concerning their orbital period and orbital stability. The escape velocity of an object, particularly at the position of the solar system, has been similarly calculated.

The author thanks W. F. Filter, retired Sandia National Laboratories physicist, for reviewing the draft of this paper and making many valuable suggestions for its improvement.

The author of this paper declares that he has no conflict of interest of any sort, particularly as described by the journal under “potential conflict of interest.”

Weiss, J.D. (2018) Certain Aspects of the Gravitational Field of a Disk. Applied Mathematics, 9, 1360-1377. https://doi.org/10.4236/am.2018.912089