Astrophyics topic 11, Slides of Astrophysics

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.] TOPIC 11 ASTROPHYSICS AND COSMOLOGY Gravity is one of the most obvious scientific phenomena and we have studied it for a very long time. When you drop a pen you can see gravity and its effects clearly and we know how to utilise these in very precise ways. In November 2014, the European Space Agency landed a probe on the comet 67P/Churyumov-Gerasimenko. This may not sound remarkable. We have landed probes on the Moon, Mars, other planets and moons of other planets. However, the size and mass of this comet means that the gravitational field is so weak that it would be difficult to create a centripetal force of attraction on the probe which would allow orbiting. The ESA had to ensure that the lander was travelling at less than one metre per second, and even then the lander bounced significantly. However, gravity remains one of the least well-understood areas of physics. There are a number of experiments currently ongoing to try and observe some of the theoretical explanations of how gravity works ona fundamental level. As yet, none have been conclusive, but as their sensitivity improves, scientists hope that we will be able to collect enough experimental evidence to confirm how gravity works. Gravity waves, quantum gravity and so-called strong gravity are current areas of interest to cutting-edge physicists. Solving algebraic equations (e.g. finding a satellite’s orbital height above the surface of the Earth) Changing the subject of a non-linear equation (e.g. finding the separation of objects using Newton’s universal law of gravitation) Sketching and interpreting relationships shown graphically (e.g. comparing sketch graphs of gravitational field strength and potential) Distinguishing between average and instantaneous rates of change (e.g. gravitational field strength as the rate of change of potential with distance) Weight and gravity Newton's laws of motion Gravitational potential energy Circular motion Coulomb's law Electric fields and electric potential How gravitational forces follow an inverse square relationship What is meant by a gravitational field How to calculate the strength of gravitational fields Gravitational potential Comparisons of electric and gravitational fields How gravity leads to the development of stars The production of elements in stellar nuclear fusion The gravitational field strength at the surface of a black hole The influence of gravity on the development of galaxies and the Universe as a whole Dark matter and its gravitational effects Gravitational lensing as evidence for black holes RAVITATIONAL FIELDS Newton was the first scientist to publish the equation that gives us the gravitational force between two masses, m, and m,, which are separated by a distance, r, between their centres of gravity: Gmym:; poems Pp where Gis the gravitational constant, G = 6.67 x 107!! Nm?kg™. WORKED EXAMPLE 2 (a) What force will exist between two neutrons in the Crab Nebula which are two metres apart? Fe Grin iz 6.67 «1071 x 1.67 x 107 «1.67 x 1077 2 F=4.65x10°N (b) How quickly will each accelerate towards the other (ignoring the effects of all other particles)? Coma _ 4.65 x 10° ~ 1.67 x 107 a=278x10%ms* (c) Why is it difficult to calculate how long they would take to reach each other? The acceleration is inversely proportional to the square of the distance between them. Therefore, the acceleration is constantly increasing as they get closer together. Our usual equations of motion require uniform acceleration. WEIGHING THE EARTH Newton thought about how the Moon knows that the Earth is there, when he suggested that gravity keeps the Moon in orbit around the Earth. The answer to Newton's problem is still not fully understood by scientists, but his formula can be used to great effect. It was of vital importance to NASA scientists when they made calculations in order to send the six Apollo Moon missions about fifty years ago. Even before space travel, it was possible to use data about the Moon's orbit to work out the mass of the Earth. The time period of the Moon’s orbit around the Earth is 27.3 days, or T = 2.36 x 10°s. The average orbital radius for the Moon is 384 000 km, or r= 3.84 x 10°m. From these data, we can calculate the mass of the Earth: Gmemy, Gravitational attraction between Moon and Earth, F'= wa r myv? 7 Gravity is the cause of the centripetal force, so these are equal: Centripetal force required to keep Moon in orbit, F= , eet Mat EXAM HINT Calculations about the properties of Gmc y2 satellites usually need us to put the por gravitational force expression equal to the centripetal force expression and then re-arrange. r 2 Gmyma _ 72 a r 11A.1 GRAVITATIONAL FORCES 171 The speed of the Moon comes from the time it takes to orbit: 2ar yoo T _ 27 x 3.84 x 108 2.36 x 10° v=1022ms" _ 3.84 x 108 x 1022? Mm, = - 6.67 x 1071! mz = 6.01 x 10% kg. @|> A. fig D_ Weighing the Earth can be done by observing the Moon LEARNING TIP Due to a wide variety of measuring devices and techniques, and rounding errors, you may see slight variations in the quoted mass of the Earth. They should all be close to 6 x 10%*kg. ECKPOINT INTERPRETATION 1.) Explain why the weight of an object on the Earth is found by multiplying its mass in kilograms by 9.81. 2. Calculate the gravitational force between two T” particles in deep space, if they are 8 metres apart. The mass of a neutral pion is 2.40 x 10-8 kg. 3. Calculate the gravitational force between the Earth and the Moon if the Moon's mass is 7.35 x 10”? kg. 4. Calculate the average distance of the Earth from the Sun, if the mass of the Sun is 2.0 x 102°kg. 5. Draw a sketch graph to show how the magnitude of the gravitational attraction, F, between two masses varies with the . _ 1 inverse square of the separation: —. r SUBJECT VOCABULARY accrete to grow slowly by attracting and joining with many small pieces of rock and dust, due to the force of gravity gravitational field a region of spacetime which is curved. This curvature will cause particles to experience an accelerating force uniform acceleration acceleration that always has the same value; constant acceleration orbit the curved path that a planet, satellite or similar object takes around another object in space 11A 2 GRAVITATIONAL FIELDS RNING OBJECTIVES lM Derive and use the equation g = ror the gravitational field strength due to a point mass. HM Use the equation Vera = - hile radial gravitational field. ™@ Compare electrical and gravitational fields. RADIAL FIELDS ‘A. fig A The radial gravitational field around a point mass. Any mass will generate a gravitational field, which will then exert a force on any mass within the field. As gravity is always attractive, the field produced by a point mass will be radial in shape and the field lines will always point towards the mass (fig A). GRAVITATIONAL FIELD STRENGTH The radial field produced by a point mass naturally has its field lines closer together nearer the mass, as a result of its geometry (see fig A). This means that the strength of the field decreases with increasing distance from the mass causing it. The decrease is significant. In outer space, where it is the furthest possible from a galaxy or other particles, there are regions where there is almost no gravity. This can be explained mathematically by the formula which tells us the strength of a gravitational field at a certain distance, r, from amass, M. We have already seen the force on a mass, m, caused by a gravitational field is F= mg. Also, the gravitational force on a mass, m, because of another mass, M, is given by Newton's expression: _ GMm Fi 2 These two expressions are calculating the same force, so must themselves be equal: pe Mm. pm ie. the field strength is independent of the object being acted upon. SPECIFICATION REFERENCE 5.6.157 5.6.158 5.6.159 5.6.160 WORKED EXAMPLE 1 Calculate the gravitational field strength caused by the Earth at the distance of a geostationary satellite, which orbits at a height, h, of 35 800 km above the surface of the Earth. The radius of the Earth, Re, is 6400 km. From Section 11A.1 we have the mass of the Earth as 6.01 x 10%*kg. Distance from the Earth's centre: r= Re+h=42200km = 4.22 x 107m GM ge _ 6.67 x 101! x 6.01 x 10%4 7 (4.22 x 10" og =0.23Nkg™ GRAVITATIONAL POTENTIAL Potential energy is the stored energy that an object has due to its position. The potential at a point in any type of field is the amount of energy which is needed to get to that position in the field for any object which is affected by the field. Therefore for a gravitational field it is expressed as an amount of energy per unit mass (J kg‘), as mass is what is affected by the field. The definition of gravitational potential is the amount of work done per unit mass to move an object from an infinite distance to that point in the field. As gravitational fields are always attractive, objects will always gain energy on moving into a point in the field, and so gravitational potential is always a negative quantity. Gravitational potential can be calculated at a distance r from a mass M from the equation: cM Vea = WORKED EXAMPLE 2 Calculate the gravitational potential caused by the Earth, at the distance of a geostationary satellite, which orbits at a height, h, of 35 800 km above the surface of the Earth. The radius of the Earth, Re, is 6400 km. r=Re+h=42200km = 4.22 x 107m GM Veray =

1 8 is nebula iE a \-2 40000 20000 10000 5000 2500 Temperature, T/K A. fig F The Sun’ life cycle will move it around the H-R diagram CHECKPOINT CRITICAL THINKING The mass of the Sun is 2.0 x 10*°kg. If we assume that it is completely composed of protons, how many protons is this? If the Sun fuses all of its protons during its estimated nine billion year lifetime, how many protons are undergoing nuclear fusion every second? INTERPRETATION 2.) Draw a flow chart showing the life cycles of: (a) a star which starts out with six times the mass of the Sun (b) a star which starts out with twice the mass of the Sun. 3. Calculate the gravitational field strength at the surface of a black hole, that has five times the mass of the Sun and a diameter of 10cm. How does this compare to the Earth's gravitational field strength? Ez INTERPRETATION, INTEGRITY 4. (a) From memory, make a quick sketch of the Hertzsprung- Russell diagram and mark on it the path taken by a star with the same mass as our Sun as it develops through the various stages of its life cycle. (b) Mark on your sketch the life cycle path taken by a blue supergiant if it starts out with a mass that is 8 times that of the Sun. 5. Why do the nuclear fusion processes within stars usually produce elements with a mass number which is a multiple of four, such as carbon-12, oxygen-16 and silicon-20? 6. (a) Why can't the nuclear fusion in stars produce elements higher in the periodic table than iron-56? (b) How do heavier elements than iron-56 exist, when the Big Bang produced an initial Universe composed of only hydrogen and helium? SUBJECT VOCABULARY Hertzsprung-Russell diagram a plot of stars, showing luminosity (or absolute magnitude) on the y-axis, and temperature (or spectral class) on the x-axis main sequence a diagonal line from top left to bottom right of a Hertzsprung-Russell diagram which marks stars that are in a generally stable phase of their existence main sequence star a stable star whose core performs hydrogen fusion and produces mostly helium protostar the mass of dust and gas clumping together under the force of gravity prior to the start of nuclear fusion in its core, which will go ‘on to become a star neutron star one of the possible conclusions to the life of a large mass star; small and very dense, composed of neutrons black hole one of the possible conclusions to the life of a large mass star; a region of space-time in which the gravity is so strong that it prevents anything from escaping, including EM radiation planetary nebula the remains of an explosion at the end of the life cycle of a low-mass star; material which may eventually join together into new planets black dwarf the final stage of the life cycle of a small mass star, when nuclear fusion has ceased and it has cooled so that it no longer emits visible light supernova the explosion of a large mass star at the end of its lifetime, when it becomes extremely unstable 11B 3 DISTANCES TO THE STARS ‘SPECIFICATION REFERENCE 5.6.164 5.6.165 5.6.166 EARNING OBJECTIVES ™@ Determine astronomical distances using trigonometric parallax. ™ Use the equation for the intensity of a star, | = = And? ™@ Measure astronomical distances using standard candles. BIG DISTANCE UNITS Astronomical distances are very large. Our nearest neighbour is the Moon, and even that is nearly 400 000 000 m away. The Sun is 150 000 000 000 m away, and the distance to the orbit of Neptune is 4500 000 000 000 m. Measuring across space generates very large values. Using standard form notation helps with this, but astronomers have defined a number of alternative distance units to cut down the magnitudes of the numbers involved. You have probably already heard of the light year. This is the distance that light can travel in one year, which is about 10'°m, We also use the astronomical unit (AU), which is the radius of the Earth’s orbit around the Sun: 1 AU = 1.5 x 10''m. Much of our understanding of the structure and formation of galaxies depends on being able to measure the distances to stars accurately. Astronomers have developed a number of techniques for doing this, but all have their limitations. These limitations can be overcome, or at least minimised, by comparing the results from the different techniques on the same star, and refining the techniques to improve accuracy. LEARNING TIP Some of the techniques explained in this section are used on individual stars, and some are used on large groups of stars, such as galaxies or star clusters. TRIGONOMETRIC PARALLAX To measure the distance to relatively close stars, astronomers use a method which is commonly used in surveying, known as trigonometric parallax. As the Earth moves around the Sun, a relatively close star will appear to move across the background of more distant stars. This optical illusion is used to determine the distance of the star. The star itself does not move significantly during the course of the observations. To determine the trigonometric parallax you measure the angle to a star, and observe how that changes as the position of the Earth changes. We know that in six months the Earth will be exactly on the opposite side of its orbit, and therefore will be two astronomical units from its location today. Using observations of the star which is to be measured against a background of much more distant stars, we can measure the angle between the star and the Earth in these two different positions in space, six months apart. As we know the size of the Earth’s orbit, geometry allows calculation of the distance to the star.