28 Physical Properties of Stars

V. B. Bhatia

1. Learning Outcomes

After studying this module, you should be able to

Understand the concept of space velocities of stars
Explain the need for a Local Standard of Rest (LSR)
Grasp how LSR is defined
Account for the motion of the Sun with respect to the LSR
Describe the terms Solar Apex and Antapex
Understand the notion of peculiar velocities of Star
Account for the long term change in the shape of stellar constellations
Grasp how the distances of stellar clusters are measured using their space velocities

2. Introduction

In the last module we introduced the proper motion of stars. It must be borne in mind that the stars appear fixed because of their large distances; actually they are in constant motion. The proper motion is the motion transverse to the line of sight of the star. The proper motion is such a small angular displacement of stars that in most cases it amounts to a mere arc – second, or even smaller, in a year. That is why we need to compare photographs of the star field containing the star separated by decades. The proper motion of stars being so small, it is not possible to measure it for stars in other galaxies.

It must be remembered that the proper motion depends on the distance of the star from the Sun. Two stars having the same displacement normal to the line of sight in a given time will have different proper motions depending on their distances (Fig. 7.1). However, smaller proper motion does not necessarily imply a larger distance. An important effect of the proper motion is that it changes slowly the coordinates of a star. Therefore, whenever we state the coordinates of a star, we also state the epoch when the coordinates were measured.

Fig. 7.1. Transverse displacement of two stars, S1 and S2 in a given time are equal. However, their proper motions ?1 and ?2 are not equal because of unequal distances.

If the star is attended by an invisible companion, say a satellite, then the proper motion shows a periodicity. In fact, this an important method by which the presence of extrasolar planets is detected.

Combined with the velocity along the line of sight of the star, the radial velocity, the proper motion gives the space velocity of the star, which we discuss in this module.

3. Space Velocity of Stars

The product of the proper motion of a star with its distance from the Sun gives its transverse velocity ??. If we express ??in km/s, the distance ?in pc and the proper motion in ˝/yr, then we get the following relation:

??= 4.74?d= 4.74?/?km/s, (7.1)

where ?is the parallax of the star in arc-second. This is a particularly useful relation, for it enables us to find stellar distances when the annual parallaxes cannot be directly measured. If we wait long enough, ?can be accurately determined and then a knowledge of ??give space velocity. Application of this relation to stellar clusters will be discussed later.

The velocity of a star in the line of sight is called its radial velocity, denoted by ??. It is inferred from the Doppler shifts of the lines in the star’s spectrum. Combining the radial velocity with the transverse velocity, we get the space velocity of the star (Fig. 7.2). Typical space velocities of stars range from 20 to 100 km/s.

Fig. 7.2. Radial velocity, transverse velocity, space velocity and proper motion of a star.

Clearly,

4. Peculiar Velocity of Stars

So far we have described the motion of a star relative to the Sun. But it is quite conceivable that the Sun itself may be moving in some direction. This, indeed, is the case. We, therefore, define a Local Standard of Rest (LSR), a sort of origin, with respect to which we state the motion of the Sun and stars in its neighbourhood.

4.1. Local Standard of Rest (LSR)

The solar neighbourhood, is a region roughly of radius ~ 100 pc centred on the Sun. These stars are supposed to share the motion of the Sun round the centre of the Galaxy, so that their motion due to differential rotation of the Galaxy does not become conspicuous. (Differential rotation implies that the various subsystems in the Galaxy rotate with their own characteristic speed round the centre of the Galaxy.) We have seen earlier that the Sun is situated at a

Fig. 7.3. Local Standard of Rest orbits the galactic centre in a circular orbit at the distance of the Sun and completes one revolution in the same time as the Sun.

distance of about 8.5 kpc from the centre of the Galaxy. The Sun and its neighbourhood move round the centre of the Galaxy, completing one revolution in about 250 million years, the orbital velocity being ~ 220 km/s. The orbit of the Sun round the galactic centre is elliptic. Therefore, we define the Local Standard of Rest a point which orbits the galactic centre in a circular orbit at the galacto-centric distance of the Sun and completing one revolution in the same time as the Sun (Fig. 7.3).

Now, if we plot all the velocity vectors of stars in the solar neighbourhood, we find that they all converge towards the Sun and in the general direction of the constellation Columba with a resultant velocity of about 20 km/s (Fig. 7.4). This is obviously a reflection of the Sun’s own

Fig. 7.4. A sketch of the velocity vectors of stars in the solar neighbourhood. All these vectors converge towards the Sun.

motion in the opposite direction, that is, in the direction of the star Vega in the constellation of Lyra with a velocity of 20 km/s.

4.2. Solar Apex

The direction in which the Sun is moving with respect to the LSR is called Solar Apex. The direction diametrically opposite to that of Apex in which the stars in the solar neighbourhood are moving is called Antapex. As stated above, Apex is in the direction of star Vega and Antapex is in the direction of the constellation Columba.

Consider Fig. 7.5. The radial velocity vector of all the stars in the solar neighbourhood add to zero. Their space velocities all point in the direction of Antapex. Equivalently, the Sun motion with respect to the LSR is in the direction of the Solar Apex.

Fig. 7.5. Velocities of stars surrounding the Sun. Red arrows indicate radial velocities, which all add to zero. The black arrows are the space velocities of star pointing in the direction of Antapex. This is due to the Sun’s own motion in the direction of Apex at a speed of ~ 20 km/s.

Space velocities corrected for the Sun’s own motion are called the peculiar velocities. It is found that the peculiar velocities of stars are essentially random. Note that the net gravitational field due to all the stars surrounding a given star is zero. Therefore, being under no force, the stars move in random directions like the particles of a perfect gas. The typical magnitude of the peculiar velocities is such that in a time interval of ~ 105 year the stars will switch their nearest neighbours. Thus, in a period of about a million years the present constellations will be thoroughly shuffled (Fig. 7.6).

Fig. 7.6. Changing shape of the familiar constellation due to the random motion of stars.

5. Distances of Star Clusters

In the case of some star clusters it is found that the space velocity vectors of their individual members converge in a certain direction (Fig. 7.7). This could only mean that the cluster as a whole is moving in that direction. Let �⃗ be the resultant velocity of the stars of the cluster.

From Fig. 7.8 it is obvious that the parallax of the cluster is given by ?

Fig. 7.7. The space velocity vectors of the members of a stellar cluster converge towards the point of convergence.

using Equation (7.1). Here ?is the angle between the line of sight and the direction in which the velocity vectors of the members of the cluster converge. Thus, a knowledge of ?and ? leads to the distances of these clusters. This is the moving cluster method for finding cluster distances.

The differential rotation of the Galaxy (to be discussed later) has a systematic effect on the proper motions. It will be shown that for a star at galactic longitude ??�, the transverse velocity produced by the galactic rotation is given by

Fig. 7.8. The velocity vectors of the stars of a cluster converge in a direction which makes an angle ?with the line of sight.

6. Summary

Proper motion gives the transverse velocity, which together with the radial velocity (found by using Doppler Effect) gives the space velocity of a star.
The Sun’s own motion is defined in terms of the Local Standard of Rest (LSR).
LSR is a point which orbits the centre of the Galaxy in a circular orbit at the galacto-centric distance of the Sun and completes one revolution in the same time as the Sun.
The convergence of velocity vectors of stars in the solar neighbourhood shows that the Sun is moving in a certain direction with respect to the LSR with a velocity of about 20 km/s.
The point towards which the Sun is moving in space is called the solar apex.
The velocity of a star transverse to the line of sight is called the transverse velocity. It is obtained by multiplying the proper motion by the distance of the star.
Velocity along the line of sight is called the radial velocity.
Transverse and radial velocities together give the space velocity of a star.
Space velocity of stars corrected for the Sun’s motion is called the peculiar velocity.
Peculiar velocities of stars are purely random. In a time of the order of 100,000 years all the constellations will be thoroughly shuffled.
Stars in some clusters show that their velocity vectors converge in a certain direction. This observation gives the space velocity of the cluster.
Armed with the space velocities of clusters and their radial velocities, we can determine their distances.

You can view video on Physical Properties of Stars

The following are the references for more-material on the topics covered in this module:

http://www.cosmos.esa.int/web/hipparcos
http://hyperphysics.phy-astr.gsu.edu/hbase/astro/para.html
http://cseligman.com/text/stars/propermotion.htm
https://inspirehep.net/record/818899/plots
https://medium.com/starts-with-a-bang/how-fast-are-we-moving-through-space-985bf470378d#.gvhngkinf