Figure 2.3 The solar nebula surrounding the proto-Sun. The proto-Sun is too small to show on this scale.

The starting point is, however, ill defined in one important respect: we do not know the initial mass of the disc. In some solar nebula theories the mass of the disc is about 1% of the present solar mass M0. At the other extreme are versions in which the initial mass of the disc is comparable with M0. A disc mass of about 1% M0 is called the minimum mass solar nebula, MMSN. This is calculated from the estimate that there are about 65 Earth masses of heavy elements in the Solar System today, mainly in the interiors of the giant planets. To this is added the hydrogen and helium necessary to achieve solar composition. Much of the hydrogen and helium has been lost, mainly through the T Tauri wind. Some indication of an appropriate choice of mass is obtained by considering the angular momentum in the Solar System.

2.2.1 Angular Momentum in the Solar System

The magnitude I of the angular momentum of a body with mass m moving at a speed v with respect to an axis, as in Figure 2.4, is given by

Figure 2.4 A body of mass m moving at a speed v at a perpendicular distance r from an axis perpendicular to the page.

where r is the perpendicular distance from the path of the body to the axis. The angular momentum of m is with respect to this axis. In the solar nebula a natural choice of axis is the rotation axis in Figure 2.3 - through the centre of the proto-Sun and perpendicular to the plane of the disc. In the Solar System today the natural choice is for the axis to go through the centre of mass of the Solar System and to be perpendicular to the ecliptic plane. It is the angular momenta with respect to these natural choices of axes that are of concern here.

Equation (2.1) applies to the mass m when its dimensions are small compared with r so that the whole of m can be regarded as being the same distance from the axis. This is closely approximated by a planet in orbit around the centre of mass of the Solar System, and the quantity is called the orbital angular momentum. If this condition is not met then the body is notionally subdivided into many small masses 8m and the magnitude of its angular momentum is then a combination of the quantities 8mvr. A simple case is when the angular momentum of a rotating planet or the Sun is calculated, as in Figure 2.5. The natural choice of axis is again the rotation axis, and because the paths of the 8m around this axis are all circular and in the same set of parallel planes, the combination is simply the sum of 8mvr over the whole body. The quantity in this case is called the rotational (or spin) angular momentum.

In the Solar System today about 85% of the angular momentum is in the orbital motion of Jupiter and Saturn, and only about 0.5% is in the rotation of the Sun. Less than 0.5% is in the orbital motion of the Sun around the centre of mass of the Solar System. This is in sharp contrast to the Sun having about 99.8% of the mass of the Solar System. Thus today, 'where the mass is, the angular momentum is not'. The Sun's rotational angular momentum is small because it rotates slowly, about once every 26 days. Its orbital angular momentum is small for two reasons. First, the centre of mass of the Solar System is just outside the Sun, so r in equation (2.1) is small (call it r0). Second, the orbital period P0 for its small orbit is about 12 years, so its speed v0(= 2^r0/P0) is very small. For a planet, the average orbital angular momentum is well approximated by mva, where m is the mass of the planet, v is its average orbital speed, and a is the semimajor axis of the orbit (strictly, the distance from the centre of mass of the system should be used). By combining Kepler's third law

Figure 2.5 The rotation of an element 8m of a spherical body.

with equation (2.1), and using 2va/P for the average speed, where P is the planet's orbital period, we get lorb = 2rma1'2 (2.2)

k where 1orb is the average orbital angular momentum of the planet (see Question 2.4). □ So, why are the orbital angular momenta of Jupiter and Saturn large? They have large orbits, and they are by far the most massive of the planets.

This distribution of angular momentum today is in sharp contrast with that calculated for a contracting cloud fragment. The proto-Sun rotates rapidly, and has a correspondingly large fraction of the total angular momentum. Therefore we need to explain how most of the angular momentum of the proto-Sun could have been lost. One of the more convincing explanations involves turbulence in the disc at the time it still blended with the outer proto-Sun. Turbulence is the random motion of parcels of gas and dust and is expected to have been a feature of the contracting nebula. (Note that though the parcels can be small, they are much larger than atomic scale - this is not the random thermal motion that occurs at the atomic level.) Turbulent motions are superimposed on the orderly swirl of circular orbital motion around the proto-Sun. Turbulence transfers parcels radially, and it can be shown that the net transfer of disc mass is outwards in the outer part of the disc and inwards in the inner part of the disc. The associated net transfer of angular momentum is from the proto-Sun to the disc, and it is carried by a small fraction of the disc mass.

Further transfer arises from the solar wind. The ions that constitute the wind get snared by the Sun's magnetic field. Therefore, as they stream outwards they are forced to rotate with the Sun, and slow its rotation. There is thus a transfer of angular momentum from the Sun to the wind.

Note that the transfer of disc mass leads to the loss of mass from the disc. In the outer disc this loss is to interstellar space. In the inner disc it is to the proto-Sun, and also to interstellar space via an outflow along the rotation axis of the disc, perhaps enhanced by mass loss from the polar regions of the proto-Sun. Such bipolar outflow is observed from protostars, as in Figure 2.6,

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