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

Gravitational and Space Research
Volume 1 (2013): Issue 1 (July 2013)
By:
Open Access
|Jul 2013

Figures & Tables

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
0Sarcoma cell9.107×10-100.09107 fJ
30 9.126×10-100.09126 fJ
45509.145×10-100.09145 fJ
60 9.164×10-100.09164 fJ
90 9.183×10-100.09183 fJ
Human EggNace’s result 1.5x10-8dyne cm
0 1.118×10-81.200 fJ
30891.120×10-81.120 fJ
45 1.123×10-81.123 fJ
60 1.125×10-81.125 fJ
90 1.128×10-81.128 fJ
Gallus gallus eggNace’s result 0.85 dyne cm
0 0.81881.860 nJ
30310000.82082.031 nJ
45 0.82282.204nJ
60 0.82482.380 nJ
90 0.82582.550 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 = 0.2Cell length ℓ0 [μm]Torque L [dyne cm]Equivalent Energy
i = 0°Sarcoma cell9.045×10-100.0904 fJ
i = 45° 9.067×10-100.0906 fJ
i = 90°509.089×10-100.0908 fJ
e = 0.2
i = 0°Human Egg1.110×10-81.110 fJ
i = 45° 1.113 × 10-81.113 fJ
i = 90°891.116× 10-81.116 fJ
e = 0.2
i = 0°Gallus gallus egg0.80080.00 nJ
i = 45° 0.80380.30 nJ
i = 90°310000.80480.40 nJ

UT1

asorbital semimajor axis.
eeccentricity of the orbit.
uargument of latitude.
ftrue anomaly.
Gconstant of universal gravitation.
iorbital inclination.
r′radial orbital distance of the spacecraft from the center of the Earth.
J2oblateness coefficient of the Earth.
Mthe mass of the Earth.
Mpmass of any planet.
VEtotal gravitational potential of the Earth.
xω, yω, zωdefine a right handed coordinate system.
Fapplied force.
rdistance of the center of mass that the force is applied to the axis of rotation.
ℓdistance between the centers of the heavy and light masses in the cell.
ℓ0total cell length.
rH = brBHwhere, where rH is the radius of the heavy mass.
rBHthe radius of the cell.
ρmdensity of the medium.
ρHdensity of the heavy mass.
ρLdensity of light mass.
a, b, c, dconstants in the range [0,1].
gtotcorrected gravitational acceleration.
Ltorque.
REradius of the Earth.
Reqequatorial radius of the Earth.
Rpolpolar radius of the Earth.
f′flattening of the Earth.
fMflattening of Mars.
fpplanetary flattening.
Morbital mean anomaly.
nspacecraft mean angular velocity.
φEgeocentric latitude.
ωEangular velocity of the Earth.
θEcolatitude.
ωargument of the perigee.
Ωargument of the ascending node.
λgeocentric longitude.
θis the angle between F and r.
L = M + Ω + ωmean 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 = 0.01Cell length ℓ0 [μm]Torque L [dyne cm]Equivalent Energy to torque
i = 0°Sarcoma cell8.340×10-100.0834 fJ
i = 45° 8.360×10-100.0836 fJ
i = 90°508.376×10-100.8376 fJ
e = 0.01
i = 0°Human Egg1.024×10-81.024 fJ
i = 45° 1.026×10-81.026 fJ
i = 90°891.028×10-81.028 fJ
e = 0.01
i = 0°Gallus gallus egg0.75075.000 nJ
i = 45° 0.75175.100 fJ
i = 90°310000.75275.200 fJ

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

Orbital Eccentricitye = 0Cell length ℓ0 [μm]Torque L [dyne cm]Equivalent Energy to torque
i = 0°Sarcoma cell8.340×10-100.0834 fJ
i = 45° 8.356×10-100.0835 fJ
i = 90°508.375×10-100.0837 fJ
e = 0
i = 0°Human Egg1.024×10-81.024 fJ
i = 45° 1.026×10-81.026 fJ
i = 90°891.028×10-81.028 fJ
e = 0
i = 0°Gallus gallus egg0.75075.000 nJ
i = 45° 0.75175.100 nJ
i = 90°310000.75375.300 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 = 0.4Cell length ℓ0 [μm]Torque L [dyne cm]Equivalent Energy
i = 0°Sarcoma cell1.1809×10-90.1180 fJ
i = 45° 1.1846×10-90.1184 fJ
i = 90°501.1883 × 10-90.1188 fJ
e = 0.2
i = 0°Human Egg1.4499×10-81.4499 fJ
i = 45° 1.4546×10-81.4546 fJ
i = 90°891.4591×10-81.4591 fJ
e = 0.2
i = 0°Gallus gallus egg1.0615106.150 nJ
i = 45° 1.0648106.480 nJ
i = 90°310001.0681106.810 nJ
DOI: https://doi.org/10.2478/gsr-2013-0006 | Journal eISSN: 2332-7774
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Language: English
Page range: 59 - 78
Published on: Jul 1, 2013
Published by: American Society for Gravitational and Space Research
In partnership with: Paradigm Publishing Services
Publication frequency: 2 issues per year
Keywords:
Torque in Cells,
Orbital Experiments,
Oblateness,
Spherical Harmonics
Related subjects:
Life sciences,
Life sciences, other,
Materials sciences,
Materials sciences, other,
Physics,
Physics, other

© 2013 Ioannis Haranas, Ioannis Gkigkitzis, George D. Zouganelis, published by American Society for Gravitational and Space Research
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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