# National Aeronautics and Space Administration

## Mathematics Problems about Black Holes

Black holes are completely described in terms of their matter and how fast they are spinning. Although this makes them among the simplest objects in the universe, they continue to amaze us because of the many peculiar things that happen to space and time near them. Astronomers have considered them the most obvious explanation for the most energetic phenomena that we can study in the universe because of their extreme concentration of gravity. Because matter moves more quickly in strong gravity fields, black holes are powerful engines for accelerating matter to very high speeds. This produces high temperature gas that can emit enormous amounts of x-ray and gamma-ray light; a common feature of many of the most exotic objects that we see across the universe.

## The Event Horizon

Problem 613: Measuring the Speed of Gas Near a Black Hole
Students use a graph of intensity and time to estimate thhe orbit period of matter around a black hole. [Grade: 6-8 | Topics: time; graph analysis] (PDF)

Problem 475: Exploring Tidal Forces: Black holes and Saturns rings
Students use the equation for tidal disruption to explore the stability of a star encountering a black hole, and a satellite of Saturn. Why are there no large satellites of Saturn inside the ring system? [Grade: 9-12 | Topics: Evaluating equations; scientific notation] (PDF)

Problem 427: A Black Hole - Up Close
Students explore how the color of a light bulb changes as it gets close to a black hole, demonstrating the principle of the gravitational 'red shift'. [Grade: 9-12 | Topics: Evaluating an equation with one variable; square roots; metric units; nanometers] (PDF)

Problem 426: Black Holes - Hot Stuff!
Students explore the temperature of matter falling into a black hole using a simple equation to calculate the gas temperature at different distances. [Grade: 9-12 | Topics: Evaluating an equation with one variable; fractional exponents] (PDF)

Problem 425: Exploring a Full-sized Black Hole
Students explore how the speed of an orbiting satellite changes if it were near a black hole with 5 times the mass of our Earth. [Grade: 6-8 | Topics: Evaluating an equation with one variable; square roots; speed = distance/time; circumference of a circle] (PDF)

Problem 424: Exploring Black Holes
Students compare the sizes of the planets in our solar system if they were actually black holes. They use a compass and metric ruler to create circles that are the actual sizes of the 'black hole' planets. [Grade: 3-5 | Topics: working with a compass and metric ruler] (PDF)

Problem 423: The Moon as a Black Hole
Students draw a life-sized model of the Earth and Moon as two black holes to explore the actual sizes of these exotic astronomical bodies. [Grade: 3-5 | Topics: Working with a compass; metric ruler] (PDF)

Problem 421: The Lense-Thirring Effect Near the Sun and a Neutron Star Students work with a formula for the Lense-Thirring Effect and estimate how large it will be in orbit around our sun, and in the intense gravitational field of a dense neutron star. [Grade: 9-12 | Topics: algebra; scientific notation, angular measure] (PDF)

Problem 420: Gravity Probe B: Testing Einstein again! Students learn about the Lense-Thirring Effect, and calculate its magnitude near Earth's orbit using an algebraic equation with integer and fractional exponents. [Grade: 9-12 | Topics: algebra; scientific notation, angular measure] (PDF)

Problem 418: Supercomputers: Modeling colliding neutron stars! Students use a series of time-lapse images calculated using a supercomputer to determine the speed of collision of two neutron stars, and whether they will form a black hole afterwards. [Grade: 8-10 | Topics: distance=speed x time; scale model; triangle and circle geometry; circumference] (PDF)

Problem 417: Estimating the Size and Mass of a Black Hole Students use a simple formula to estimate the size of a black hole located 3.8 billion light years from Earth, recently studied by NASA's Chandra and Swift satellites. [Grade: 8-10 | Topics: distance=speed x time] (PDF)

Problem 389: Estimating the diameter of the SN1979c black hole Students use simple equations to learn about the various definitions for the sizes of black holes in terms of their event horizons, last photon orbit, and last stable particle orbit radii, and apply this to the recently discovered 'baby' black hole in the galaxy M-100 [Grade: 6-8 | Topics: evaluating linear functions; integer math; metric units] (PDF)

Problem 128 : Event Horizons Students learn about the most basic component to a black hole - the event horizon. Using a simple formula, and scientific notation, they examine the sizes of various kinds of black holes. [Grade: 8 - 12 | Topics:Simple linear equations; scientific notation]

Problem 145: What's Inside a Black Hole? Students work with the Pythagorean Theorem for black holes and investigate what happens to space and time on the other side of an Event Horizon. [Grade:9 - 11 | Topics: Scientific Notation; distance; time calculations; algebra]

Problem 147: Light Fade-out Students calculate how long it takes light to fade away as an object falls into a black hole. [Grade: 9 - 11 | Topics: Scientific Notation; exponential functions]

Problem 140: Falling Into a Black Hole If you fell into a black hole, how fast would you be traveling? Students use a simple equation to calculate the free-fall speed as they pass through the event horizon. [Grade: 7 - 10 | Topics:Scientific Notation; Working with equations in one variable to first and second power.]

Problem 138: The Intense Gravity of a Black Hole Tidal forces are an important gravity phenomenon, but they can be lethal to humans in the vicinity of black holes. This exercise lets students calculate the tidal acceleration between your head and feet while standing on the surface of Earth...and falling into a black hole. [Grade: 7 - 10 | Topics:Scientific Notation; Working with equations in one variable to first and second power.]

,b>Problem 132: Black Holes and Time Distortion Students learn about how gravity distorts time near a black hole and other massive bodies. [Grade: 8 - 12 | Topics:Simple linear equations; scientific notation]

Problem 130: Gravity and Time Distortion Near Earth Students learn about how gravity distorts time and causes problems even for the Global Positioning System satellites and their timing signals. [Grade: 8 - 12 | Topics:Simple linear equations; scientific notation]

## Energy Emitted by Infalling Matter

Problem 390: X-rays from hot gases near the black hole SN1979c Students use two functions to estimate the size of a black hole from the gas emitting x-rays which is flowing into it. [Grade: 8-10 | Topics: Functions; substitution; evaluation] (PDF)

Problem 146: Black Hole Power - I Students calculate how much power is produced as matter falls into a rotating and a non-rotating black hole including solar and supermassive black holes. [Grade: 9 - 11 | Topics:Scientific Notation; Spherical shells; density; power]

Problem 291: Black Hole Power - II Students use a simple formula to calculate how much power is produced by black holes of various sizes as they absorb matter from nearby stars and gas clouds. [Grade: 9-12 | Topics: Scientific Notation; evaluating simple formulas; unit conversion]

Problem 136: Black Hole Power - III Students explore how much energy is generated by stars and gas falling into black holes. The event horizon radius is calculated from a simple equation, R = 2.95 M, and energy is estimated from E = mc^2. [Grade: 7 - 10 | Topics:Scientific Notation; Working with equations in one variable to first and second power.]

Problem 142: Accretion Disks Matter that falls into a black hole heats up in an accretion disk, which can emit x-rays and even gamma rays visible from Earth. In this problem, students use a simple algebraic formula to calculate the temperature at various places in an accretion disk. [Grade: 7 - 10 | Topics:Scientific Notation; Working with equations in one variable to first and second power.]

## The Mass of a Black Hole

Problem 137: Black Hole Mass Students explore how Kepler's Third Law can be used to determine the mass of a black hole, or the mass of the North Star: Polaris. [Grade: 7 - 10 | Topics:Scientific Notation; Working with equations in one variable to first and second power.]