Is Weightlessness Without Gravity?

Abstract

A body upon which no force other than the force of gravity (or weightiness), is acting is in free-fall state, i.e. that it is falling freely in an external inertial reference frame, while it is weightless, i.e. its weight is null in a non-inertial reference frame attached to the body. This simple concept, sometime difficult to grasp, is introduced gradually in this chapter by introducing firstly, the notions of free fall and weightlessness, secondly, the various free fall trajectories and thirdly, the free fall in the universe. Other notions on ballistic movement, escape velocity, and perturbations inside a satellite are addressed in the appendices. Finally, incorrect or approximative ideas on weightlessness and its causes are corrected as it is important to not only say what weightlessness is, but also what it is not.

Appendices

Appendix 7: Ballistic Movement and Escape Velocity

A body thrown upwards with an initial velocity inclined on the horizontal describes a certain trajectory and falls back on Earth. One shows simply that this trajectory is parabolic if:

• the travelled horizontal distance is sufficiently small in front of the curvature radius of Earth’s surface for the gravitational field (more precisely the gravity (weightiness) field) to be approximated by a parallel field, i.e. the direction of the vectorial acceleration g stays parallel to itself in all points of the trajectory; and

• the travelled vertical distance is sufficiently small in front of Earth’s radius to consider that the gravitational field is uniform, i.e. that the magnitude of the acceleration g varies in a negligible way.

After the initial impulsion, if air drag is neglected, the only force that acts on the body is its own weight coming from the phenomenon of gravity (weightiness).

As the ballistic movement is described in a vertical plane that includes the whole of the trajectory, one chooses a local reference frame such that its origin is at the injection point (the point where the initial impulsion stops), the X axis horizontal, the Z axis along the ascending local vertical and the Y axis perpendicular to the vertical plane formed by the X and Z axes.

Projecting on the horizontal X and vertical Z axes the constant initial velocity v₀ making an angle θ with horizontal (see Fig. 10), one obtains a constant horizontal velocity v_x and a vertical velocity v_z function of time t:

$$ v_{x} = v_{0} .{ \cos }\theta = {\text{constant}} $$

(A7.1)

$$ v_{z} = v_{0} .{ \sin }\theta - {\text{g}}.{\text{t}} $$

(A7.2)

Fig. 10

Ballistic parabolic trajectory in a gravitational field considered as uniform

Integrating velocities with respect to time, one obtains the components along the X and Z axes of the body position:

$$ x\left( t \right) = v_{0} .\cos \theta .t $$

(A7.3)

$$ z\left( t \right) = v_{0} .\sin \theta .t - \frac{{g.t^{2} }}{2} $$

(A7.4)

(where integration constants are null because of the choice of the local reference frame). Eliminating time between (A7.3) and (A7.4) yields the equation of the parabola shaped trajectory:

$$ z\left( x \right) = - \frac{g}{{2v_{0}^{2} .{ \cos }^{2} \theta }}.x^{2} + x.tg\theta $$

(A7.5)

The total duration T of the movement on the parabolic trajectory corresponds to the non-zero value of t in the second-degree Eq. (A7.4) in t² after posing z = 0 (i.e. when the parabola intersects again the X axis):

$$ T\,\, = \,\,\frac{{2v_{0} .\sin \theta }}{g} $$

(A7.6)

As the parabolic trajectory is symmetrical, the trajectory maximum height h is reached for ${\text{t}}\,\,{ = }\,\,\frac{\text{T}}{ 2}$ in Eq. (A7.4), yielding

$$ {\text{h}}\,\,{ = }\,\,\frac{{{\text{v}}_{ 0}^{ 2} . {\text{sin}}^{ 2} \theta }}{{ 2 {\text{g}}}} $$

(A7.7)

The distance d travelled horizontally during the parabola is found from Eq. (A7.3) by replacing t by T, yielding

$$ d\,\, = \,\,\frac{{v_{0}^{2} .\sin 2\theta }}{g} $$

(A7.8)

and is maximum for θ = 45°.

These equations define the ballistic movement, i.e. unpropelled. Let us note that the duration, the horizontal distance and the maximum height of this ballistic movement are independent from the body mass.

A football or a rugby ball thrown by hand with an angle of 45° and an initial velocity of 20 m/s travels a horizontal distance of approximately 40 m. The Airbus A310 ‘ZERO-G’ (see Fig. 11) initiating a parabolic trajectory with an initial velocity of 500 km/h inclined by 50° on the horizontal describes a parabola of 21.7 s duration, travels a horizontal distance of 1936 m and climbs approximately 580 m above the altitude of the parabola injection.

Fig. 11

(Credit: Novespace)

Flight profile of the Airbus A310 ZERO-G during a parabolic flight: (1) starting from steady horizontal flight at an altitude of approximately 6100 m, the aircraft climbs for about 20 s with accelerations between 1.8 and 2 g; this phase is called the pull-up; (2) when the climbing angle reaches 50° at an altitude of approximately 7600 m, the pilots strongly reduce all engine thrust for about 20–25 s, keeping enough thrust to balance the air drag; the sum of all forces other than gravity is null, the aircraft is in a parabolic ballistic free fall; (3) when the diving angle reaches 42° below the horizontal, it is time for the pilots to increase again all engine thrust to dive downward accelerating around 1.8 g for approximately 20 s, to come back to a steady horizontal flight; this phase is called the pull-out.

One can also verify that this parabola is actually an arc of ellipse, but the approximation committed is so small (in the order of 10⁻⁴, see Ref. 25) that it is perfectly acceptable.

If one throws upward a body from ground with a vertical initial velocity v₀ (θ = 90°), the maximum height reached is

$$ h_{\hbox{max} } \,\, = \,\,\frac{{v_{0}^{2} }}{2g} $$

(A7.9)

before falling back to Earth. However, beyond a certain initial velocity, the body does not fall back anymore on Earth. The limit initial velocity beyond which the body escape from the planet gravitational attraction is called escape velocity (or first cosmic velocity in the Russian literature).

It can be calculated easily by finding the work required to fight against the gravitational attraction and kinetic and potential energies.

To move against the force of gravity a body initially on Earth’s surface of an elementary vertical distance dr, one must expand an elementary work dW

$$ {\text{d}}W = F_{gr} {\text{d}}r = \frac{{G.M_{E} .m}}{{r^{2} }}{\text{d}}r $$

(A7.10)

that is transformed in potential energy. The maximum potential energy is equivalent to the total work expanded to bring the body from Earth’s surface to an infinite distance for the body to escape terrestrial gravitational attraction (if one neglects the influence of all other masses in the Universe):

$$ E_{pot} = W_{pot} = \mathop \smallint \limits_{{r_{E} }}^{\infty } F_{gr} {\text{d}}r = \mathop \smallint \limits_{{r_{E} }}^{\infty } \frac{{G.M_{E} .m}}{{r^{2} }}{\text{d}}r = \frac{{G.M_{E} .m}}{{r_{E} }} $$

(A7.11)

On the other hand, a velocity v induces a kinetic energy in a body of mass m

$$ E_{cin} \, = \,\,\frac{{m\,.\,v^{2} }}{2} $$

(A7.12)

The escape velocity v_esc is the initial velocity necessary to give the kinetic energy to the body to attain the state of maximum potential energy. In other words, the escape velocity must give the body a sufficiently large kinetic energy to balance the potential energy due to Earth’s gravitational attraction.

If one neglects the dissipation of kinetic energy in heat by friction in the atmosphere, this velocity is calculated by the total energy conservation between the initial state (where kinetic and potential energies are respectively maximum and null) and the final state (where kinetic and potential energies are respectively null and maximum), yielding:

$$ \frac{{m.v_{esc}^{2} }}{2} = \frac{{G.M_{E} .m}}{{r_{E} }} $$

(A7.13)

or with relation (5) $g_{gr} = \frac{{G.M_{E} }}{{r^{2} }}$ (see Chapter 1) of gravity acceleration at Earth’s surface

$$ v_{esc} = \sqrt {\frac{{2G.M_{E} }}{{r_{E} }}} = \sqrt {2g_{gr} r_{E} } $$

(A7.14)

At Earth’s surface, the escape velocity is

$$ v_{esc\,E} = 11 186\;{\text{m}}/{\text{s }} \approx 11.2\;{\text{km}}/{\text{s}} $$

(A7.15)

The interplanetary spacecraft have initial velocities greater than this value when leaving the low Earth orbit.

Appendix 8: Perturbations Inside a Satellite

Let us consider the ideal case of a satellite having a spherical shape and mass distribution, such as its inertia centre is at the satellite geometrical centre. In its orbital movement around the Earth, it is the inertia centre that describes the circular or elliptical free fall orbital trajectory. An object inside the satellite located at a certain distance from the satellite’s inertia centre is subject to a small perturbation force due to this distance. The order of magnitude of this force can be calculated with the following hypotheses:

• Earth is spherical of mass M_E;

• the satellite has a mass m_S negligible compared to Earth’s mass;

• the satellite orbit is circular and external perturbation forces (such as e.g. atmospheric drag) are negligible;

• the satellite presents always the same side to Earth, i.e. the satellite performs a complete rotation on itself while it describes a revolution around Earth (the two angular velocity vectors of orbital revolution and satellite’s rotation are parallel and have the same magnitude).

One considers the two following reference frames (see Fig. 12):

• a celestial geocentric reference frame R_E whose origin is at Earths’ centre O and that one supposes inertial, in which the satellite movement is expressed;

• a reference frame R_S whose origin is at the satellite inertia centre C and axes are fixed with respect to the satellite, one of them being along the radius vector from Earth’s to satellite’s centres. As the satellite always shows the same side to Earth, in its orbital movement, the satellite remains immobile in this reference frame. This reference frame R_S is non-inertial as it is in rotation with respect to the geocentric reference frame R_E and one describes the movement of an object in the satellite with respect to this reference frame R_S.

Fig. 12

Forces applied to an object located in P away from the inertia centre C of a spherical satellite on a circular orbit around a spherical Earth. The celestial geocentric reference frame R_E is inertial; the reference frame R_S attached to the satellite inertia centre is non-inertial

In the inertial reference frame R_E, the only force that acts on the satellite is Earth’s gravitational attraction; there are no other forces, in particular no centrifugal inertia force. In the satellite reference frame R_S, two forces act on all points of the satellite, in particular on the satellite inertia centre C:

• (1) the gravity force due to Earth gravitational attraction

  $$ \vec{F}_{gr} = - \frac{{G.M_{E} .m_{S} }}{{r_{oc}^{2} }}.\left( {\frac{{\vec{r}_{oc} }}{{r_{oc} }}} \right) $$

  (A8.1)

• (2) the centrifugal inertia force due to the satellite orbital movement

  $$ \vec{F}_{in} = m_{S} .\omega_{orb}^{2} .\vec{r}_{oc} $$

  (A8.2)

  where ω_orb is the satellite orbital angular velocity.

These two forces are equal in magnitude but of opposite direction. One deduces easily the expression of the orbital angular velocity:

$$ \omega_{orb} = \sqrt {\frac{{G.M_{E} }}{{r_{oc}^{3} }}} $$

(A8.3)

Consider an object of mass m_p released without initial velocity from a point P inside the satellite and in the same orbital plane as the satellite inertia centre C. In the inertial reference frame R_E, the only force that acts on the object is Earth’s gravitational attraction (one neglects the attraction due to the satellite mass) and there is no centrifugal inertia force. In the reference frame R_S attached to the satellite, the following forces are acting:

• (1) Earth’s gravitational force

  $$ \vec{F}_{gr} = - \frac{{G.M_{E} .m_{p} }}{{r_{op}^{2} }}.\left( {\frac{{\vec{r}_{op} }}{{r_{op} }}} \right) $$

  (A8.4)

• (2) the centrifugal inertia force due to two rotation movements, the whole satellite orbital revolution movement and the satellite rotation on itself. This force reads

  $$ \overrightarrow {F}_{in} \, = \,m_{p} \,.\,\,\omega_{orb}^{2} \,.\,\,\overrightarrow {r}_{oc} + m_{\text{p}} \,.\,\,\omega_{\text{rot}}^{2} \,.\,\,\overrightarrow {r}_{\text{pc}} $$

  (A8.5)

As the satellite always shows the same side to Earth, its rotation velocity is equal to the satellite orbital velocity:

$$ \omega_{rot} = \omega_{orb} $$

(A8.6)

In the reference frame R_S attached to the satellite, the resultant of the two forces given in (A8.4) and (A8.5) is no longer exactly null and a small perturbing inertia force $\vec{F}_{p}$ exists due to the distance CP of the object from the satellite inertia centre. The norm of this force is with a good approximation

$$ F_{p} \approx 3\frac{{G.M_{E} .m_{p} }}{{r_{oc}^{3} }}. r_{cp} = 3\frac{{G.M_{E} .m_{p} }}{{r_{oc}^{2} }}.\left( {\frac{{r_{cp} }}{{r_{oc} }}} \right) $$

(A8.7)

It can be understood as the gravity force applied to the object mass m_p placed at the satellite inertia centre C, corrected by a factor equals to three times the ratio of the distance r_cp of the object in P to the satellite inertia centre C and the distance r_oc of this inertia centre C to Earth’s centre. The object, as soon released from a point connected to the satellite (therefore moving before the release at the satellite orbital angular velocity ω_orb) starts to describe its own orbit around Earth inside the satellite. Seen from the satellite inertia centre, the object drifts slowly until it touches a satellite wall.

For an orbit at 500 km altitude, an object inside a satellite is subjected to a parasitic acceleration of 4 × 10⁻⁶ m/s² per metre of distance.

The cases of non-rotating non-spherical satellites are treated similarly, but with a more complicated formalism. The magnitude orders of perturbations however are similar.

Appendix 9: Incorrect Ideas About Weightlessness

We correct here some incorrect or approximative ideas on weightlessness and its causes, but that are unfortunately very common and widespread. The informed reader will understand that these clarifications are necessary. Indeed, it is important to not only say what weightlessness is, but also what it is not. Experience shows that some of these incorrect ideas die hard.

3.1 1 Weightlessness is not Due to the Nulling or Disappearance of Gravitation

One never insists enough on this point: the disappearance of gravitation or gravity is impossible. The only theoretical cases (fortunately!) are the simultaneous nulling of all masses in the universe, or being at an infinite distance of all masses in the Universe, both being physically meaningless.

3.2 2 Weightlessness is not Due to Absence of Atmosphere

Evacuating a closed volume would not mean that a proof mass released in this volume would not fall anymore. To the contrary, it would fall even more easily and slightly faster as there is no longer air friction.

3.3 3 Weightlessness is not Due to a Large Distance Away from Earth or Any Other Mass

Weightlessness is currently generated on Earth. It is sufficient to put oneself in a state of free fall. Jump two steps from the staircase and, for a fraction of seconds, you are in a state of weightlessness (if air friction is neglected). Being away from all masses in the universe is of course impossible. This incorrect idea is close to the one of nulling gravitation: “if you are far enough from all attractive masses, you do not feel the attraction anymore or very little”. This is wrong of course: the diminishing of the force of gravity does not cause weightlessness. To the contrary, it is because the masses in the Universe that gravitation exists and that free fall can exist. The state of free fall is a dynamic phenomenon: the body must be in motion in a gravitational field. It does not help to try to get away from gravity. To the contrary, one must abandon oneself to it.

3.4 4 Weightlessness is not Due to a Very High Velocity

This idea is probably caused by an analogy with an airplane at take-off that must reach a certain speed to be airborne and to be able to fly freely in the air. If it is true that sustentation or lift of an airplane is a function of velocity by Bernoulli’s principle, it has nothing to do with weightlessness. Moreover, the high orbital velocities of satellites in orbit around Earth (approximatively 28 000 km/h or 8 km/s) can contribute to the confusion. Let us recall that weightlessness is immediately achieved even with a null initial velocity. An object is directly in free fall as soon as it is released from the hand and therefore, in a weightless state in its own reference frame.

3.5 5 Weightlessness is not Only Obtained at a Point of Equal Attraction Between Two Masses

This idea very unfortunately was popularized by the French Author Jules Verne in his book “Around the Moon”. He describes weightlessness as the state that only lasts the instant of passage to the point of equal gravitational attraction between Earth and the Moon. If it is exact that weightlessness does exist when passing by this point (if there are no other non-gravitational forces), it also exists at every moment of the unpropelled ballistic flight between Earth and Moon, as demonstrated by the Apollo Moon missions.

3.6 6 Weightlessness is not Simply Due to a Balance Between Gravity and Centrifugal Forces

Here, the situation is more delicate and one must be careful. The statement, often made as is, of the simple balance between two forces, one directed “inside” the movement, the other one directed “outward” of the movement, is incorrect because incomplete. The reference frame in which this statement is made is not specified and it makes it wrong. In a reference frame attached to the body in free fall, the statement is correct as the reference frame is non-inertial: inertia forces therefore exist in this reference frame to balance the gravity force. In an external reference frame, that can be considered as inertial most of the time, the statement is wrong because there are no inertia forces, and the body only falls freely due solely to the force of gravity (if there are no perturbing forces).

Let us mention as well that there is no way to recreate weightlessness in centrifuges (by making them turn “the other way around” for example). Weightlessness can be simulated on Earth by certain methods that will be described in Chapter 3. Weightlessness can also be created on ground, but always following the principle of free fall, as it will be explained in Chapter 3.