How frequently will a Supernova dangerous to life on Earth explode in our galaxy?

Michael Richmond July 25, 1998

Table of Contents

1. Introduction

2. Type II Supernovae in the disk of the galaxy If the stars were uniformly distributed in the disk If the stellar density were higher in the inner regions of the disk ... How frequently do “dangerous” Type II supernovae occur?

3. Type Ia Supernovae in the halo of the galaxy If the stars were uniformly distributed in the spheroid ... If the stars were centrally concentrated in the spheroid ... How frequently do “dangerous” Type Ia supernovae occur?

4. Conclusion

1. Introduction

I try to show herein how one might calculate the interval between explosions of supernovae near enough to the earth to present a hazard to life. I use very simple models for the distributions of stars in the Milky Way galaxy; in real life, stars occur grouped together in clusters, moving groups, and spiral arms. I also use very rough guesses for the rate at which stars explode. It is likely that the uncertainties in supernova rates dominate the overall calculations.

This document must be written in TeX and translated into Postscript or PDF because it includes many equations and integrals; it just isn’t possible to write it out clearly in HTML or plain ASCII text. Sorry.

2. Type II Supernovae in the disk of the galaxy

If the stars were uniformly distributed in the disk ...

First, let’s consider Type II (and probably Type Ib/c) supernovae. These occur when a young, massive star runs out of fuel at its center; the core collapses into a black hole or neutron star, and a small fraction of the gravitational energy released in the collapse suffices to heat up the outer layers of the star and blow them off. The progenitors of these explosions are young. In our own Milky Way galaxy, we see young stars distributed in a disk, with a high density near the center and fewer stars out near the “edge” of the disk.

The simplest approximation we can make is that the distribution of stars in the galaxy is completely uniform. The ratio of total stars in the galaxy to stars within the “killing radius” κ is simply the ratio of the total volume of the galaxy to the volume of a sphere with radius κ. Suppose we assume that the disk of the galaxy is a simple, flat cylinder, like a pancake. Given a radius R and thickness t, the total volume of the galaxy is

V_{galaxy} = πR^{2}t

whereas the volume of a sphere of radius κ is

V_{danger} = \frac{4}{3}πκ^{3}

Therefore, the fraction of all stars in the galaxy which would be dangerously close would be

dangerous fraction = \frac{V_{danger}}{V_{galaxy}}

= \frac{4κ^{3}}{3R^{2}t}

If we know the size of the galaxy, and the “killing radius”, we can calculate the value of this fraction. Using numbers

radius of galactic disk R = 20 kpc
thickness of disk t = 300 pc
killing radius κ = 10 pc

we find dangerous fraction = 1.1 × 10^{−8}

This simple calculation suggests that only a tiny fraction of the stars in our galaxy are close enough to present a danger to life on Earth, should they undergo a Type II supernova explosion. However, stars in the disk of the galaxy are not distributed in a uniform fashion. Instead, stars are concentrated near the center, and become more scarce in the outer regions. Since the Sun is (probably) closer to the center than the edge of the Milky Way’s disk, we should expect more stars in our local neighborhood than the average. Let’s perform a slightly more sophisticated calculation, and see how the result compares to the simple one above.

If the stellar density were higher in the inner regions of the disk ...

Let us assume that the disk component of the Milky Way can be described by a simple disk which has a uniform thickness t, and a uniform distribution in the vertical direction. In the radial direction, let the stellar density be described by an exponential distribution

stellar density ρ(r) = Ae^{−r/R}

where A is the central stellar density, in stars per cubic parsec, and R is the “scale length” of the disk. Finally, let’s assume that the disk extends radially a distance L from the center, then stops suddenly.

Given this model, we can calculate the probability that a particular star is within some distance (the “kill radius” κ) of the Sun. First, we’ll calculate the total number of stars in the disk; then, we’ll figure out the total number of stars within the “kill radius.” The ratio of the two numbers yields the probability that a star, chosen at random from the entire disk, is close enough to harm us.

So, how many stars are there within the disk? We need to integrate the surface density over the entire area of the disk. Call this number N_{tot}:

N_{tot} = ∫_{0}^{L} 2πrtρ(r)dr
= ∫_{0}^{L} 2πrtAe^{−r/R}dr
= 2πAt ∫_{0}^{L} re^{−r/R}dr

We can do the integral by making the substitution x = r/R:

N_{tot} = 2πAt ∫_{0}^{L/R_{0}} Rxe^{−x}Rdx
= 2πAtR^{2} ∫_{0}^{L/R} xe^{−x}dx
= 2πAtR^{2}[e^{−x}(−x − 1)]_{0}^{L/R}

Plugging in the values x = L/R and x = 0, we calculate a total number of stars in the disk

N_{tot} = 2πAtR^{2}(1 − e^{L/R}[1 + L/R])

The second part of the problem is to calculate how many stars surround the Sun within the “killing radius” κ of a Type II supernova. We make the simplifying assumptions that the killing radius is much smaller than the Sun’s distance from the center of the galaxy:

κ ≪ R

and also smaller than half the vertical thickness of the disk

κ < t/2

This second assumption means that the “killing radius” around a Type II supernova doesn’t reach the vertical edges of the disk. It’s possibly incorrect, but by making it, we increase the number of stars around our Sun which may be considered dangerous. Therefore, the probability we derive with this assumption will turn out to be an overestimate of the truth; we will overstate the danger slightly.

Given these assumptions, we can calculate the number of stars in the disk of our Milky Way galaxy which are within the “killing radius” of our Sun. Let d be the distance of our Sun from the Milky Way’s center; since κ ≪ R, we can treat the stellar density around the Sun to be constant.

N_{kill} = (volume of space) × (stellar density)

= (4πκ^{3}) × (ρ(d))

= (\frac{4}{3}πκ^{3}) × (Ae^{−d/R})

= \frac{4}{3}πAκ^{3}e^{−d/R}

Now we’re ready to answer the question “What fraction of the stars in the disk of the Milky Way are close enough to the Sun to present a danger?” We simply divide the number of stars closer than κ by the total number of stars in the disk:

dangerous fraction = \frac{N_{kill}}{N_{tot}}

= \frac{\frac{4}{3}πAκ^{3}e^{−d/R}}{2πAtR^{2}[1 − e^{−L/R}(1 + L/R)]}

= \frac{2κ^{3}e^{−d/R}}{3tR^{2}[1 − e^{−L/R}(1 + L/R)]}

Well, okay — but what’s the numerical value of this fraction? The smaller the better, from our point of view. In order to evaluate this expression, we need to know the values for the numerical parameters in our model for the disk of the galaxy. These numerical parameters must be determined from careful surveys of the stars in our galaxy. They tend to change by small amounts every few years, as more comprehensive and sensitive surveys are made. One simple model comes from a paper by Gates et al. (ApJ 500, 145 [1998]; also available as astro-ph 9711110) in which the thin disk – source of young stars which might explode as Type II supernovae – is characterized as follows:

vertical thickness t = 0.3 kpc
disk scale length R = 3.5 kpc
central surface density A = 0.042M_{⊙} pc^{−3}

Let us choose the distance from the Sun to the center of the Milky Way galaxy to be

d = 8 kpc

and the radial extent of the disk to be

L = 20 kpc

Evaluating the expression above for the fraction of all stars in the Milky Way which are close enough to be “dangerous,” we find

dangerous fraction = 1.75 × 10^{−10}κ^{3} pc^{−3}

Or, for several values of κ,

if κ = 5 pc, then fraction = 2.2 × 10^{−8}
if κ = 10 pc, then fraction = 1.8 × 10^{−7}

Compare these values with the one we calculated near the start of this section; as expected, when we take into account properly the concentration of stars towards the center of the galaxy, the “dangerous fraction” is larger than if we assume a perfectly uniform distribution of star. In fact, the more sophisticated model yields a fraction about 10 times larger than the uniform model.

How frequently do “dangerous” Type II supernovae occur?

What does this mean? Suppose that a Type II supernova exploded in our galaxy once a year, and that the progenitors of all such explosions were in the disk. Then the calculations above imply that most of those supernovae would occur so far from the sun that they wouldn’t endanger life on Earth. In fact, we would need to worry only if one of the stars in the tiny “dangerous fraction” close to us were to explode. If the “killing radius” is κ = 10 pc, then on average, we’d have to wait for

\frac{1 year}{fraction} = 5, 700, 000 years

for that to happen.

But Type II supernovae don’t occur so frequently in the Milky Way. It has been over 300 years since the last one was seen from Earth. Some may occur far away from our location in the galaxy, and hide behind thick clouds of interstellar dust. Based on observations of supernovae in other spiral galaxies, we can make rough guesses for the rate at which supernovae should occur in our own galaxy. Current work suggests that there might be a Type II supernova in our galaxy once every 100 years, not every year. In that case, the interval between explosions within the “killing radius” would be

\frac{100 years}{fraction} = 570, 000, 000 years

Weve made so many approximations along the way that it’s probably unwise to quote a result with 2 significant figures; realistically, I might expect this calculation to be correct to within a factor of 2 or 3. However, it is clear from the value’s order of magnitude that Type II supernovae don’t occur dangerously close to the Earth very often at all. If the progenitors of Type II supernova are distributed throughout the Milky Way’s disk at random, then a few nearby explosions every billion years is a ballpark estimate.

3. Type Ia Supernovae in the halo of the galaxy

Type Ia supernovae (probably) occur when a white dwarf accretes so much material from a companion star that it exceeds the Chandrasekhar limit. A wave of thermonuclear fusion passes through the star, burning it into iron-group elements and liberating huge amounts of energy. The star explodes and showers its surroundings with high-energy photons from the radioactive decay of some fusion products. These gammarays and X-rays may endanger any planet within a very large distance – perhaps 100 parsecs; certainly a Type Ia supernova presents a greater danger than a Type II supernova at an equal distance. Let’s see how frequently a Type Ia supernova explosion might occur close enough to the Earth to be considered dangerous.

If the stars were uniformly distributed in the spheroid ...

The progenitors of Type Ia supernovae are thought to be older stars of low mass: white dwarfs in a close binary system. We believe that most such systems may be in the spherical component of our galaxy called the spheroid, which incorporates the so-called “halo”, extending to great distances (there are some indications that a fraction of the Type Ia progenitors may live in the disk, but that fraction isn’t known with any precision).

Now, if the spheroid were filled uniformly with stars, then it would be easy to compute the fraction of stars which might be close enough to the Earth to present a danger. We would simply compare the total volume of the spheroid to the volume of a sphere with the “killing radius” κ:

volume of spheroid V_{galaxy} = \frac{4}{3}πL^{3}
volume of danger zone V_{kill} = \frac{4}{3}πκ^{3}

where L is the total radius of the spheroidal component of the Milky Way. So the fraction of all stars close enough to be dangerous would be

dangerous fraction = \frac{V_{kill}}{V_{galaxy}}

= \frac{κ^{3}}{L^{3}}

The halo extends very far from the center of our galaxy; let’s take its maximum radius to be

L = 100 kpc

(note that this implies the Magellanic Clouds orbit within the halo). The “killing radius” of a Type Ia supernova is uncertain, but a reasonable guess is about 50 parsecs. The dangerous fraction would then be

dangerous fraction = 10^{−15}

which is miniscule – much smaller than the comparable numbers for Type II supernovae.

However, stars in the spheroid of the Milky Way are not uniformly distributed. They are concentrated very strongly towards the central regions. Since our solar system is not very far from the center, this means that the true fraction should be larger.

If the stars were centrally concentrated in the spheroid ...

Observations of stars belonging to the spheroid of our galaxy show that they are more densely packed near the galactic center. A reasonable approximation for the density is a power law, with a central core:

density of spheroid ρ(r) = B \frac{b^{3}}{b^{3} + L^{3}}

where B is the central stellar density, b is the radius of the core (inside of which the density is uniform), and L is the total radial extent of the spheroid.

In order to find the “dangerous fraction,” we must first calculate the total number of stars in the spheroid for this model. This is an integration of density times volume:

N_{tot} = ∫_{0}^{L} 4πr^{2}ρ(r)dr

= 4πB ∫_{0}^{L} \frac{b^{3}r^{2}}{(b^{3} + r^{3})}dr

= 4πBb^{3} ∫_{0}^{L} \frac{r^{2}}{(b^{3} + r^{3})}dr

At this point, we make the substitution x = r^{3} which yields

\begin{aligned}N_{tot} & = 4πBb^{3}(\frac{1}{3} ln(b^{3} + r^{3}))_{0}^{L} \\ & = \frac{4}{3}πBb^{3}[ln(b^{3} + L^{3}) − ln(b^{3})] \end{aligned}

Let us assume that the “killing radius” κ is small compared to the Sun’s distance from the center of the galaxy d:

κ ≪ d

so that we can approximate the stellar density throughout the “killing sphere” as uniform, and equal to the density ρ(d) at the Sun’s position. Then the number of stars within this “killing radius” is

\begin{aligned}N_{kill} & = \frac{4}{3}πκ^{3}ρ(d) \\ & = \frac{4}{3}πκ^{3}B(\frac{d^{3}}{b^{3} + d^{3}} ) \end{aligned}

Under these assumptions, the fraction of all spheroid stars which are close enough to be dangerous (should they explode as Type Ia supernovae) would be

dangerous fraction = \frac{N_{kill}}{N_{tot}}

= \frac{\frac{4}{3}πκ^{3}B(\frac{d^{3}}{b^{3}+d^{3}} )}{\frac{4}{3}πBb^{3}[ln(b^{3} + L^{3}) − ln(b^{3})]}

= \frac{κ^{3}(\frac{d^{3}}{b^{3}+d^{3}} )}{b^{3}[ln(b^{3} + L^{3}) − ln(b^{3})]}

Given parameters for our model of the spheroid, we can evalulate this expression to find the “dangerous fraction.” Using again the parameters listed in Gates et al. (ApJ 500, 145 [1998]; also available as astro-ph 9711110), we have

core radius b = 2 kpc
Sun′s distance from center of galaxy d = 8 kpc
radial extent of spheroid L = 100 kpc

and adopting a “killing radius” of κ = 50 parsecs, we find

dangerous fraction = 1.3 × 10^{−6}

Note that this is much, much larger than the fraction we calculated for a uniform distribution of stars in the spheroid. This fraction is also considerably larger than that for Type II supernovae (calculated in Section 2 above). The main reason for the difference is the larger “kill radius” we adopted for Type Ia explosions: 50 parcsecs, versus only 10 parsecs for a Type II explosion.

How frequently do “dangerous” Type Ia supernovae occur?

Once again, we can calculate the typical interval between explosions within the “danger zone.” We need to divide the rate of explosions in our galaxy by the fraction of stars which are close enough to be hazardous.

Above, I adopted a rate of one Type II supernova per 100 years in our galaxy. Type Ia supernovae are less common in spiral galaxies; Cappellaro et al. (A&A 322, 431 [1997]) find that they occur about half as often in Sb galaxies like the Milky Way. So, if we adopt a rate of 1 Type Ia supernova every 200 years in our galaxy, we find an interval between “dangerous Type Ia supernovae” of

interval between Type Ia SN = \frac{200 years}{fraction}

= \frac{200 years}{1.3 × 10^{−6}}

= 150, 000, 000 years

Formally, this interval is smaller than the one we derived between “dangerous Type II supernovae.” However, given our sweeping approximations of the stellar distributions in the disk and the halo, and the uncertainties in our knowledge of the relative rates of supernova of different types in the Milky Way, it is probably safer to conclude that “dangerous” instances of each type of supernova occur equally often – to within an order of magnitude.

Conclusion

Even if one combines both types of supernovae, one is left with intervals of something like one hundred million years (or more) between explosions which might be close enough to endanger life on Earth.