The earliest method of enabling astronomers to determine the distance to remote objects is the parallax method. As Earth orbits the Sun, we see an apparent shift in the positions of stars relative to much more distant stars, called parallax. For nearby stars, the parallax is larger and for more distant stars the parallax is smaller.
The baseline used by the parallax method is fixed by the size of Earth's orbit around the Sun. Recall that like all orbits, that of the Earth is an ellipse rather than a circle. The mean Earth–Sun distance, or Astronomical Unit (AU), is approximately 1.5 * 10 to the power 9 metres. Earth's orbit is so nearly circular that using the mean distance is accurate enough for these purposes.
As shown in Figure 1, if we observe a star's position with respect to the distant stellar background between two observations that are six months apart in time we will see the parallax shift in our data. The observation baseline is 2 AU.
At this point, let us to define a unit of distance called the parsec. This unit of measure is defined as the distance at which 1AU subtends an angle of 1 arcsecond to the observer.
Hence referring to Figure 2, the distance to a remote object is given by:
Some examples are shown in Table 1, where stellar distances are derived from parallax angle using equation (1).
Limitations of the parallax method
Looking at the results above, we can see that distance values for the first two stars agree quite closely with recognized values. However, the distances for Rigel and Betelgeuse do not agree very well with accepted values. These stars are both "nearby" in that they are both very much naked-eye objects. So where does the problem lie?
Parallax angles of less than 10 milliarcseconds (mas) are very difficult to measure using Earth based telescopes due to atmospheric effects.
Using equation (1) and considering a parallax angle of 10 milliarcseconds:
Hence, distances greater than 100pc pose increasing difficulty for ground based instrumentation. In 2008, researchers using the Very Large Array (VLA) produced a radio solution of 5.07±1.10 mas for the parallax angle of Betelgeuse, corresponding to a distance of 197±45 pc or 643±146 lyr (Harper et al, 2008). In this paper, Harper et al also point out that the angular size of Betelgeuse is greater than it's parallax angle, creating further problems when observing such large stellar objects at this distance range.
The European Space Agency (ESA) Hipparcos satellite mission made it possible to measure the parallax displacements with an accuracy of up to 0.1 mas a big improvement allowing parallax measurements to be theoretically useful up to
The follow-on ESA mission, Gaia can measure parallax angles to an accuracy of 0.01 mas giving useable distance measurements up to:
So we can see that the use of space-based telescopes has extended the range at which the parallax method is of practical use by about 1,000 times.
What part does distance measurement play when observing stars?
We'll consider three cases where knowing the distance to a remote object is key in undertanding that object's properties. In each case, the reason this is so important is because we can't measure a remote object's luminosity directly.
The luminosity of a remote object is observationally determined by measuring its magnitude. Here we are essentially examining ratios of flux density. Whereas luminosity is independent of distance, flux scales inversely with distance according to Newton's inverse square law, so an accurate determination of luminosity requires an accurate knowledge of distance.
In practical terms, the inverse square law states that if we have two objects of the same luminosity, one twice as distant than the other, then the flux density of the more distant object will be one quarter of that of the nearer object.
Case 1 - the size of a star
We can estimate stellar radius by using the Stephan-Boltzmann Law:
We can measure the luminosity of the star, L, by photometry. That is, provided we know the distance from us. We can measure and its photospheric temperature, T, by spectrometry. Some simple algebra allows us to determine the star’s radius in terms of the solar radius is:
Those who saw my webinar on the Hertzsprung-Russel diagram last year may recall this equation. Measuring stellar parameters in terms of Solar units is commonplace in astrophysics as it simplifies calculations considerably. In this case, we don't even need to know the value of the Stephan-Boltzmann constant. But we do need to know the distance.
Case 2 - the size of an exoplanet
When an exoplanet (a planet orbiting a star other than the Sun) crosses the face of the star as seen by an Earthly observer, it causes a dip in the observed magnitude of the remote star. The characteristic light curve of the star will appear similar to Figure 4, in this case of the star WASP-2A being transited by its planet WASP-2A-b:
HOPS is made available as part of a pro-am collaboration project called Exoclock. The goal of the Exoclock project is to make as many observations of confirmed exoplanet transits as possible. Amateur participation is valued as there is simply not enough time available on professional telescopes.
Because the cross sectional area of both the star and the planet scale by radius squared, the transit depth, is related to the squares of the ratio of stellar radius and planetary radius:
So, in this case also we can infer the size of the exoplanet in terms of the solar radius, but to know the value of this in metres, as in case 1, we must again know the distance.
In the case of WASP2-A-b, it turns out that -according to this light curve- the planetary radius is 1.23*RJ where 1.23*RJ is equal to the radius of Jupiter.
The published value in the exoplanet.eu/catalog is 1.079 ± 0.033 RJ, so we are reasonably close. Nevertheless we can classify WASP2-A-b as a Jupiter-size exoplanet.
Case 3 - the mass of a star
The mass-luminosity ratio is stated in relation to solar units as:
However, the relation is highly empirical in the sense that both the coefficient a and the exponent b depend on the mass of the star. Almost as if you have to guess the mass first!
For a very small star (e.g. a red dwarf), a = 0.23 and b = 2.3.
For a star comparable to a solar mass, a = 1 and b = 4.
For a star with a mass comparable to Betelgeuse, a = 1 and b = 3.5.
In this case also, to calculate the mass of another star in kilograms (or any other units of mass), we need to know the distance as accurately as possible so that we can calculate its luminosity.
The mass of a star also determine its ultimate fate. There was great excitement in the popular press at the end of 2020 to the effect that "Betelgeuse was about to go supernova". This was prompted by the fact that Betelgeuse had dimmed considerably. In fact, this is now believed to be because Betelgeuse had expelled a great deal of its atmosphere as dust, causing the apparent dimming of magnitude (Kidger, 2020).
Table 2 is based on much more detailed modeling published in 2015 (Woosley and Heger, 2015 - refer to Table 1, page 3)
1. The parallax method is earliest method used todetermine the distance to remote objects. Space-based observatories such as Gaia have extended the effective usefulness of the method out to ~10 to the power 5 pc.
2. In order to calculate parameters such as size and mass, it is necessary to have an accurate value for distance. Note that we can calculate the mass of a star in abinary system, but we then need the mass of the other star.
3. Measuring stellar parameters in terms of Solar units is commonplace in astrophysics. For example we know that M⊙ = 1.99*10 to the power 30 kg. How we determine soar parameters will be covered in a future blog.
4. Betelgeuse is in many respects an enigmatic object. Because its distance is not known precisely, so neither is its luminosity. That also means its mass is not known precisely and hence the ultimate fate of Betelgeuse is uncertain.
Harper G et al. (2008). A new vla–hipparcos distance to Betelgeuse and its implications https://iopscience.iop.org/article/10.1088/0004-6256/135/4/1430/pdf, Accessed April 14, 2021. ApJ, 135:1430–1440, 2008 April.
Woosley, S and Heger, A (2015). The remarkable deaths of 9–11 solar mass stars https://iopscience.iop.org/article/10.1088/0004-637X/810/1/34/pdf, Accessed April 14, 2021. ApJ,810:34(20pp), 2015 September 1
Kidger, M (2020) Supernova Betelgeuse https://britastro.org/jbaa/pdf_cut/jbaa_25295.pdf , Accessed April 14, 2021. Journal of the British Astronomical Society, 2020 December.
I am quite surprised that this month’s blog will complete a year of them, meaning that we have been all the way round the night sky. By the nature of the subject there will be a bit of repetition from now on but there is always something new happening and who can ever tire of looking at a lovely starry sky even if you have seen it before. We passed the spring equinox on the 20th March so we are getting more daylight and the clocks went forward an hour on the 28th March so it will be later before the sky darkens. The good news is that we have had some clear nights and seeing Mars, Aldebaran, the Pleiades and a crescent Moon all together just after the middle of the month was particularly pleasing. Since we are making a fresh start I thought that it would be a good idea to repeat the bit about the celestial sphere and how the sky changes in appearance from night to night and month to month. I hope this will prove useful to any newcomers to the subject and any youngsters who are hopefully embarking on observing the skies as a lifetime’s hobby.
The Celestial Sphere
Before we venture outside let us recall some helpful facts. It is useful to think of the sky as a hollow sphere which has the Earth at its centre and to which all the heavenly objects are attached. This sphere is known as the celestial sphere. Just like when you visit a planetarium. The celestial sphere also has north and south poles directly above the corresponding poles on Earth and a celestial equator directly above the Earth’s equator. Far away objects such as stars and galaxies are in more or less ‘fixed positions’ on the celestial sphere whereas the Sun, Moon and planets continually shift their positions but stay close to a circular path on the sphere’s surface called the ‘ecliptic’ which is tilted to the celestial equator because the Earth’s axis is tilted by 23.5 degrees to the plane of its orbit. In reality of course the Earth revolves round the Sun and the ecliptic is where the plane of the Earth’s orbit cuts the celestial sphere. This makes sense because when we observe the Sun we are looking along the radius of the Earth’s orbit and hence in the plane of its orbit.
The recent equinox marks the point where the path round the ecliptic crosses the celestial equator. This is when the Sun is overhead at the equator and it continues to travel further north until the summer solstice when it is overhead at the Tropic of Cancer. We see from the diagram that the ecliptic is north of the celestial equator during this period of time.
For us in the northern hemisphere we see the stars rotate about the north celestial pole. Don’t worry about some of the additional information on the diagram. The yellow line is the ecliptic and it shows the signs of the zodiac (representing the constellations) and how the Sun appears to pass in front of them as the Earth revolves around the Sun. Remember we are using a model for what we see and this is governed by the movement of the Earth. The Earth spins about its axis from West to East once a day (ie 360 degrees in 24 hours or 15 degrees per hour) and that is why we see the Sun move across the sky daily from East to West. It may not be so obvious that the stars are doing the same thing at night and they move across the sky from East to West at 15 degrees per hour as well. Of course, they also do it during the day, but we cannot see them for the glare of the Sun.
The Earth also revolves about the Sun once a year (ie 360 degrees in 365 days or about 1 degree per day or 15 degrees in 15 days) which is why the sky at 10.00pm one day will look like the sky at 9.00pm 15 days later. If you wait till 10.00pm again the celestial sphere has moved on by 15 degrees or 1 hour and all the stars have moved that amount further west.
Okay, it is time to look at the stars. The following charts represent the night sky at 10.00pm BST on the 8th of April and at 9.00pm BST on the 23rd April. To use the chart, face south at the appropriate time with the bottom of the chart towards the southern horizon and you will see the stars in the chart. From ancient times the stars have been put into groups called constellations with names supposedly indicating what they represent but this is seldom clear.
The fact that some stars appear in a group does not indicate that they are close together and their distances can vary by very large amounts. Some groups of stars stand out but may be only part of a constellation and such groupings are called ‘asterisms’.
So facing south and going up from the horizon you will see the constellation Leo- The Lion. Fortunately it does look like a crouching lion facing towards the right with the brightest star Regulus (the 15th brightest seen from the northern hemisphere) being its front paw and the curve of stars above that representing its head and mane. This latter grouping of stars is an example of an asterism called ‘the Sickle’, looking like a backwards question mark with Regulus being the dot at the bottom.
Now raise your eyes upward to your zenith (the point directly above where you are standing) and you will see what must be the best known asterism in the night sky- The Plough. It contains seven stars and the chart shows three of them named. The Plough is part of the constellation – Ursa Major- The Great Bear, but it takes a lot of imagination to see a bear and that region is mostly referred to as The Plough. In North America it is called the Big Dipper and perhaps here in the UK a better name in modern times would be ‘The Pan’. We said in the introduction that the stars rotate about the celestial North Pole and stars close to there never set but are visible all year round when the skies are dark. Stars like this are said to be circumpolar and Ursa Major is a circumpolar constellation. But note The Plough’s orientation carefully because as it continues on its circular journey it will appear upside down in six months’ time.
The constellations are used as signposts in the sky and enable us to engage in a fun activity called ‘star hopping’. Now let’s look at the second chart.
The two stars in the Plough, Merak and Dubhe, are called the pointers and a line from Merak to Dubhe continued onwards leads to Polaris- the Pole Star. The distance is about x5 the distance between Merak and Dubhe. Polaris is very close to the celestial north pole and easily found because although not very bright it is the only star visible in that area. Polaris is in the constellation- Ursa Minor- The Little Bear. Now consider a line from the star Alioth in the Plough, through Polaris and continued onwards for about the same distance again until you see a bright star. It will be the central star of a W formation, an asterism in the constellation Cassiopeia- Queen Cassiopeia in Greek mythology. Most people see the W shape and call it Cassiopeia. The bright star was never given a name in Western or Middle Eastern culture so is referred to as gamma (g) Cas. The convention is to name stars using the letters of the Greek alphabet and an abbreviated form of the constellation. Generally this is done in the order of brightness of the star but it is not a hard and fast rule.However this star has been given the name Navi, allegedly by the American astronaut Virgil (Gus) Ivan Grissom as an anagram of his middle name because it was used for navigation in the early space missions. A fitting tribute to someone who made the ultimate sacrifice for space exploration. The constellation Cassiopeia is also circumpolar and because it is directly opposite the Plough across the North Celestial Pole the two will have exchanged positions in six months so we will see Cassiopeia much better in November. Just imagine the two of them at the ends of a long pole rotating about the North Pole.
Something to look out for
There will be a close approach of a five day old Moon and Mars on Saturday 17th April. They will be separated by about 4 degrees initially but will come within a quarter of a degree of each other at their closest. The pair will be visible after 8.30pm as dusk fades above your western horizon.
At the end of the month we welcome back Venus to the evening sky and though it is still close to the Sun it will be visible for a short time after sunset above the western horizon. It will have a close approach with Mercury on Sunday the 25th April but you will have to let the dusk sky fade before they become visible. Clear skies.