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

Ioannis Haranas
Ioannis Haranas
, 
Ioannis Gkigkitzis
Ioannis Gkigkitzis
 and 
George D. Zouganelis
George D. Zouganelis
   | Jul 01, 2013
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Article Category: Research Article
Published Online: Jul 01, 2013
Page range: 59 - 78
DOI: https://doi.org/10.2478/gsr-2013-0006
© 2013 Ioannis Haranas et al., published by Sciendo
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Figure 1.

Explanation of the orbital elements: inclination i, argument of latitude u = ω + f, and the radial vector r′ of the spacecraft, and λ = 0 is the zero longitude point on the Earth’s equator, and xω, yω, zω define a right handed coordinate system.
Explanation of the orbital elements: inclination i, argument of latitude u = ω + f, and the radial vector r′ of the spacecraft, and λ = 0 is the zero longitude point on the Earth’s equator, and xω, yω, zω define a right handed coordinate system.

Figure 2.

Nace’s floating cell used in this study to illustrate examples of torque. H and L are the heavy and light masses of the cell. r is the radius of the heavy mass and ℓ0 is the total length of the cell, and ℓ is the distance between the light and the heavy masses; dH and dL are the distance from these centers respectively, to the center of mass c.m. of the cell. VH and VL are the volumes of the heavy and light masses, and VBH and VBL are the volumes of the buoyant spheres surrounding VH and VL. FH and FL are the force of gravity, and FBH and FBL are the buoyant forces respectively, acting on H and L. g shows the direction of gravity, and θ is the direction between F and ℓ.
Nace’s floating cell used in this study to illustrate examples of torque. H and L are the heavy and light masses of the cell. r is the radius of the heavy mass and ℓ0 is the total length of the cell, and ℓ is the distance between the light and the heavy masses; dH and dL are the distance from these centers respectively, to the center of mass c.m. of the cell. VH and VL are the volumes of the heavy and light masses, and VBH and VBL are the volumes of the buoyant spheres surrounding VH and VL. FH and FL are the force of gravity, and FBH and FBL are the buoyant forces respectively, acting on H and L. g shows the direction of gravity, and θ is the direction between F and ℓ.

Figure 3.

Torque exerted on a human egg in an experiment taking place on the surface of the Earth as a function of geocentric latitude ϕE and for various angles θ = 90° (blue), 88° (red), 86° (yellow brown), 84° (light blue), 82° (purple), 80° (light green) between the force and distance d.
Torque exerted on a human egg in an experiment taking place on the surface of the Earth as a function of geocentric latitude ϕE and for various angles θ = 90° (blue), 88° (red), 86° (yellow brown), 84° (light blue), 82° (purple), 80° (light green) between the force and distance d.

Figure 4.

Torque exerted on a human egg in an experiment taking place on the surface of Mars as a function of areocentric latitude ϕE and for various angle θ = 90° (blue), 88° (red), 86° (yellow brown), 84° (light blue), 82° (purple), 80° (light green) between the force and distance d.
Torque exerted on a human egg in an experiment taking place on the surface of Mars as a function of areocentric latitude ϕE and for various angle θ = 90° (blue), 88° (red), 86° (yellow brown), 84° (light blue), 82° (purple), 80° (light green) between the force and distance d.

Figure 5.

Plot of the torque exerted on cells of increasing length having the same properties as those given by Nace (1983) on the surface of the Earth as a function of geocentric latitude φE and the length of the cell ℓ0, assuming θ = 90 °.
Plot of the torque exerted on cells of increasing length having the same properties as those given by Nace (1983) on the surface of the Earth as a function of geocentric latitude φE and the length of the cell ℓ0, assuming θ = 90 °.

Figure 6.

Plot of the torque exerted on a human egg in the scenario given by Nace (1983), for an experiment that is taking place aboard a spacecraft in an elliptical polar orbit around the Earth, with e = 0.01 and at the orbital altitude h = 300 km, as a function of orbital semimajor axis a and the orbital true anomaly f, assuming θ = 90 °.
Plot of the torque exerted on a human egg in the scenario given by Nace (1983), for an experiment that is taking place aboard a spacecraft in an elliptical polar orbit around the Earth, with e = 0.01 and at the orbital altitude h = 300 km, as a function of orbital semimajor axis a and the orbital true anomaly f, assuming θ = 90 °.

Figure 7.

Plot 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.
Plot 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.

Figure 8.

Plot 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 = 0.4 as a function of the angle θ and the orbital true anomaly f.
Plot 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 = 0.4 as a function of the angle θ and the orbital true anomaly f.

Figure 9.

Plot of the torque exerted on a human egg in the scenario given by Nace (1983) for an experiment that is taking place in a spacecraft in polar circular orbit at h = 300 km as a function of the angle θ and the orbital true anomaly f.
Plot of the torque exerted on a human egg in the scenario given by Nace (1983) for an experiment that is taking place in a spacecraft in polar circular orbit at h = 300 km as a function of the angle θ and the orbital true anomaly f.

Figure 10.

Plot of the variation of gravitational orbital acceleration as a function of orbital time t of a full orbit at the orbital altitude of 300 km and for various eccentricities.
Plot of the variation of gravitational orbital acceleration as a function of orbital time t of a full orbit at the orbital altitude of 300 km and for various eccentricities.

Torque geocentric latitude effect exerted on various test objects.

Geocentric Latitudeϕ [°] Cell lengthℓ0 [μm] Torque L [dyne cm] Equivalent to torque energy
0 Sarcoma cell 9.107×10-10 0.09107 fJ
30 9.126×10-10 0.09126 fJ
45 50 9.145×10-10 0.09145 fJ
60 9.164×10-10 0.09164 fJ
90 9.183×10-10 0.09183 fJ
Human Egg Nace’s result 1.5x10-8 dyne cm
0 1.118×10-8 1.200 fJ
30 89 1.120×10-8 1.120 fJ
45 1.123×10-8 1.123 fJ
60 1.125×10-8 1.125 fJ
90 1.128×10-8 1.128 fJ
Gallus gallus egg Nace’s result 0.85 dyne cm
0 0.818 81.860 nJ
30 31000 0.820 82.031 nJ
45 0.822 82.204nJ
60 0.824 82.380 nJ
90 0.825 82.550 nJ

Torque effect exerted on various test objects in an experiment taking place in a spacecraft in an elliptical orbit (e = 0.2) around Earth.

Orbital Eccentricitye = 0.2 Cell length ℓ0 [μm] Torque L [dyne cm] Equivalent Energy
i = 0° Sarcoma cell 9.045×10-10 0.0904 fJ
i = 45° 9.067×10-10 0.0906 fJ
i = 90° 50 9.089×10-10 0.0908 fJ
e = 0.2
i = 0° Human Egg 1.110×10-8 1.110 fJ
i = 45° 1.113 × 10-8 1.113 fJ
i = 90° 89 1.116× 10-8 1.116 fJ
e = 0.2
i = 0° Gallus gallus egg 0.800 80.00 nJ
i = 45° 0.803 80.30 nJ
i = 90° 31000 0.804 80.40 nJ

UT1

as orbital semimajor axis.
e eccentricity of the orbit.
u argument of latitude.
f true anomaly.
G constant of universal gravitation.
i orbital inclination.
r′ radial orbital distance of the spacecraft from the center of the Earth.
J2 oblateness coefficient of the Earth.
M the mass of the Earth.
Mp mass of any planet.
VE total gravitational potential of the Earth.
xω, yω, zω define a right handed coordinate system.
F applied force.
r distance of the center of mass that the force is applied to the axis of rotation.
ℓ distance between the centers of the heavy and light masses in the cell.
ℓ0 total cell length.
rH = brBH where, where rH is the radius of the heavy mass.
rBH the radius of the cell.
ρm density of the medium.
ρH density of the heavy mass.
ρL density of light mass.
a, b, c, d constants in the range [0,1].
gtot corrected gravitational acceleration.
L torque.
RE radius of the Earth.
Req equatorial radius of the Earth.
Rpol polar radius of the Earth.
f′ flattening of the Earth.
fM flattening of Mars.
fp planetary flattening.
M orbital mean anomaly.
n spacecraft mean angular velocity.
φE geocentric latitude.
ωE angular velocity of the Earth.
θE colatitude.
ω argument of the perigee.
Ω argument of the ascending node.
λ geocentric longitude.
θ is the angle between F and r.
L = M + Ω + ω mean longitude.

Torque effect exerted on various test objects in an experiment taking place in a spacecraft in a slightly elliptical orbit (e = 0.01) around Earth.

Orbital Eccentricity e = 0.01 Cell length ℓ0 [μm] Torque L [dyne cm] Equivalent Energy to torque
i = 0° Sarcoma cell 8.340×10-10 0.0834 fJ
i = 45° 8.360×10-10 0.0836 fJ
i = 90° 50 8.376×10-10 0.8376 fJ
e = 0.01
i = 0° Human Egg 1.024×10-8 1.024 fJ
i = 45° 1.026×10-8 1.026 fJ
i = 90° 89 1.028×10-8 1.028 fJ
e = 0.01
i = 0° Gallus gallus egg 0.750 75.000 nJ
i = 45° 0.751 75.100 fJ
i = 90° 31000 0.752 75.200 fJ

Torque effect exerted on various test objects in an experiment taking place in a spacecraft in circular orbit around Earth.

Orbital Eccentricitye = 0 Cell length ℓ0 [μm] Torque L [dyne cm] Equivalent Energy to torque
i = 0° Sarcoma cell 8.340×10-10 0.0834 fJ
i = 45° 8.356×10-10 0.0835 fJ
i = 90° 50 8.375×10-10 0.0837 fJ
e = 0
i = 0° Human Egg 1.024×10-8 1.024 fJ
i = 45° 1.026×10-8 1.026 fJ
i = 90° 89 1.028×10-8 1.028 fJ
e = 0
i = 0° Gallus gallus egg 0.750 75.000 nJ
i = 45° 0.751 75.100 nJ
i = 90° 31000 0.753 75.300 nJ

Torque effect exerted on various test objects in an experiment taking place in a spacecraft in an elliptical orbit (e = 0.4) around Earth.

Orbital Eccentricitye = 0.4 Cell length ℓ0 [μm] Torque L [dyne cm] Equivalent Energy
i = 0° Sarcoma cell 1.1809×10-9 0.1180 fJ
i = 45° 1.1846×10-9 0.1184 fJ
i = 90° 50 1.1883 × 10-9 0.1188 fJ
e = 0.2
i = 0° Human Egg 1.4499×10-8 1.4499 fJ
i = 45° 1.4546×10-8 1.4546 fJ
i = 90° 89 1.4591×10-8 1.4591 fJ
e = 0.2
i = 0° Gallus gallus egg 1.0615 106.150 nJ
i = 45° 1.0648 106.480 nJ
i = 90° 31000 1.0681 106.810 nJ
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