A Computational Study of the Mechanics of Gravity-induced Torque on Cells

Gravity is an important force that affects the life of a cell in various ways. When the gravitational force is exerted upon cells and similar objects, it results in cellular deformations and also in changes of the cell/object’s orientation. This may be due to compression caused by hydrostatic pressure that can affect the tensegrity of load bearing structures within cells, such as actin microfilaments and other cytoskeletal components (Vassy et al., 2001). In addition, it could be due to the influence it exerts on the physico-chemical self-organization of microtubules that comprise the cytoskeleton (Papaseit et al., 2000; Tabony et al., 2001). For example, Hughes-Fulford (2002) has demonstrated that serum deprived osteoblasts exhibit reduction in cell proliferation in microgravity conditions while these observations were confirmed by Vassy et al. (2001) in human breast cancer cells. In plant cells microgravity permits the enhancement of proliferation, perhaps due to reduced interactions in the absence of gravity (Matía et al., 2010).

The goal of this contribution is to reexamine the mechanics of gravity-induced torque in cells and similar objects, in surface and orbital experiments. For that we have considered three biological systems, i.e., sarcoma cells, human egg, and the Gallus gallus egg. The choice of these cells and similar objects is based on the fact that they vary increasingly in size. In addition, there is a scientific interest related to the gravitational effects of cancerous type of cells, along with the effect of gravitation at the basal stages of embryonic development. For that, our paper re-examines Nace’s result (Nace, 1983) that assumes a constant gravitational acceleration g and considers the gravitational acceleration on the surface of the Earth that is now corrected for geocentric latitude φ_E, the J₂ harmonic coefficient, and the rotation of the Earth via its angular velocity ω_E. These corrections are needed for accurate planet-based experiments. Similarly, the same calculations are repeated at the orbital altitude of an orbiting spacecraft, with the J₂ harmonic as the only correction included, accounting for the Earth’s oblateness. At the orbital altitude of the spacecraft the effect of the rotational potential of the Earth is not included since it does not affect the spacecraft, which is not in any physical contact with the Earth’s surface but is governed by the rotational potential of its orbit. Our calculations can be applied in the case where more accurate surface and orbital values of the gravitational potential are required. Our goal is to also derive and investigate the mathematical relations that relate biological experimental parameters to various Earth surface parameters as well as the orbital elements of the spacecraft. These planetary geophysical parameters include the J₂ harmonic, the flattening of the Earth f′ and finally the angular velocity of the Earth’s rotation ω_E. Furthermore, the idea of using biological space experiments as an alternative method of calculating important planetary parameters in more advanced future experiments to come is also introduced.

In our effort to calculate the effect of gravity on cells and similar objects let us consider the gravitational potential V(r′) at the surface of the Earth, where an experiment with cells and similar objects is taking place under controlled conditions. We can write the total potential as the sum of three different potential terms, namely: (1) Vtot (r′)=−GMEr′+GMERE2J22r′3(3sin2⁡ϕE−1)−12ωE2r′2cos2⁡ϕE. V_{\text {tot }}\left(r^{\prime}\right)=-\frac{G M_{E}}{r^{\prime}}+\frac{G M_{E} R_{E}^{2} J_{2}}{2 r^{\prime 3}}\left(3 \sin ^{2} \phi_{E}-1\right)-\frac{1}{2} \omega_{E}^{2} r^{\prime 2} \cos ^{2} \phi_{E}

where the first term in the right-hand side (RHS) represents the central Newtonian potential, the second term represents the potential correction due to the J₂ spherical harmonic of the Earth’s gravitational potential, and the third term represents the rotational potential, acting only on the surface of the Earth. Following Haranas et al. (2012), we write the magnitude of the total of all accelerations on the surface of the Earth to be: (2) gtot =GMEr′2−3GMERE2J22r′4(3sin2⁡ϕE−1)−ωE2r′cos2⁡ϕE, g_{\text {tot }}=\frac{G M_{E}}{r^{\prime 2}}-\frac{3 G M_{E} R_{E}^{2} J_{2}}{2 r^{\prime 4}}\left(3 \sin ^{2} \phi_{E}-1\right)-\omega_{E}^{2} r^{\prime} \cos ^{2} \phi_{E},

where r′ is the radial distance from the center of the Earth to an external surface point, M_E is the mass of the Earth, R_E is the radius of the Earth, J₂ is the zonal harmonic coefficient that describes the oblateness of the Earth, ω_E the angular velocity of the Earth, and φ_E is the geocentric latitude of the designed experiment. Zonal harmonics are simply bands of latitude, whose boundaries are the roots of a Legendre polynomial. This particular gravitational harmonic coefficient is a result of the Earth’s shape and is about 1000 times larger than the next harmonic coefficient J₃ and its value is equal to J₂ = -0.0010826260 (Kaula, 2000). At the orbital point of the spacecraft the rotational potential on the surface of the Earth does not affect the orbit of spacecraft. Therefore the gravitational acceleration as it is given by Eq. (2) can be transformed as a function of orbital elements, using standard transformations given by Kaula (2000) and Vallado and McClain (2007), namely sin = φsin isin(u_E = )cos θ, where, φ_E is the geocentric latitude, measured from the Earth’s equator to the poles, and θ_E is the corresponding colatitude measured from the poles down to the equator (θ_E = 90 – φ_E), u = ω + f is the argument of latitude that defines the position of a body moving along a Kepler orbit, i its orbital inclination, ω is the argument of the perigee of the spacecraft (not to be confused with angular velocity, which we write with subscripts – see nomenclature section below), f is its true anomaly (an angle defined between the orbital position of the spacecraft and its perigee). The orbital elements are shown in Figure 1 below.

Figure 1.

To familiarize the reader with the orbital elements used here, let us define the orbital elements appearing in our Eq. (4) below. a_s is the semi-major axis that defines the size of the orbital ellipse. It is the distance from the center of the ellipse to an apsis, i.e., the point where the radius vector is maximum or minimum (i.e., apogee and perigee points). Similarly, e is the eccentricity, which defines the shape of the orbital ellipse (minor to major axis ratio), i is the inclination of the orbit defined as the angle between the orbital and equatorial planes, and ω is the argument of the perigee, the angle between the direction of the ascending node (the point on the equatorial plane at which the satellite crosses from south to north) and the direction of the perigee. Finally, the true anomaly f is the angle that locates the satellite in the orbital ellipse and is measured in the direction of motion from the perigee to the position vector of the satellite. Assuming an elliptical orbit, the geocentric orbital distance r′ is given by (Vallado and McClain, 2007): (3) 6r′=as(1−e2)(1+ecos⁡f), 6 r^{\prime}=\frac{a_{s}\left(1-e^{2}\right)}{(1+e \cos f)},

6 and therefore Eq. (2) becomes a function of the spacecraft orbital elements and therefore we have: (4) gtot =GME(1+ecos⁡f)2as2(1−e2)2−3GMERE2J2(1+ecos⁡f)42as4(1−e2)4(3sin2⁡isin2⁡f−1). g_{\text {tot }}=\frac{G M_{E}(1+e \cos f)^{2}}{a_{s}^{2}\left(1-e^{2}\right)^{2}}-\frac{3 G M_{E} R_{E}^{2} J_{2}(1+e \cos f)^{4}}{2 a_{s}^{4}\left(1-e^{2}\right)^{4}}\left(3 \sin ^{2} i \sin ^{2} f-1\right) .

In relation to the torque we say that torque is defined as the moment of a force that measures the tendency of a particular force to produce rotation about an axis, and is defined according to the formula below (Meirovitch, 1998): (5) L→=r→×F→=Frsin⁡θ \vec{L}=\vec{r} \times \vec{F}=F r \sin \theta

Figure 2.

where F is the applied force, r is the distance from the center of mass that the force is applied to the axis of rotation, and θ is the angle between F and r. Following Nace (1983) and adopting exactly the same experimental scenario of a floating model cell composed of light and heavy masses, we have that the net torque exerted on the cell is given by Nace (1983) to be: (6) Lcell =4π3aℓ0c3rBH3ρmsin⁡θ(b3ρH−d3ρLb3ρH+c3ρLd3)gtot L_{\text {cell }}=\frac{4 \pi}{3} a \ell_{0} c^{3} r_{B H}^{3} \rho_{m} \sin \theta\left(\frac{b^{3} \rho_{H}-d^{3} \rho_{L}}{b^{3} \rho_{H}+c^{3} \rho_{L} d^{3}}\right) g_{\text {tot }}

where ℓ aℓ₀ and ℓ is the distance between the centers of the heavy and light masses in the cell, ℓ₀ is the total cell length, and r_H = br_BH where, where r_H is the radius of the heavy mass and r_BH the radius of the cell, ρ_m is the density of the medium, ρ_H and ρ_L is the density of the heavy and light mass, θ is the angle between the applied force F and the distance r in Eq. (5), and a, b, c, d scaling constants given in Nace’s paper that take values in the range 0 < a < 1, 0 < b < 1, 0 < c < 1, 0 < d < 1. Quoting Nace (1983) Eq. (6) is an equation that: “disregards Stoke’s law which gives the rate of fall of a small sphere in a viscous fluid under the force of gravity. It omits viscosity which considers the internal friction in the system, and also surface tension which considers the tension in the surface of separation between the cell and the medium. This is possible because, although these parameters influence the displacement and the response time of the cell, ‘they do not change the torque attributed to gravity, at the instant under consideration.’ ” (Nace, 1983). Taking into account Eq. (4) and substituting in Eq. (6) we obtain the following modified equation for the torque exerted on a cell, in an experiment that takes place in a spacecraft in orbit around a planet, Earth in our case, to be: (7) Lcell =4πaℓ0c3rBH3ρmsin⁡θ3(b3ρH−d3ρLb3ρH+c3ρLd3)[GME(1+ecos⁡f)2as2(1−e2)2−3GMERE2J2(1+ecos⁡f)42as4(1−e2)4(3sin2⁡fsin2⁡i−1)]. L_{\text {cell }}=\frac{4 \pi a \ell_{0} c^{3} r_{B H}^{3} \rho_{m} \sin \theta}{3}\left(\frac{b^{3} \rho_{H}-d^{3} \rho_{L}}{b^{3} \rho_{H}+c^{3} \rho_{L} d^{3}}\right)\left[\begin{array}{l} \frac{G M_{E}(1+e \cos f)^{2}}{a_{s}^{2}\left(1-e^{2}\right)^{2}} \\ -\frac{3 G M_{E} R_{E}^{2} J_{2}(1+e \cos f)^{4}}{2 a_{s}^{4}\left(1-e^{2}\right)^{4}}\left(3 \sin ^{2} f \sin ^{2} i-1\right) \end{array}\right] .

Therefore, Eq. (7) is Nace’s equation transformed as a function of the orbital elements. Similarly, using Eq. (2) for the acceleration of gravity on the surface of the Earth becomes: (8) gs=GMERE2−3GMEJ22RE2(3sin2⁡ϕE−1)−REωE2cos2⁡ϕE, g_{s}=\frac{G M_{E}}{R_{E}^{2}}-\frac{3 G M_{E} J_{2}}{2 R_{E}^{2}}\left(3 \sin ^{2} \phi_{E}-1\right)-R_{E} \omega_{E}^{2} \cos ^{2} \phi_{E},

furthermore, we approximate the Earth as an ellipsoid of revolution so we can write that the radius of the Earth ellipsoid becomes (Kaula, 2000): (9) RE=Req(1−(f′+32f′2)sin2⁡ϕE+32f′2sin4⁡ϕE−…)≈Req(1−f′sin2⁡ϕE) R_{E}=R_{e q}\left(1-\left(f^{\prime}+\frac{3}{2} f^{\prime 2}\right) \sin ^{2} \phi_{E}+\frac{3}{2} f^{\prime 2} \sin ^{4} \phi_{E}-\ldots\right) \approx R_{e q}\left(1-f^{\prime} \sin ^{2} \phi_{E}\right)

where R_eq is the equatorial radius of the Earth ellipsoid and where f′ is the Earth’s flattening given by, (Stacey, 1977): f′=Req−RpolReq=3J22+RE3ωE22GME f^{\prime}=\frac{R_{e q}-R_{p o l}}{R_{e q}}=\frac{3 J_{2}}{2}+\frac{R_{E}^{3} \omega_{E}^{2}}{2 G M_{E}}

where R_pol is the Earth’s polar radius. Therefore with the help of Eq. (9) Eq. (7) becomes: (10) Lcell=4πaℓ0c3rBH3ρmsin⁡θ3(b3ρH−d3ρLb3ρH+c3ρLd3)(GMEReq2(1−f′sin2⁡ϕE)2−3GMEJ22Req2(1−f′sin2⁡ϕE)2(3sin2⁡ϕE−1)−Req(1−f′sin2⁡ϕE)ωE2cos2⁡ϕE). L_{c e l l}=\frac{4 \pi a \ell_{0} c^{3} r_{B H}^{3} \rho_{m} \sin \theta}{3}\left(\frac{b^{3} \rho_{H}-d^{3} \rho_{L}}{b^{3} \rho_{H}+c^{3} \rho_{L} d^{3}}\right)\left(\begin{array}{l} \frac{G M_{E}}{R_{e q}^{2}\left(1-f^{\prime} \sin ^{2} \phi_{E}\right)^{2}}-\frac{3 G M_{E} J_{2}}{2 R_{e q}^{2}\left(1-f^{\prime} \sin ^{2} \phi_{E}\right)^{2}}\left(3 \sin ^{2} \phi_{E}-1\right) \\ -R_{e q}\left(1-f^{\prime} \sin ^{2} \phi_{E}\right) \omega_{E}^{2} \cos ^{2} \phi_{E} \end{array}\right) .

Using Eq. (10) on the surface of the Earth and for the geocentric latitudes φ_E = 0°, 45°, and 90°, we obtain the following three corresponding torque expressions: (11) Lcell =4πaℓ0c3rBH3ρmsin⁡θ3(b3ρH−d3ρLb3ρH+c3ρLd3)(GMEReq2+3GMEJ22Req2−Req2ωE2), L_{\text {cell }}=\frac{4 \pi a \ell_{0} c^{3} r_{B H}^{3} \rho_{m} \sin \theta}{3}\left(\frac{b^{3} \rho_{H}-d^{3} \rho_{L}}{b^{3} \rho_{H}+c^{3} \rho_{L} d^{3}}\right)\left(\frac{G M_{E}}{R_{e q}^{2}}+\frac{3 G M_{E} J_{2}}{2 R_{e q}^{2}}-R_{e q}^{2} \omega_{E}^{2}\right), (12) Lcell=4πaℓ0c3rBH3ρmsin⁡θ3(b3ρH−d3ρLb3ρH+c3ρLd3)(GMEReq2(1−f′2)2−3GMEJ24Req2(1−f′2)2−12(1−f′2)2Req2ωE2), L_{c e l l}=\frac{4 \pi a \ell_{0} c^{3} r_{B H}^{3} \rho_{m} \sin \theta}{3}\left(\frac{b^{3} \rho_{H}-d^{3} \rho_{L}}{b^{3} \rho_{H}+c^{3} \rho_{L} d^{3}}\right)\left(\frac{G M_{E}}{R_{e q}^{2}\left(1-\frac{f^{\prime}}{2}\right)^{2}}-\frac{3 G M_{E} J_{2}}{4 R_{e q}^{2}\left(1-\frac{f^{\prime}}{2}\right)^{2}}-\frac{1}{2}\left(1-\frac{f^{\prime}}{\sqrt{2}}\right)^{2} R_{e q}^{2} \omega_{E}^{2}\right), (13) Lcell=4πaℓ0c3rBH3ρmsin⁡θ3(b3ρH−d3ρLb3ρH+c3ρLd3)(GMEReq2(1−f′)2−3GMEJ2Req2(1−f′)2). L_{c e l l}=\frac{4 \pi a \ell_{0} c^{3} r_{B H}^{3} \rho_{m} \sin \theta}{3}\left(\frac{b^{3} \rho_{H}-d^{3} \rho_{L}}{b^{3} \rho_{H}+c^{3} \rho_{L} d^{3}}\right)\left(\frac{G M_{E}}{R_{e q}^{2}\left(1-f^{\prime}\right)^{2}}-\frac{3 G M_{E} J_{2}}{R_{e q}^{2}\left(1-f^{\prime}\right)^{2}}\right).

We first consider circular orbits, i.e., eccentricity e = 0 with inclinations i = 0,°45°, and 90°, and semimajor axis a_s. For circular orbits the true anomaly is undefined because the orbits do not possess a uniquely determined periapsis (perigee point), and therefore the argument of latitude u is used instead. In particular, circular equatorial orbits i = 0° do not have an ascending node point from which u is defined. In this case, Earth’s equatorial plane is used as the reference plane, and the First Point of Aries as the origin of longitude and the mean longitude is used instead. This is the longitude that an orbiting body would have if its orbit were circular and its inclination i = 0. Mean longitude is given by the following relation L = M ω+ Ω+, where M is the mean anomaly defined as the angle between the perigee and the satellite radius vector assuming that the satellite moves with a constant angular velocity. Similarly, Ω is the right ascension of the node. On Earth it is measured positively (counter clockwise) in the equatorial plane from the longitude zero meridian (ℓ = 0) and the point of the orbit at which the satellite crosses the equator from south to north (ascending node). In this case Eq. (7) must have mean longitude used rather that argument of latitude. Using Eq. (7) for L = 0° we obtain Eq. (14), and we find the following expressions for the torque at orbital inclinations of 0 °, 45 °, and 90°: (14) Lcell=4πaℓ0c3rBH3ρmsin⁡θ3(b3ρH−d3ρLb3ρH+c3ρLd3)(GMEas2+3GMERE2J22as4), L_{c e l l}=\frac{4 \pi a \ell_{0} c^{3} r_{B H}^{3} \rho_{m} \sin \theta}{3}\left(\frac{b^{3} \rho_{H}-d^{3} \rho_{L}}{b^{3} \rho_{H}+c^{3} \rho_{L} d^{3}}\right)\left(\frac{G M_{E}}{a_{s}^{2}}+\frac{3 G M_{E} R_{E}^{2} J_{2}}{2 a_{s}^{4}}\right), (15) Lcell =4πaℓ0c3rBH3ρmsin⁡θ3(b3ρH−d3ρLb3ρH+c3ρLd3)(GMEas2−3GMERE2J22as4(3sin2⁡u2−1)), L_{\text {cell }}=\frac{4 \pi a \ell_{0} c^{3} r_{B H}^{3} \rho_{m} \sin \theta}{3}\left(\frac{b^{3} \rho_{H}-d^{3} \rho_{L}}{b^{3} \rho_{H}+c^{3} \rho_{L} d^{3}}\right)\left(\frac{G M_{E}}{a_{s}^{2}}-\frac{3 G M_{E} R_{E}^{2} J_{2}}{2 a_{s}^{4}}\left(\frac{3 \sin ^{2} u}{2}-1\right)\right), (16) Lcell =4πaℓ0c3rBH3ρmsin⁡θ3(b3ρH−d3ρLb3ρH+c3ρLd3)(GMEas2−3GMERE2J22as4(3sin2⁡u−1)). L_{\text {cell }}=\frac{4 \pi a \ell_{0} c^{3} r_{B H}^{3} \rho_{m} \sin \theta}{3}\left(\frac{b^{3} \rho_{H}-d^{3} \rho_{L}}{b^{3} \rho_{H}+c^{3} \rho_{L} d^{3}}\right)\left(\frac{G M_{E}}{a_{s}^{2}}-\frac{3 G M_{E} R_{E}^{2} J_{2}}{2 a_{s}^{4}}\left(3 \sin ^{2} u-1\right)\right) .

From Eq. (14) we see that in circular equatorial orbits the calculated cell torque depends on only one orbital parameter, i.e., the orbital semimajor axis a_s, and the geophysical parameters such as the mass of the Earth M_E, its radius R_E, and finally the J₂ harmonic coefficient of the gravitational field. Orbits of inclinations i = 45°, 90° will depend additionally on the argument of latitude u. Similarly, for elliptical orbits we use Eq. (7) varying only the inclination i = 0°, 45°, and 90° so that we obtain the following corresponding torques: (17) Lcell =4πaℓ0c3rBH3ρmsin⁡θ3(b3ρH−d3ρLb3ρH+c3ρLd3)(GME(1+ecos⁡f)2as2(1−e2)2+3GMERE2J2(1+ecos⁡f)42as4(1−e2)4). L_{\text {cell }}=\frac{4 \pi a \ell_{0} c^{3} r_{B H}^{3} \rho_{m} \sin \theta}{3}\left(\frac{b^{3} \rho_{H}-d^{3} \rho_{L}}{b^{3} \rho_{H}+c^{3} \rho_{L} d^{3}}\right)\left(\frac{G M_{E}(1+e \cos f)^{2}}{a_{s}^{2}\left(1-e^{2}\right)^{2}}+\frac{3 G M_{E} R_{E}^{2} J_{2}(1+e \cos f)^{4}}{2 a_{s}^{4}\left(1-e^{2}\right)^{4}}\right) . (18) Lcell=4πaℓ0c3rBH3ρmsin⁡θ3(b3ρH−d3ρLb3ρH+c3ρLd3)(GME(1+ecos⁡f)2as2(1−e2)2−3GMERE2J2(1+ecos⁡f)42as4(1−e2)4(3sin2⁡f2−1)), L_{c e l l}=\frac{4 \pi a \ell_{0} c^{3} r_{B H}^{3} \rho_{m} \sin \theta}{3}\left(\frac{b^{3} \rho_{H}-d^{3} \rho_{L}}{b^{3} \rho_{H}+c^{3} \rho_{L} d^{3}}\right)\left(\frac{G M_{E}(1+e \cos f)^{2}}{a_{s}^{2}\left(1-e^{2}\right)^{2}}-\frac{3 G M_{E} R_{E}^{2} J_{2}(1+e \cos f)^{4}}{2 a_{s}^{4}\left(1-e^{2}\right)^{4}}\left(\frac{3 \sin ^{2} f}{2}-1\right)\right), (19) Lcell=4πaℓ0c3rBH3ρmsin⁡θ3(b3ρH−d3ρLb3ρH+c3ρLd3)(GME(1+ecos⁡f)2as2(1−e2)2−3GMERE2J2(1+ecos⁡f)42as4(1−e2)4(3sin2⁡f−1)). L_{c e l l}=\frac{4 \pi a \ell_{0} c^{3} r_{B H}^{3} \rho_{m} \sin \theta}{3}\left(\frac{b^{3} \rho_{H}-d^{3} \rho_{L}}{b^{3} \rho_{H}+c^{3} \rho_{L} d^{3}}\right)\left(\frac{G M_{E}(1+e \cos f)^{2}}{a_{s}^{2}\left(1-e^{2}\right)^{2}}-\frac{3 G M_{E} R_{E}^{2} J_{2}(1+e \cos f)^{4}}{2 a_{s}^{4}\left(1-e^{2}\right)^{4}}\left(3 \sin ^{2} f-1\right)\right) .

From Eq. (17)-(19) we see that in elliptical equatorial orbits the calculated cell torque depends on the following orbital parameters, namely, the orbital semimajor axis a_s, eccentricity e, the true anomaly f of the orbit, and the same geophysical parameters as the circular orbits above.

Today’s methods for the modelling and calculation of various geophysical parameters involve the continuous acceleration component monitoring of orbiting satellites. In this section we propose the use of biological and biophysical experimental parameters for the calculation, at least in principle, of parameters such as the J₂. As a first step, we want to establish analytical mathematical expressions that relate torque cell experimental parameters to the J₂ harmonic coefficient. For that we use Eq. (7) that is a modified Nace equation that calculates the torque exerted on a floating object in an experiment that takes place aboard an orbiting spacecraft. Solving Eq. (7) for J₂ and simplifying we obtain: (20) J2=2as2(1−e2)3RE2(1+ecos⁡f)2(3sin2⁡isin2⁡f−1)[1−3as2(1−e2)2(b3ρH+c3d3ρL)4GMπc3aℓ0rBH3ρm4(1+ecos⁡f)2(b3ρH−d3ρL)Lcellcsc⁡θ]. J_{2}=\frac{2 a_{s}^{2}\left(1-e^{2}\right)}{3 R_{E}^{2}(1+e \cos f)^{2}\left(3 \sin ^{2} i \sin ^{2} f-1\right)}\left[1-\frac{3 a_{s}^{2}\left(1-e^{2}\right)^{2}\left(b^{3} \rho_{H}+c^{3} d^{3} \rho_{L}\right)}{4 G M \pi c^{3} a \ell_{0} r_{B H}^{3} \rho_{m} 4(1+e \cos f)^{2}\left(b^{3} \rho_{H}-d^{3} \rho_{L}\right)} L_{c e l l} \csc \theta\right].

This is a general equation relating the parameters of a cell torque experiment taking place aboard an orbiting spacecraft to the J₂ harmonic coefficient. Equation (20) can be also used in the case of circular orbits when e = 0. Similarly, for an experiment that takes place on the surface of Earth, solving Eq. (10) we obtain the following expression: (21) J2=16GME(1−f′sin4⁡ϕE)(3sin2⁡ϕE−1)[−4GME(1−18GMERE3ωE2cos2⁡ϕE(2−f′(1+ecos⁡f)3))(1−f′sin2⁡ϕE)2+3RE2csc⁡θπc3aℓ0rBH3ρm(b3ρH+c3d3ρLb3ρH−d3ρL)Lcell] J_{2}=\frac{1}{6 G M_{E}} \frac{\left(1-f^{\prime} \sin ^{4} \phi_{E}\right)}{\left(3 \sin ^{2} \phi_{E}-1\right)}\left[-\frac{4 G M_{E}\left(1-\frac{1}{8 G M_{E}} R_{E}^{3} \omega_{E}^{2} \cos ^{2} \phi_{E}\left(2-f^{\prime}(1+e \cos f)^{3}\right)\right)}{\left(1-f^{\prime} \sin ^{2} \phi_{E}\right)^{2}}+\frac{3 R_{E}^{2} \csc \theta}{\pi c^{3} a \ell_{0} r_{B H}^{3} \rho_{m}}\left(\frac{b^{3} \rho_{H}+c^{3} d^{3} \rho_{L}}{b^{3} \rho_{H}-d^{3} \rho_{L}}\right) L_{c e l l}\right]

Equation (21) can be used at any geocentric latitude, and both equations (20)-(21). Application of Eq. (21) presupposes the ability to measure the torque applied on the floating object L_cell, something that might be possible in future Earth and space experiments.

The above derived torques are obviously changing as a function of orbital position. In order to obtain the time rate of change of the torque acting on a cell during a spacecraft orbiting experiment, let us assume that the distance ℓ₀ and the angle θ within the cells set up as given by Nace (1983) remain constant for a spacecraft revolution around the planet (Earth in our case). Let us also assume that the rest of the orbital elements in Eq. (4) do not significantly change in one spacecraft revolution, and consider that the acceleration of gravity g is a function of time t along the orbit. Thus, taking the derivative w.r.t. time t of Eq. (7) we obtain the time rate of change of the cell torque to be: (22) dLcell dt=4πa3c3rBH3Q0(g˙tot ℓ0sin⁡θ), \frac{\mathrm{d} L_{\text {cell }}}{\mathrm{d} t}=\frac{4 \pi a}{3} c^{3} r_{B H}^{3} Q_{0}\left(\dot{g}_{\text {tot }} \ell_{0} \sin \theta\right),

where, for convenience, (23) Q0=(b3ρH−d3ρL)(b3ρH+c3d3ρL)ρm. Q_{0}=\frac{\left(b^{3} \rho_{H}-d^{3} \rho_{L}\right)}{\left(b^{3} \rho_{H}+c^{3} d^{3} \rho_{L}\right)} \rho_{m}.

Next, to proceed with the time rate of change of the torque exerted on the cell around the orbit we note that the orbit element that changes with time is the true anomaly f that is included in Eq. (4). This is considered as a function of time using the relation between the time-dependent true anomaly f and the true anomaly M (Murray and Dermott, 1999): (24) f≈M+2esin⁡M+5e24sin⁡2M=nt+2esin⁡(nt)+5e24sin⁡(2nt)..≈nt+2esin⁡(nt), f \approx M+2 e \sin M+\frac{5 e^{2}}{4} \sin 2 M=n t+2 e \sin (n t)+\frac{5 e^{2}}{4} \sin (2 n t) . . \approx n t+2 e \sin (n t),

Eventually the relation of the mean anomaly M to time t is through the equation M = n(t – t_p) (Murray and Dermott, 1999), defined as the angle between the perigee and the satellite radius vector, assuming that the satellite moves with a constant angular velocity. t_p is the time at which the satellite crosses its perigee, and n is its mean angular velocity that is defined according to the equation: n=(GME/as3)1/3n=\left(G M_{E} / a_{s}^{3}\right)^{1 / 3}, and, it is a measure of how fast a satellite progresses around its elliptical orbit. Therefore, taking the time derivative of Eq. (4) after the substitution of f as a function of time t from equation (24), the derivative of the orbital acceleration g˙tot\dot{g}_{t o t} to O(e) becomes: (25) g˙tot =[2GMEBsin⁡Aas2(1−e2)2−9GMERE2J2Bsin⁡2Asin2⁡i2as4(1−e2)4−6GMERE2J2sin⁡Aas4(1−e2)4−36GMERE2J2sin⁡Acos2⁡Asin2⁡ias4(1−e2)4+18GMERE2J2Bsin3⁡Asin2⁡ias4(1−e2)4]e+O(e2)…., \dot{g}_{\text {tot }}=\left[\begin{array}{l} \frac{2 G M_{E} B \sin A}{a_{s}^{2}\left(1-e^{2}\right)^{2}}-\frac{9 G M_{E} R_{E}^{2} J_{2} B \sin 2 A \sin ^{2} i}{2 a_{s}^{4}\left(1-e^{2}\right)^{4}}-\frac{6 G M_{E} R_{E}^{2} J_{2} \sin A}{a_{s}^{4}\left(1-e^{2}\right)^{4}} \\ -\frac{36 G M_{E} R_{E}^{2} J_{2} \sin A \cos ^{2} A \sin ^{2} i}{a_{s}^{4}\left(1-e^{2}\right)^{4}}+\frac{18 G M_{E} R_{E}^{2} J_{2} B \sin ^{3} A \sin ^{2} i}{a_{s}^{4}\left(1-e^{2}\right)^{4}} \end{array}\right] e+O\left(e^{2}\right) \ldots .,

where: (26) A=nt+esin⁡(nt), A=n t+e \sin (n t), (27) B=n+encos⁡(nt) B=n+e n \cos (n t)

O(e²) represents negligible higher-order terms.

For circular orbits of inclinations i = 0°, 45°, and 90° we obtain, respectively: dLcell dt=0 \frac{\mathrm{d} L_{\text {cell }}}{\mathrm{d} t}=0 (28) dLcell dt=−6πac3rBH3ℓ0Q0sin⁡θ(GMERE2J2Bsin⁡2Aas4) \frac{\mathrm{d} L_{\text {cell }}}{\mathrm{d} t}=-6 \pi a c^{3} r_{B H}^{3} \ell_{0} Q_{0} \sin \theta\left(\frac{G M_{E} R_{E}^{2} J_{2} B \sin 2 A}{a_{s}^{4}}\right) (29) dLcell dt=−36πac3rBH3ℓ0Q0sin⁡θ(GMERE2J2Bsin⁡2Aas4) \frac{\mathrm{d} L_{\text {cell }}}{\mathrm{d} t}=-36 \pi a c^{3} r_{B H}^{3} \ell_{0} Q_{0} \sin \theta\left(\frac{G M_{E} R_{E}^{2} J_{2} B \sin 2 A}{a_{s}^{4}}\right)

and therefore for elliptical orbits of the same inclinations i = 0°, 45°, and 90° as above, we obtain: (30) dLcell dt=4πa3ℓ0c3rBH3Q0sin⁡θ[2GMEBsin⁡Aas2(1−e2)2−6GMERE2J2sin⁡Aas4(1−e2)4]e+O(e2), \frac{\mathrm{d} L_{\text {cell }}}{\mathrm{d} t}=\frac{4 \pi a}{3} \ell_{0} c^{3} r_{B H}^{3} Q_{0} \sin \theta\left[\frac{2 G M_{E} B \sin A}{a_{s}^{2}\left(1-e^{2}\right)^{2}}-\frac{6 G M_{E} R_{E}^{2} J_{2} \sin A}{a_{s}^{4}\left(1-e^{2}\right)^{4}}\right] e+O\left(e^{2}\right), (31) dLcelldt=4πa3ℓ0c3rBH3Q0sin⁡θ⌊−2GMEBsin⁡Aas2(1−e2)2−9GMERE2J2Bsin⁡2A4as4(1−e2)4−6GMERE2J2Bsin⁡Aas4(1−e2)4−9GMERE2J2Bsin⁡2Acos⁡Aas4(1−e2)4+9GMERE2J2Bsin3⁡Aas4(1−e2)4⌋e+O(e2), \frac{\mathrm{d} L_{c e l l}}{\mathrm{~d} t}=\frac{4 \pi a}{3} \ell_{0} c^{3} r_{B H}^{3} Q_{0} \sin \theta\left\lfloor\begin{array}{l} -\frac{2 G M_{E} B \sin A}{a_{s}^{2}\left(1-e^{2}\right)^{2}}-\frac{9 G M_{E} R_{E}^{2} J_{2} B \sin 2 A}{4 a_{s}^{4}\left(1-e^{2}\right)^{4}}-\frac{6 G M_{E} R_{E}^{2} J_{2} B \sin A}{a_{s}^{4}\left(1-e^{2}\right)^{4}} \\ -\frac{9 G M_{E} R_{E}^{2} J_{2} B \sin 2 A \cos A}{a_{s}^{4}\left(1-e^{2}\right)^{4}}+\frac{9 G M_{E} R_{E}^{2} J_{2} B \sin ^{3} A}{a_{s}^{4}\left(1-e^{2}\right)^{4}} \end{array}\right\rfloor e+O\left(e^{2}\right), (32) dLcelldt=4πa3ℓ0c3rBH3Q0sin⁡θ⌊−2GMEBsin⁡Aas2(1−e2)2−9GMERE2J2Bsin⁡2A2as4(1−e2)4−6GMERE2J2Bsin⁡Aas4(1−e2)4−18GMERE2J2Bsin⁡2Acos⁡Aas4(1−e2)4+18GMERE2J2Bsin3⁡Aas4(1−e2)4⌋e+O(e2). \frac{\mathrm{d} L_{c e l l}}{\mathrm{~d} t}=\frac{4 \pi a}{3} \ell_{0} c^{3} r_{B H}^{3} Q_{0} \sin \theta\left\lfloor\begin{array}{l} -\frac{2 G M_{E} B \sin A}{a_{s}^{2}\left(1-e^{2}\right)^{2}}-\frac{9 G M_{E} R_{E}^{2} J_{2} B \sin 2 A}{2 a_{s}^{4}\left(1-e^{2}\right)^{4}}-\frac{6 G M_{E} R_{E}^{2} J_{2} B \sin A}{a_{s}^{4}\left(1-e^{2}\right)^{4}} \\ -\frac{18 G M_{E} R_{E}^{2} J_{2} B \sin 2 A \cos A}{a_{s}^{4}\left(1-e^{2}\right)^{4}}+\frac{18 G M_{E} R_{E}^{2} J_{2} B \sin ^{3} A}{a_{s}^{4}\left(1-e^{2}\right)^{4}} \end{array}\right\rfloor e+O\left(e^{2}\right) .

Equations (28) to (32) demonstrate that the time rate of change of the torque exerted on the cell around the orbit of a spacecraft is a periodic function of orbital time t.

For the numerical evaluation of our formulas regarding cell torques in experiments that take place on the surface of the Earth corrected for the J₂ harmonic coefficient, the rotation of the Earth, at its angular velocity ω_E following Nace (1983) we consider the following biological objects: First, a sarcoma cell with length ℓ₀ = 6.4 μm in peritoneal fluid for which ρ_m = 1.012 g/cm³, with glycogen granules ρ_H = 1.510 g/cm³ as m_H and oil vacuoles ρ_L = 0.918 g/cm³ as m_L. Similarly, the masses m_H and m_L are separated by 3/4 length of the cell ℓ₀ and a = ¾. The two masses are of equal volumes and fill the buoyant volume b = c = d = 1, are spherical with radii 1/8 of the length of the cell r_BH = ℓ₀/8 (Nace, 1983), and we take θ = 90° at the instant of observation. Furthermore, we consider a human egg of diameter 89 μm, for which ρ_m 1.007 g/cm³, ρ_H = 1.100 g/cm³ as m_H and ρ_L = 0.950 g/cm³ (Nace, 1983), assuming a = 0.75, b = c = d = 1 (Nace, 1983), and finally we consider the oocyte, yolk of the chicken, Gallus gallus egg, which has a polar diameter ℓ₀ = 31 mm with the blastodisc and nucleus of Pander occupying a volume with a radius r_L = 3 mm and a ρ_L = 1.027 g/cm³, respectively. The remainder of the yolk has a radius r_L = 12.5 mm and ρ_H = 1.032 g/cm³. This yolk floats in an albumin-containing medium with ρ_m = 1.040 g/cm³; furthermore, the constants a, b, c, d are 0.5, 1.0, 0.24, 1.0 (Nace, 1983), respectively. For the mass/radius of the Earth we use M_E, = 5.94×10²⁴ kg, and R_E = 6378.1363 km, and the angular velocity of rotation is ω_E = 7.292115 × 10^–5 rad/s (Vallado and McClain, 2007). The oblateness coefficient is J₂ = -0.0010827 (Kaula, 2000), and the period of an orbital revolution of the spacecraft at 300 km is T_rev = 5463.282 s, and its mean motion n = 0.001149492 rad/s. Our results for the calculated torque on the surface of the Earth and in orbit around the Earth are given in tables 1, 2, 3, and 4.

Next, considering the Earth as an ellipsoid we derive formulas for the calculation of the torque exerted on a human egg on the surface of the Earth as a function of the geocentric latitude φ_E. For that, using the parameters above, we obtain: (33) Lnetegg=sin⁡θ∣+−3.941×10−11+3.954×10−11sin2⁡ϕE−1.321×10−13sin4⁡ϕE(1−0.00352sin2⁡ϕE)2−1.825×10−11(1−0.00352sin⁡ϕE)4+5.476×10−11sin2⁡ϕE(1−0.00352sin⁡ϕE)4⌋, \left.L_{n e t_{e g g}}=\sin \theta \mid+\frac{-3.941 \times 10^{-11}+3.954 \times 10^{-11} \sin ^{2} \phi_{E}-1.321 \times 10^{-13} \sin ^{4} \phi_{E}}{\left(1-0.00352 \sin ^{2} \phi_{E}\right)^{2}}-\frac{1.825 \times 10^{-11}}{\left(1-0.00352 \sin \phi_{E}\right)^{4}}+\frac{5.476 \times 10^{-11} \sin ^{2} \phi_{E}}{\left(1-0.00352 \sin \phi_{E}\right)^{4}}\right\rfloor,

Similarly, extending this on the Martian surface ellipsoid we similarly derive: (34) Lnetggg=sin⁡θ[−1.982×10−11+1993×10−11sin2⁡ϕM−1.167×10−13sin4⁡ϕM+1.270×10−11(1−0.00588sin⁡ϕM)4−4.310×10−9(1−0.00588sin2⁡ϕM)2−3.810×10−11sin2⁡ϕM(1−0.00588sin⁡ϕM)4]sin⁡θ L_{n e t_{g g g}}=\sin \theta\left[\begin{array}{l} -1.982 \times 10^{-11}+1993 \times 10^{-11} \sin ^{2} \phi_{M}-1.167 \times 10^{-13} \sin ^{4} \phi_{M} \\ +\frac{1.270 \times 10^{-11}}{\left(1-0.00588 \sin \phi_{M}\right)^{4}}-\frac{4.310 \times 10^{-9}}{\left(1-0.00588 \sin ^{2} \phi_{M}\right)^{2}}-\frac{3.810 \times 10^{-11} \sin ^{2} \phi_{M}}{\left(1-0.00588 \sin \phi_{M}\right)^{4}} \end{array} \mid \sin \theta\right.

where φ_M is the areocentric latitude, defined north and south of the Martian equator, from 0° to 90° to 0° to -90°. In Figure 3 below we plot the effect of the geocentric latitude φ_E effect on the torque exerted on a human egg, for various values of the angle = θ90,°88,°86,°84,°82,°80 °. Table 1 tabulates the results for the net torque for an experiment that takes place on the surface of the Earth.

Figure 3.

Higher θ angles result in higher torques. Similarly, Figure 4 demonstrates torques exerted on a human egg in an experiment taking place on the surface of Mars as a function of areocentric latitude ϕ_M resulting in behavior similar to but less by approximately an order of magnitude when compared to the torque applied in an Earth experiment. Higher torques are expected if the experiment takes place in a relatively low orbit scenario (i.e., around Earth or Mars). In very deep space and far away from the Earth g does not become identically zero and therefore there will always be torque acting on the cells. Finally, the imparted torque is highly sensitive to the size of the cell to the fourth power, but also to the distribution of its subcellular particulates. Figure 5 demonstrates the torque exerted on cells of increasing size in the scenario having the same properties as those cells given by Nace (1983) on the surface of the Earth, as a function geocentric latitude φ_E, and cell’s length ℓ₀, assuming θ = 90. We see that the torque increases simultaneously as the cell’s length ℓ₀ and geocentric (planetocentric) latitude φ_E increase.

Figure 4.

Figure 5.

In Nace’s paper the author obtains results for a sarcoma cell experiment on the surface of the Earth in which g is taken as 980.0 cm/s², a near-average value that was considered suitable for the purpose. We obtain results using our modified Nace expression for the torque, corrected for the J₂ harmonic and also the rotation of the Earth. We find that this introduces a smaller torque with a % difference in the range 2.24%≤|ΔL⟨L⟩|sarsur≤1.40%2.24 \% \leq\left|\frac{\Delta L}{\langle L\rangle}\right|_{s a r}^{s u r} \leq 1.40 \% if the corresponding geocentric latitude range is in the range 0° ≤ ϕ_E ≤ 90°. At this point we remind the reader about the peculiarity of our first inequality, sequenced to match the second, since lower latitudes result in higher percentage differences and higher latitudes result in lower percentage differences when comparing to Nace’s result. Our pole calculated value results in a 1.40% difference less than the result given by Nace for the same cell that has length equal of 6.4 μm. At the same time our calculations demonstrate that higher torque is exerted on the cell at the poles due to the higher value of g. In the case of a human egg and for the same geocentric latitude range, we find a % difference in the range of 29%≤|ΔL⟨L⟩|eggsur≤28%29 \% \leq\left|\frac{\Delta L}{\langle L\rangle}\right|_{\operatorname{egg}}^{s u r} \leq 28 \%. This results from the fact that in our calculations we have used r_BH = ℓ₀/5, where Nace seems to have used r_BH = ℓ₀/4.54. It appears that the torque results in Nace’s original formula and also in the one derived in this paper greatly depend on the choice of parameter r_BH, on the surface of the Earth and also in space. Our polar torque experiment value exhibits a 28% difference less than the one given by Nace, but still consistent with a higher torque value, for higher polar g scenario. Finally, for the same geocentric latitude range the Gallus gallus egg cell results in percentage difference in the range 3.8%≤|ΔL⟨L⟩|Galsur ≤2.9%3.8 \% \leq\left|\frac{\Delta L}{\langle L\rangle}\right|_{G a l}^{\text {sur }} \leq 2.9 \%. Next, and for the same type of cells aboard an orbiting spacecraft in circular (e = 0) and elliptical orbits (e = 0.2) and inclinations i = 0°, 45°, and 90°, we obtain the following % differences between surface and orbital values: Nace’s “sarcoma” cell 11%≤|ΔL⟨L⟩|sare=0≤10%11 \% \leq\left|\frac{\Delta L}{\langle L\rangle}\right|_{s a r}^{e=0} \leq 10 \%, and also for a 50 μm sarcoma cell, we also obtain 8.8%≤|ΔL⟨L⟩|sare=0≤9.2%8.8 \% \leq\left|\frac{\Delta L}{\langle L\rangle}\right|_{s a r}^{e=0} \leq 9.2 \%, assuming that all the parameters remain the same. Similarly, for Nace’s “sarcoma” cell in an elliptical orbit, we obtain 2.9%≤|ΔL⟨L⟩|sar e=0.2≤2.4%2.9 \% \leq\left|\frac{\Delta L}{\langle L\rangle}\right|_{\text {sar }}^{e=0.2} \leq 2.4 \%, and also for the 50 μm sarcoma cell 0.6%≤|ΔL⟨L⟩|sar e=0.2≤1.0%0.6 \quad \% \leq\left|\frac{\Delta L}{\langle L\rangle}\right|_{\text {sar }}^{e=0.2} \leq 1.0 \%, respectively. Next, for the human egg in the same orbital scenarios we obtain 38%≤|ΔL⟨L⟩|logge=0≤37%38 \% \leq\left|\frac{\Delta L}{\langle L\rangle}\right|_{\operatorname{logg}}^{e=0} \leq 37 \%, and 30%≤|ΔL⟨L⟩|egg e=0.2≤29%30 \% \leq\left|\frac{\Delta L}{\langle L\rangle}\right|_{\text {egg }}^{e=0.2} \leq 29 \%, respectively. Finally, for the Gallus gallus egg we obtain 12.5%≤|ΔL⟨L⟩|egge=0≤12.1%12.5 \% \leq\left|\frac{\Delta L}{\langle L\rangle}\right|_{e g g}^{e=0} \leq 12.1 \%, and 6%≤|ΔL⟨L⟩|egge=0.2≤5.6%6 \% \leq\left|\frac{\Delta L}{\langle L\rangle}\right|_{e g g}^{e=0.2} \leq 5.6 \%, respectively. We find that elliptical orbits and highly elliptical orbits result to a less % difference, when compared with Nace’s results. This is due to the fact that elliptical orbits exhibit higher values of orbital acceleration g. It is possible by adjusting the eccentricity of the orbit, to achieve a torque effect similar to the torque exerted on the cell under a constant gravitational acceleration on the surface of the Earth as given by Nace (1983). For example, an elliptical polar orbit i = 90° at the orbital altitude of 300 km would need to achieve eccentricities e = 0.211 (elliptical), and e = 1.398 (hyperbolic) in order that the torque felt by a Gallus gallus egg in the orbital experiment around the Earth at the given eccentricities would result in a torque exerted on the cell that is equal to that exerted at the surface of the Earth. Figure 5 demonstrates the torque exerted on a human egg in the scenario given by Nace (1983) for an experiment that takes place in elliptical polar orbit (eccentricity e = 0.01) at the orbital altitude h = 300 km, as a function of semimajor axis a, and the orbital true anomaly f, assuming θ = 90°. Similarly, Figure 7 demonstrates of the torque exerted on a human egg in the scenario given by Nace (1983) for an experiment that is taking place in a polar orbiting spacecraft at h = 300 km and eccentricity e = 0.1 as a function of the angle θ and the orbital true anomaly f. Similarly, figures 8 and 9 plot the torque exerted on a human egg in an experimental scenario taking place aboard a polar orbiting spacecraft at h = 300 km and eccentricity e = 0.4, as a function of the angle θ and the orbital true anomaly f. Finally, Figure 10 demonstrates the variation of g at h = 300 km along the orbit of the spacecraft for the time of one orbital period. The exerted torque exhibits a periodic effect in the true anomaly, with a minimum around f = 180° and reduces as the semimajor axis of the spacecraft increases. Finally, for the Gallus gallus egg we find that 0.818 dynes cm ≤ L_Gal ≤ 0.825 dynes cm. On the other hand, as another example we quote Berg (2003) and comparing we say that the torque that a human egg feels on the surface of the Earth at geocentric latitude ϕ_E = 45° in Nace’s scenario, when compared to the torque produced by a flagellar motor of an E. coli bacterium, is 245 to 414 times smaller or L_egg = (245 – 414)L_E.Coli. Similarly, for a sarcoma cell we have that L_sar ≈ (20 – 34)L_E.Coli, where both torques have been expressed in pN nm. Finally, for a Gallus gallus egg we obtain that L_Gal = (0.303-1.80) × 10¹⁰ L_E.coli, respectively. Equations (35), (36), (37) give the torques exerted on a human egg, a Gallus gallus egg, and a sarcoma cell on the surface of a planetary body in the solar system, as a function of the body’s planetary mass M_p, oblateness coefficient J₂, planetocentric latitude φ_p, and also its planetary flattening coefficient, and finally the angle θ to be:

Figure 6.

Figure 7.

Figure 8.

Figure 9.

Figure 10.

(35) Legg=sin⁡θ[7.750×10−19MpReq2(1−fpsin2⁡ϕp)2−1.162×10−18MpJ2(3sin2⁡ϕp−1)Req2(1−fpsin⁡ϕp)4−1.162×10−11Req2ωp2cos2⁡ϕp(1−fpsin2⁡ϕp)], L_{e g g}=\sin \theta\left[\frac{7.750 \times 10^{-19} M_{p}}{R_{e q}^{2}\left(1-f_{p} \sin ^{2} \phi_{p}\right)^{2}}-1.162 \times 10^{-18} \frac{M_{p} J_{2}\left(3 \sin ^{2} \phi_{p}-1\right)}{R_{e q}^{2}\left(1-f_{p} \sin \phi_{p}\right)^{4}}-1.162 \times 10^{-11} R_{e q}^{2} \omega_{p}^{2} \cos ^{2} \phi_{p}\left(1-f_{p} \sin ^{2} \phi_{p}\right)\right] \text {, } (36) LGal−gal=sin⁡θ[5.882×10−11MpReq2(1−fpsin2⁡ϕp)2−8.823×10−11MpJ2(3sin2⁡ϕp−1)Req2(1−fpsin⁡ϕp)4−8.820×10−4Req2ωp2cos2⁡ϕp(1−fpsin2⁡ϕp)] L_{G a l-g a l}=\sin \theta\left[\frac{5.882 \times 10^{-11} M_{p}}{R_{e q}^{2}\left(1-f_{p} \sin ^{2} \phi_{p}\right)^{2}}-\frac{8.823 \times 10^{-11} M_{p} J_{2}\left(3 \sin ^{2} \phi_{p}-1\right)}{R_{e q}^{2}\left(1-f_{p} \sin \phi_{p}\right)^{4}}-8.820 \times 10^{-4} R_{e q}^{2} \omega_{p}^{2} \cos ^{2} \phi_{p}\left(1-f_{p} \sin ^{2} \phi_{p}\right)\right] (37) Lsar=sin⁡θ[6.312×10−20MpReq2(1−fpsin⁡ϕp)2−9.467×10−20MpJ2(3sin2⁡ϕp−1)Req2(1−fpsin⁡ϕp)4−9.463×10−13Req2ωp2cos2⁡ϕp(1−fpsin2⁡ϕp)]. L_{s a r}=\sin \theta\left[\frac{6.312 \times 10^{-20} M_{p}}{R_{e q}^{2}\left(1-f_{p} \sin \phi_{p}\right)^{2}}-\frac{9.467 \times 10^{-20} M_{p} J_{2}\left(3 \sin ^{2} \phi_{p}-1\right)}{R_{e q}^{2}\left(1-f_{p} \sin \phi_{p}\right)^{4}}-9.463 \times 10^{-13} R_{e q}^{2} \omega_{p}^{2} \cos ^{2} \phi_{p}\left(1-f_{p} \sin ^{2} \phi_{p}\right)\right] .

Next, we consider the same experiment as above in a spacecraft in circular/elliptical orbits around Earth with inclinations i = 0°, 45°, 90°, at the altitude h = 300 km. Results are tabulated in tables 2-5:

In this paper we have studied the effect of orbit parameters on the torque exerted by gravity on cells and eggs. In particular we have calculated the torque effects, in the human egg, sarcoma, and Gallus gallus egg. We have extended Nace’s model on the surface of the Earth by considering the J₂ harmonic, the rotation of the Earth, and its flattening. This is for an experiment that takes place on the surface of the Earth at various geocentric latitudes. The torque acting in cells is proportionally related to the acceleration of gravity g, and the fourth power of the cell’s length. Furthermore, we have also extended Nace’s result for an experiment that takes place in orbit around a planetary body. We have found that in orbit around Earth, the effect of torque is less when compared to that of the surface, because gravity is less but not zero at the orbital altitude of the spacecraft, and that the effect of the J₂ harmonic can be calculated for possible future experiments. We have derived expressions for the applied cell torques by Earth and Mars gravity, taking their flattening into account. We have also found that elliptical polar orbits result in higher torques, and that high eccentricities increase the torque drastically. Depending on the size of the cell, percentage differences due to the correction of J₂ harmonic and rotation of the Earth that are less than 1% can be ignored, where higher effects should be included.