Why distance matters Determining distance using the parallax method by Gordon Dennis

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)

Conclusions 
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.

 
References
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.