Chapter 6. The Tidal Force

Transcription

1 Chapter 6. The Tidal Force The so-called tidal force is not a separate force from gravity it is simply the differential gravitational force. That is, if two parts of an object have different distances from some mass (e.g. the side of the Earth facing the Moon is somewhat closer to it that the other side of the Earth) then they feel a different force. The part further away feels less force. This leads to a differential force across the body that can stretch it out of shape or, in extreme cases, even shatter and destroy it. There are many interesting effects of this Tidal force throughout astronomy and here we touch on the basics and some examples. The tidal force is normally defined as the difference between the force acting at some point on an object due to some external object and the force from that external object acting on the center of mass. Hence, F tidal = F x,y,z F centerofmass 1. Differential Gravitational Force Consider a point mass M that is exerting a gravitational force on another (initially spherical) mass m whose radius is r. Note that we need to consider the finite radius of at least one of the objects because for two point masses there is no tidal force all locations within the point mass are exactly the same distance from the other mass! If M and the center of m are separated by a distance d, then the magnitude of the force between their centers of mass is F = GMm d 2. Taking the derivative along the direction between their centers of mass we have df dd = 2GMm d 3 so along this line, that is the rate at which the tidal force grows with distance from the center of mass. The magnitude of the tidal force at the surface of the object of radius r along the line between the two objects, would be (as long as r<<d) F tidal = 2GMmr d 3 Notice that the tidal force falls off much more rapidly with distance than does the gravitational force i.e. it goes as d 3, not d 2. This means that even relatively small mass objects can have a large tidal impact if they are very close to you. For example, the Moon

2 2 dominates the tides on Earth, even though the Sun is so much more massive. Essentially, this is because opposite sides of the Earth do not have a very great difference (percentage wise) in terms of their distance to the Sun, whereas they have a much greater percentage difference in terms of their distance to the Moon. It is important to note also that the tidal force, like all forces, is a vector quantity and the equation of the first paragraph defining it is a vector equation (hence the bold-face type). When one subtracts two vectors of different magnitude, pointing in the same direction, one can end up with a vector pointing the other direction. A common example is two cars headed down the highway in the same direction. As seen from one, the other may seem to be going backwards because of the vector subtraction. When one calculates the direction of the tidal force acting on the surface it is towards the perturbing object for points closer to it and away from the perturbing object for points further away than the center of mass. This leads to tidal stretching and the existence of tidal bulges on opposite sides of the perturbed object (see diagrams from class presentation). A second effect of the vector subtraction is that for points that are not along the line between the centers of mass of the two objects there is a net force that points towards the center of mass. There is a sort of squeezing of the object, as if someone had it in a vice grip and was squeezing down on the poles (again see diagram used in class). This is sometimes called the toothpaste effect it is like squeezing a tube of toothpaste it compresses in at one location and bulges out at another. 2. Tides on Earth The Moon is the main tidal perturber on the Earth but the Sun has an effect also, at about one-third of the amplitude of the Moon, due to its much greater distance. Basically, the Earth has two tidal bulges at any one time, one more or less facing the Moon and one on the other side. As the Earth rotates, the tidal bulges slide around the surface. Since water responds better than land to the tidal force, being more malleable, the tides are noticed most at the coasts where the water rises much more than the land. Since there are two tidal bulges on the Earth there are about two high tides each day...not exactly two because the Moon and Earth also orbit each other so during one day there alignment changes. The effect of the Sun is to either add to or subtract from the effect of the Moon. When the Sun and moon act at along the same line (i.e. at new moon or full moon) the high tide is higher than usual and the low tide lower than usual meaning the tidal range is greater. These are referred to as Spring Tides. During the quarter phases of the Moon,

3 3 the direction to the Sun and Moon is roughly a right angle so the tidal effect of the two bodies is counter to each other. This results in a reduced tidal range. This is called Neap Tides. If a hurricane hits a coastal area during full or new moon its effect can be more devastating because the tidal range is already higher then. The details of how tides appear at any coastal location on Earth or even when, exactly, high tide occurs, depend on the details of the coast and how water flows. In some places, like the Bay of Fundi in Canada, there can be a long funnel that basically channels water into it and causes a huge tidal range. In other places the effect is more muted. But tides do occur constantly on Earth and they can even be a source of energy if tapped as is done in a few locations around the Earth. 3. Tidal Locking The tidal force affects the rotation of an object because rotation carries the tidal bulges past the line connecting the centers of the interacting objects. This results in a net retarding force (torque) which acts to slow down the rotation of the object. The Moon, for example, has already become tidally locked to the Earth. It rotates with the same period that it orbits the Earth, therefore keeping one side always facing the Earth. The Earth is gradually slowing its spin such that the length of a sidereal day (spin period of Earth) is growing at the rate of about 1 second/century. This gives rise to the phenomenon of the leap second, a second or fraction of a second that is added to the civil clock every once in awhile, normally on New Year s Eve, to keep the Earth and civil time in step. Eventually (billions of years in the future) the Earth will also have one side of it that will face the Moon perpetually and the other side will face away from the Moon. The phenomenon of tidal locking is wide spread in astronomy. Many close binary stars are tidally locked so that they orbit and spin at the same rate keeping one face perpetually towards the other and one face perpetually away. Many moons are also tidally locked to their planets. In addition to our Moon, Phobos and Deimos (both moons of Mars) are tidally locked as are the four Galilean moons of Jupiter (Io, Europa, Ganymede and Callisto). Mercury is in a 3:2 tidal resonance with the Sun it makes 3 spins for every 2 orbits. It is in this resonance rather than 1:1 because its orbit is highly elliptical so the tidal force is strongest when the planet is at perihelion. It therefore got locked into the spin rate that matches its orbital rate when at perihelion. This is faster than the average spin rate, leading to its making an extra half turn each orbit how cool is that??! Some stars in highly elliptical orbits become pseudo-synchronized in this way. They

4 4 end up in some kind of resonance, such as 3:1 or 4:1, rather than 1:1. They make extra spins while near apastron and are locked to their orbital rates near periastron. 4. Distance to the Moon As a reaction to the force on the tidal bulge that is slowing the Earth s rotation, there is an acceleration of the Moon in its orbit, that is causing it to spiral away from the Earth (very slowly!). Refer to the diagram shown on class lecture notes to understand this more fully. The distance to the Moon can be measured extremely accurately (cm) using the laser mirror built at Wesleyan and placed on the Moon during an Apollo mission. Measurements over the years have confirmed that indeed the Moon is moving away from the Earth at several cm per year due to the tidal interaction. Eventually (billions of years in the future) the Earth will be tidally locked to the Moon so that a day on Earth (i.e. one spin) will be equal to a month (synodic period of Moon) but that month will be many times longer than the current month. Eventually we will be tidally locked to the Sun as well so that a day will equal a month will equal a year, simplifying our calendar enormously! 5. Tidal Destruction and the Roche Limit Sometimes the tidal force, which acts to stretch a body along the line to the perturbing object can be so strong that the tensile strength and/or self-gravity of the object cannot hold it together. When this happens, a large object can be shattered into smaller pieces which experience less tidal force (being smaller) and can survive. Comet Shoemaker-Levy, which collided with Jupiter back in the 1990 s provided a beautiful example of this happening. During an earlier passage of Jupiter the comet was broken by the tidal stress into about 23 pieces which became separate little comets all on about the same orbits. You can see pictures of this on our Web page or elsewhere on the Web by Googling it. The individual pieces of the comet then slammed into Jupiter leaving a series of black eyes where they hit vast disturbances in Jupiter s atmosphere caused by the comet piece hitting, with the explosive force of all of the Earth s thermonuclear weapons being detonated at once. One can calculate how close a moon or other large object can come to a planet before it is destroyed by the tidal force. Such a calculation was done by Roche and is referred to as the Roche limit. Check out the Wikipedia page for some nice images and a calculation of the size of the Roche limit. Note that the rings of Saturn are within the planet s Roche limit, so they may well be the remnants of a moon that wandered in too close. Of course,

5 5 nothing can actually accrete into a moon at that location so they could also be the building blocks of a moon that never got itself together. 6. Tidal Tails Galaxies interact tidally when they collide with each other and sometimes leave spectacular results in the form of tidal tails that stretch in curved fashion away from the interacting pair. An excellent example is provided by the antennae galaxy and can be seen in Web images. Of course, the time scale for interactions and motions of galaxies is millions of years, so we do not see any motions from year to year. We see merely a snapshot in time of the interaction. Computer simulations can show how this looked in the past and how it will look in the future. Another example of tidal tails that is important in astronomy is collisions between our own Milky Way galaxy and nearby companions, such as the Sagittarius galaxy and the Magellanic Clouds. These companions are tidally distorted and disrupted by the Milky Way and as they orbit they leave tidal tails of stars behind them. These stars are part of streams of stars in our Milky Way that come from the other galaxies. Basically, our own galaxy is disrupting and absorbing these other systems. This is a common process in building a galaxy, any galaxy. We can do galactic archaeology by finding these stellar streams in our own galaxy and then inferring, with the help of computer simulations, what the past interactions of the Milky Way and its companions has been. Cool!