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Chapter 25: Problem 31

If the accretion rate at the Galactic center is \(10^{-3} \mathrm{M}_{\odot} \mathrm{yr}^{-1}\) and if it has remained constant over the past 5 billion years, how much mass has fallen into the center over that period of time? Compare your answer with the estimated mass of a possible supermassive black hole residing in the center of our Galaxy.

Short Answer

Expert verified
Approximately 5 million solar masses; comparable to smaller supermassive black holes.

Step by step solution

01

Understand the Given Information

We are given an accretion rate of \(10^{-3} M_{\odot} \text{ yr}^{-1}\) and a time period of 5 billion years (\(5 \times 10^9 \text{ years}\)). We need to find out how much mass has been accreted over this time period.
02

Calculate the Total Accreted Mass

The total accreted mass \(M_t\) can be calculated by multiplying the accretion rate \(R\) by the time period \(t\). \[ M_t = R \times t = 10^{-3} M_{\odot} \text{ yr}^{-1} \times 5 \times 10^9 \text{ years} \]Solve this equation to find \(M_t\).
03

Perform the Multiplication

Carrying out the multiplication:\[ M_t = 10^{-3} \times 5 \times 10^9 = 5 \times 10^6 M_{\odot} \]This means 5 million solar masses have fallen into the center over the period.
04

Compare with a Supermassive Black Hole

Typical masses for supermassive black holes in galactic centers range from a few million to several billion solar masses. In comparison, the 5 million solar masses we calculated is on the lower end but still comparable to smaller supermassive black holes.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Galactic center
The Galactic center refers to the central point of the Milky Way galaxy. This region is often bustling with activity due to gravitational interactions between numerous stars, gas clouds, and other astronomical objects. It's a fascinating and dynamic area of study because it houses many mysterious phenomena.
At the heart of our Galactic center lies possibly the most intriguing of all, a supermassive black hole. Discovering what's happening in the Galactic center is equivalent to unraveling the cosmic secrets of our universe. Studying it helps astronomers understand the behavior and evolution of galaxies, as well as providing insights into the forces shaping the cosmos.
  • Unique in its complexity with diverse celestial objects.
  • Source of intense radiation due to various astrophysical processes.
  • Main site for astronomical observations to learn about galactic evolution.
supermassive black hole
Supermassive black holes are cosmic giants residing in the centers of most, if not all, large galaxies. They boast masses ranging from millions to billions of solar masses. These celestial behemoths exert a powerful gravitational pull, influencing nearby stars and gas clouds. Unlike smaller black holes formed from collapsing stars, supermassive black holes have origins still under scientific investigation.
For instance, Sagittarius A* is the name of the object believed to be a supermassive black hole at the center of our Milky Way galaxy. It plays a crucial role in the dynamics of our Galactic center, acting like an anchor around which stars and gas revolve.
  • Supermassive black holes contribute to the stability of galaxies.
  • They are typically surrounded by an accretion disk of hot, glowing gas.
  • Offer a field of study full of unanswered questions in astrophysics.
accretion rate
The accretion rate is the rate at which matter is gathered by celestial bodies, such as black holes, from their surroundings. It's a vital concept in understanding how objects like supermassive black holes grow over time. When matter spirals into a black hole, it's typically part of an accretion disk, producing immense light and energy as it heats up during the process.
In our Galactic center scenario, the accretion rate provided is \(10^{-3}\) solar masses per year. This means that every year, matter equivalent to a thousandth of the Sun's mass is being absorbed. Over billions of years, this process can significantly increase the mass of a black hole.
  • Essential for calculating the growth of black holes and galaxies.
  • Helps in estimating how long certain astronomical objects have been accumulating mass.
solar masses
A solar mass, denoted as \(M_{\odot}\), is a standard unit of mass equivalent to the mass of our Sun. It is approximately \(2 \times 10^{30}\) kilograms. Astrophysicists use this unit when discussing masses of stars, galaxies, and black holes because it provides a convenient scale for comparison.
For the exercise at hand, the mass accreted over time in the Galactic center is measured in solar masses for ease and comprehensibility. This allows scientists to relate the accumulated mass to that of objects like the Sun, which are fundamental reference points in astronomy.
  • Facilitates comparisons of stars and other massive astronomical objects.
  • Used universally in astrophysical calculations and communications.
  • A critical unit for expressing the scale and scope of cosmic phenomena.

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Most popular questions from this chapter

The globular cluster IAU \(\mathrm{C} 0923-545\) has an integrated apparent visual magnitude of \(V=\) +13.0 and an integrated absolute visual magnitude of \(M_{V}=-4.15 .\) It is located \(9.0 \mathrm{kpc}\) from Earth and is 11.9 kpc from the Galactic center, just 0.5 kpc south of the Galactic midplane. (a) Estimate the amount of interstellar extinction between IAU \(\mathrm{C} 0923-545\) and Earth. (b) What is the amount of interstellar extinction per kiloparsec?

Using Eq. (52) for the density profile of the dark matter halo, show that (a) \(\rho_{\mathrm{NFW}} \propto r^{-1}\) for \(r \ll a\) and \(\rho_{\mathrm{NFW}} \propto r^{-3}\) for \(r \gg a\). (b) the integral of the mass from \(r=0\) to \(r \rightarrow \infty\) is infinite. $$\rho_{\mathrm{NFW}}(r)=\frac{\rho_{0}}{(r / a)(1+r / a)^{2}}$$

Estimate the Eddington luminosity of a black hole with the mass of Sgr \(A^{*}\). What is the ratio of the upper limit of the bolometric luminosity of Sgr \(\mathrm{A}^{*}\) to its Eddington luminosity?

The \(r^{-2}\) dependence of Coulomb's electrostatic force law allows the construction of Gauss's law for electric fields, which has the form $$\oint \mathbf{E} \cdot d \mathbf{A}=\frac{Q_{\mathrm{in}}}{\epsilon_{0}},$$ where the integral is taken over a closed surface that bounds the enclosed charge, \(Q_{\text {in }}\). Because Newton's gravitational force law also varies as \(r^{-2},\) it is possible to derive a gravitational Gauss's law." The form of this gravitational version is $$\oint \mathbf{g} \cdot d \mathbf{A}=-4 \pi G M_{\mathrm{in}},$$ where the integral is over a closed surface that bounds the mass \(M_{\mathrm{in}},\) and \(\mathrm{g}\) is the local acceleration of gravity at the position of \(d \mathbf{A}\). The differential area vector \((d \mathbf{A})\) is assumed to be normal to the surface everywhere and is directed outward, away from the enclosed volume. Show that if a spherical gravitational Gaussian surface is employed that is centered on and surrounds a spherically symmetric mass distribution, Eq. ( 56 ) can be used to solve for g. The result is the usual gravitational acceleration vector around a spherically symmetric mass.

(a) Estimate \(d \Theta / d R\) in the solar neighborhood, assuming that the Oort constants \(A\) and \(B\) are +14.8 and \(-12.4 \mathrm{km} \mathrm{s}^{-1} \mathrm{kpc},\) respectively. What does this say about the variation of \(\Theta\) with \(R\) in the region near the Sun? (b) If \(A\) and \(B\) were +13 and \(-13 \mathrm{km} \mathrm{s}^{-1}\) kpe, respectively, what would the value of \(d \Theta / d R\) be? What would this say about the shape of the rotation curve in the solar neighborhood?
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