�� <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"> <html xmlns="http://www.w3.org/1999/xhtml"> <head> <meta http-equiv="Content-Type" content="text/html; charset=utf-8" /> <META NAME="ROBOTS" CONTENT="INDEX, FOLLOW"> <meta name="description" content="This website offers teachers and students authentic mathematics problems based upon NASA press releases, mission science results, and other sources. All problems are based on STEM, common core standards and real-world applications for grades 3 to 12 and beyond."> <title>Welcome to Space Math @ NASA !</title> <link rel="stylesheet" type="text/css" href="css/reset.css" media="projection, screen" /> <link rel="stylesheet" type="text/css" href="css/base.css" media="projection, screen"/> <link rel="stylesheet" type="text/css" href="css/dropdown.css" media="projection, screen"/> <link rel="stylesheet" type="text/css" href="css/BlueTheme.css" media="projection, screen"/> <script src="js/jquery-3.5.1.min.js" type="text/javascript" ></script> <script language="javascript" id="_fed_an_ua_tag" src="https://dap.digitalgov.gov/Universal-Federated-Analytics-Min.js?agency=NASA"></script> <!--[if IE 6]> <link rel="stylesheet" type="text/css" href="css/ie6.css" media="projection, screen"/> <script src="js/DD_belatedPNG_0.0.8a-min.js" type="text/javascript" ></script> <script> DD_belatedPNG.fix('div#bannerImg, li.tab1-on, li.tab1-off, li.tab-on, li.tab-off, #footer-signature, img'); </script> <![endif]--> </head> <body> <div id="page-wrapper"> <div id="page"> <div id="nasaHeader"> <div> <a href="http://www.nasa.gov" class="imageLink"><img src="images/nasa_header_logo1.gif" alt="NASA Logo, National Aeronautics and Space Administration" width="140" height="98" /></a> </div><!--class="nasaLogo"--> <div id="headerTxtWrapper"> <h1><a href="http://www.nasa.gov">National Aeronautics and Space Administration</a></h1> <h2><a href="http://www.nasa.gov/goddard/">Goddard Space Flight Center</a></h2> </div><!--headerTxtWrapper--> <div id="searchbox"> <h3>Search:</h3> <form action="searchresult.html" method="post" id="search" name="search"> <input type="text" name="srchField" id="srchField" value="" title="srchField" size="10" class="flt_left"> <input width="49" height="22" type="image" alt="Search" src="images/search_go.jpg" onclick="this.form.submit();" class="flt_left"> <input type="hidden" value="search.quick" id="fuseaction" name="fuseaction"> </form> <div class="divClear"></div> <p> <a href="http://fpd.gsfc.nasa.gov/">Flight Projects</a> <span class="listdevider">| </span> <a href="http://sciences.gsfc.nasa.gov/sed/index.cfm?fuseAction=home.main&&navOrgCode=600&navTab=nav_about_us">Sciences and Exploration</a> </p> </div> </div><!--id = nasaHeader --> <div class="hidden"> <a href="#skipping" title="Skip Navigation" accesskey="2">Skip Navigation (press 2)</a> </div> <div id="banner"> <!-----------------------> <!--TOP MENU GOES HERE --> <!-----------------------> <div id="tabs"> <ul id="menu"> <li id="tab0" class="tab-on active"><a href="media.html" >STEM Modules</a></li> <li id="tab1" class="tab-on active"><a href="ILabs.html" >Interactive Spreadsheets</a></li> <li id="tab2" class="tab-on active"><a href="books.html" >Problem Books</a></li> <li id="tab3" class="tab-on active"><a href="SpaceMath.html" >Home</a></li> </ul> </div><!--TABS SECTION END HERE--> <!--BANNER IMAGE GOES HERE--> <div id="bannerImg"> <a href="#"><img src="images/nasa-bnr-empty.png" alt="Space Math at NASA" width="900"></a> </div><!--id=bannerImg--> </div><!--id=banner--> <div class="divClear"></div> <div id="topMenu"> <ul id="menu"> <li id="tab6" class="active"><a href="news.html" >Math in Press Releases</a></li> <li id="tab7" class="active"><a href="mission.html">Math by NASA Mission</a></li> <li id="tab4" class="active"><a href="grade.html">Math by Grade Level</a></li> <li id="tab5" class="active"><a href="space.html">Math by Space Topic</a></li> <li id="tab8" class="active"><a href="Standards.html">NGSS and CCSS-M</a></li> </ul> </div><!--id="topMenu"--> <!-----------------------> <!-----Top menu block ends here ---------> <!---------------------------------------> <div id="content"> <a name="skipping" id="skipping"></a> <div id="content-wrapper"> <div class="maincontent" id="TwoColumn"> <h2>Mathematics Problems about Black Holes</h2> <p> 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. <div class="full-width-content-item"> <! ****************************************************************************************** insert text below> <h2>The Event Horizon</h2> <p> <b>Problem 613: Measuring the Speed of Gas Near a Black Hole </b> <br>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] <A href="blackh/10Page72.pdf">(PDF)</a></p> <p> <b>Problem 475: Exploring Tidal Forces: Black holes and Saturns rings </b> <br> 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] <A href="blackh/8Page46.pdf">(PDF)</a> <P> <b>Problem 427: A Black Hole - Up Close </b> <br>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] <A href=" blackh/7Page86.pdf">(PDF)</a> </p> <P> <b>Problem 426: Black Holes - Hot Stuff! </b> <br> 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] <A href=" blackh/7Page79.pdf">(PDF)</a> </p> <P> <b>Problem 425: Exploring a Full-sized Black Hole </b> <br> 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] <A href=" blackh/7Page30.pdf">(PDF)</a> </p> <P> <b>Problem 424: Exploring Black Holes </b> <br>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] <A href=" blackh/7Page28.pdf">(PDF)</a> </p> <P> <b>Problem 423: The Moon as a Black Hole </b> <br> 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] <A href=" blackh/7Page29.pdf">(PDF)</a> </p> <p> <b>Problem 421: The Lense-Thirring Effect Near the Sun and a Neutron Star</b> 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] <A href=" blackh/7Page84.pdf">(PDF)</a> <p> <b>Problem 420: Gravity Probe B: Testing Einstein again!</b> 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] <A href=" blackh/7Page83.pdf">(PDF)</a> <p> <b>Problem 418: Supercomputers: Modeling colliding neutron stars!</b> 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] <A href=" blackh/7Page82.pdf">(PDF)</a> <p> <b>Problem 417: Estimating the Size and Mass of a Black Hole</b> 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] <A href=" blackh/7Page81.pdf">(PDF)</a> <p> <b>Problem 389: Estimating the diameter of the SN1979c black hole </b> 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] <A href=" blackh/7Page53.pdf">(PDF)</a> <p> <P><FONT face="Ariel" color=black size=4></font><A href=" blackh/4Page27.pdf"><FONT face="Ariel" size=4 color=black><b>Problem 128 : Event Horizons</b> </FONT></A><FONT face="Ariel"> 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. <FONT face="Ariel" color=black size=2>[Grade: 8 - 12 | Topics:Simple linear equations; scientific notation] </FONT></font> </p> <P><FONT face="Ariel" color=black size=4> </font><A href=" blackh/4Page36.pdf"><FONT face="Ariel" size=4 color=black><b>Problem 145: What's Inside a Black Hole? </b> </FONT></A><FONT face="Ariel"> 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. <FONT face="Ariel" color=black size=2>[Grade:9 - 11 | Topics: Scientific Notation; distance; time calculations; algebra] </FONT></font> </p> <P><A href=" blackh/4Page38.pdf"><FONT face="Ariel" size=4 color=black> <b> Problem 147: Light Fade-out</b> </FONT></A><FONT face="Ariel"> Students calculate how long it takes light to fade away as an object falls into a black hole. <FONT face="Ariel" color=black size=2>[Grade: 9 - 11 | Topics: Scientific Notation; exponential functions] </FONT></font> </p> <P><A href=" blackh/4Page34.pdf"><FONT face="Ariel" size=4 color=black> <b>Problem 140: Falling Into a Black Hole</b> </FONT></A><FONT face="Ariel"> 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. <FONT face="Ariel" color=black size=2>[Grade: 7 - 10 | Topics:Scientific Notation; Working with equations in one variable to first and second power.] </FONT></font> </p> <P><A href=" blackh/4Page33.pdf"><FONT face="Ariel" size=4 color=black><b>Problem 138: The Intense Gravity of a Black Hole</b> </FONT></A><FONT face="Ariel"> 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. <FONT face="Ariel" color=black size=2>[Grade: 7 - 10 | Topics:Scientific Notation; Working with equations in one variable to first and second power.] </FONT></font> </p> <P><FONT face="Ariel" color=black size=4> </font><A href=" blackh/4Page30.pdf"><FONT face="Ariel" size=4 color=black>,b>Problem 132: Black Holes and Time Distortion</b> </FONT></A><FONT face="Ariel"> Students learn about how gravity distorts time near a black hole and other massive bodies. <FONT face="Ariel" color=black size=2>[Grade: 8 - 12 | Topics:Simple linear equations; scientific notation] </FONT></font> </p> <P><FONT face="Ariel" color=black size=4> </font><A href=" blackh/4Page29.pdf"><FONT face="Ariel" size=4 color=black><b>Problem 130: Gravity and Time Distortion Near Earth</b> </FONT></A><FONT face="Ariel"> Students learn about how gravity distorts time and causes problems even for the Global Positioning System satellites and their timing signals. <FONT face="Ariel" color=black size=2>[Grade: 8 - 12 | Topics:Simple linear equations; scientific notation] </FONT></font> </p> <h2>Energy Emitted by Infalling Matter</h2> <p> <b>Problem 390: X-rays from hot gases near the black hole SN1979c</b> 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] <A href=" blackh/7Page54.pdf">(PDF)</a> <p> <P><FONT face="Ariel" color=black size=4> </font><A href=" blackh/4Page37.pdf"><FONT face="Ariel" size=4 color=black> <b>Problem 146: Black Hole Power - I</b> </FONT></A><FONT face="Ariel"> 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. <FONT face="Ariel" color=black size=2>[Grade: 9 - 11 | Topics:Scientific Notation; Spherical shells; density; power] </FONT></font> </p> <P><A href=" blackh/6Page96.pdf"><FONT face="Ariel" size=4 color=black><b>Problem 291: Black Hole Power - II</b></FONT></A><FONT face="Ariel"> 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. <FONT face="Ariel" color=black size=2>[Grade: 9-12 | Topics: Scientific Notation; evaluating simple formulas; unit conversion] </FONT></font> </p> <P><FONT face="Ariel" color=black size=4> </font><A href=" blackh/4Page31.pdf"><FONT face="Ariel" size=4 color=black><b>Problem 136: Black Hole Power - III</b> </FONT></A><FONT face="Ariel"> 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. <FONT face="Ariel" color=black size=2>[Grade: 7 - 10 | Topics:Scientific Notation; Working with equations in one variable to first and second power.] </FONT></font> </p> <P><FONT face="Ariel" color=black size=4> </font><A href=" blackh/4Page35.pdf"><FONT face="Ariel" size=4 color=black> <b>Problem 142: Accretion Disks</b> </FONT></A><FONT face="Ariel"> 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. <FONT face="Ariel" color=black size=2>[Grade: 7 - 10 | Topics:Scientific Notation; Working with equations in one variable to first and second power.] </FONT></font> </p> <h2>The Mass of a Black Hole</h2> <P><FONT face="Ariel" color=black size=4> </font><A href=" blackh/4Page32.pdf"><FONT face="Ariel" size=4 color=black><b>Problem 137: Black Hole Mass</b> </FONT></A><FONT face="Ariel"> 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. <FONT face="Ariel" color=black size=2>[Grade: 7 - 10 | Topics:Scientific Notation; Working with equations in one variable to first and second power.] </FONT></font> </p> <! ****************************************************************************************** end of inserted text> </div><!--class="full-width-content-item">--> <div class="divClear"></div> </div><!--id = maincontent--> <!-- MAIN CONTENT ENDS HERE --> <!--RIGHT SIDEBAR BEGINS HERE--> <div id="rightsidebar"> <h2>Additional Resources</h2> <p> <img src="images/BlackHolessm.jpg"><br> <A HREF="http://www.nasa.gov/audience/foreducators/topnav/materials/listbytype/Black_Hole_Math.html">Black Holes (2008)</a> - [7.5 Mby, 53 Problems] A revised and expanded introduction to the basic properties of black holes using elementary algebra and geometry. Students calculate black hole sizes from their mass, time and space distortion, and explore the impact that black holes have upon their surroundings. </font></td></tr> <p> </div><!--full-width-content-item--> <!-- RIGHT SIDEBAR ENDS HERE --> </div><!--id="content-wrapper"--> </div><!--id="content"--> <div class="divClear"></div> <div id="footer"> <div> <a href="http://www.nasa.gov" class="imageLink"><img src="images/nasa_header_logo1.gif" alt="NASA Logo, National Aeronautics and Space Administration" width="140" height="98" /></a> </div><!--class="nasaLogo"--> <div id="footer-signature"> <a href="http://www.nasa.gov/goddard/" class="imageLink"><img src="images/goddardsignature2.png" height="68" /></a> </div><!--id="footer-signature--> <div class="nasafootlinks"> <ul> <li>Author: <a href="mailto:Sten.F.Odenwald@nasa.gov">Dr. Sten Odenwald</a></li> <li>NASA Official : <a href="mailto:Michelle.Thaller@nasa.gov">Dr. Michelle Thaller</a></li> </ul> </div><!--nasafootlinks--> <div class="nasafootlinks" id="footerRight"> <ul> <li> <a target="_blank" href="http://www.nasa.gov/about/highlights/HP_Privacy.html">Privacy Policy &amp; Important Notices</a> </li> <li> <a href="">Contact Us</a> </li> </ul> </div><!--nasafootlinks--> </div><!--id=footer--> </div><!--id="page"--> </div><!--id = page-wrapper--> </body> </html>