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Journal Club Week 8 Answers

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John A. Johnson (2014), Warm planets orbiting cool stars, Physics Today, Vol. 67, Issue 3, p.31

The article can be accessed at https://physicstoday.scitation.org/doi/10.1063/PT.3.2309 You'll need to register (which is free) to access the article, but Physics Today are kindly offering access to their entire back catalogue to everyone for free during the COVID-19 pandemic.

You can also download the answers (which are also shown below) as a PDF. Some of the best student summaries are shown below. This week we wanted to find students who gave amazing answers to all three summary questions. This actually made our job even harder, because every week we're truly astounded by the quality of your responses. We genuinely apologise for not including more of you. It's hard to put into words what stood out about the three students below, but their understanding, and the way in which they put it down on 'paper' has a genuine clarity to it. Congratulations to Angelika, Yen Li and Phinehas (all of whom have been involved in Journal Club since week 1).

SUMMARY QUESTIONS - ANGELIKA

Why do you think red dwarfs are such an interesting area of research?

At first glance, the sky reveals that our galaxy consists of mostly sun-like stars, of up to 2 solar masses. However, our limited light sensitivity and the extreme faintness of the stars causes us to miss what is actually the most abundant class: red dwarfs. Firstly, the fact that red dwarfs are so common in our galaxy gives reason to study them as they provide more chance of finding a planet that hosts life. Secondly, red dwarfs make the search for Earth-like planets, and consequently potential extra-terrestrial life, easier. If orbiting a red dwarf, an Earth-sized planet is easier to detect because of the smaller ratio of star radius to planet radius, so the dip in the light curve due to a transiting exoplanet of Earth’s size will cause a much stronger dimming signal, easier to identify. The 'habitable zone', where the temperatures allow for liquid water to exist, is an indicator of possible life. Red dwarfs have lower luminosities (since luminosity is scaled with mass4) and temperatures which shrinks the orbital radii of orbiting planets, and the habitable zone is brought closer to the star. Planets residing in this zone have more chance of transiting their host-star, from Earth’s viewpoint, and the frequency of transits is greater. Red dwarfs make transits more likely to happen, easier to detect, and more frequent, making them an interesting target in the search for life. Furthermore, red dwarfs are very different from sun-like stars and can therefore reveal more about the formation of star systems different to our Solar System, providing us with new knowledge.

Discuss the different ways in which astronomers use spectral lines.

Spectral lines are the dark lines shown on star spectra, showing emission and absorption of light at certain frequencies. These can be used to detect exoplanets by measuring the precise movement of various host-stars. Rather than looking at ‘proper motion’ against the background of more distant ‘stationary’ stars, which can be subject to systematic errors and is only applicable for relatively close stars, the oscillating radial velocity component along Earth’s line of sight can give more precise measurements of a star’s movement. An exoplanet orbiting a star will cause it to ‘wobble’ due to its orbital motion and this ‘wobble’ in the observer’s line of sight (on Earth) can be detected as small Doppler shifts. Spectral lines will be shifted to the red end of the spectrum as the star oscillates away, and towards the blue end as it oscillates back towards Earth. Another way that spectral lines can be used is to deduce the size of a red dwarf. In 2010, a graduate at Cornell University discovered features in red dwarf IR spectra that could be used to yield a star’s temperature and metallicity (abundance of heavier than helium elements), to determine its radius. This in turn allows for the radius of orbiting exoplanets to be determined when they transit the star’s disc, leading scientists to be able to decide if that exoplanet is a potential host of life. For Sun-like stars, spectral lines can also reveal temperature, surface gravity, chemical composition, and physical properties, making spectral lines a key tool for astronomers in their research.

Compare the techniques of microlensing and transit photometry for finding exoplanets.

There are a few methods for finding exoplanets, two of which are microlensing and transit photometry. Microlensing is the exploitation of optical predictions: that light from a distant star can be gravitationally bent under the influence of a closer ‘lensing’ star, as is passes along the same line of sight to the star and observer on Earth, at a distance less than the system’s Einstein-ring radius (ring into which the distorted light from the distant star is focused onto). If the ‘lensing’ star has a planet orbiting it, that will cause a waning and waxing of the background star’s apparent brightness. Better yet, if that planet aligns correctly with its host planet near the Einstein ring radius, it will add to the lensing effect. The changes in brightness of the distant star can reveal characteristics of the planetary system of the ‘lensing’ star. In contrast, transit photometry involves the observation of varying brightness of the star as an orbiting planet transits its surface. This method is more appropriate for small stars, since the light curve of a planet follows 1/R2, so a star with half the Sun’s radius will cause a dimming signal 4 times stronger than if the same planet were to transit a sun-like star. Transit photometry is also a suitable method for red dwarfs, as their lower luminosities bring the ‘habitable zone’ closer, making exoplanets worth observing (those in this zone) closer to the star, undergoing more frequent transits.

SUMMARY QUESTIONS - YEN LI

Why do you think red dwarfs are such an interesting area of research?

The data from the Kepler mission, concerning the size distribution of planets orbiting red dwarfs, suggest that planets in red dwarf systems are more commonly found to have smaller radii (between 0.5 and 1.5 times the radius of Earth). The majority of the planets in discovered red dwarf systems are of comparable size to Earth. In addition, it was calculated that “there are about 0.5 Earth-sized planets per red-dwarf habitable zone”. Red dwarfs are thus an intriguing area of research as they offer an insight into discovering exoplanets similar to Earth, with possible signs of life. Being the most common type of star in our galaxy, there is potentially a higher chance that an exoplanet harbouring life would be found orbiting a red dwarf (as opposed to a Sun-like star, for example).

Discuss the different ways in which astronomers use spectral lines.

Doppler shift occurs when a wave source moves relative to the observer or the observer moves relative to the source: the detected frequencies will be shifted up or down the spectrum, depending on whether the object’s relative movement is toward or away from the observer. By measuring the Doppler shifting of spectral lines, astronomers can detect a radial velocity component of stars; if this component is oscillating, this could suggest that it is being orbited by a planet. Also, astronomers use spectral lines to discern the properties of stars - for example, the Sun’s absorption lines can be used to produce models that are used to work out properties such as temperature, chemical composition and surface gravity. (Absorption lines occur because gas from the star’s outer layers absorbs specific wavelengths of light.) Similar methods can be used on red dwarfs if their infrared spectra are analysed - their temperature and metallicity can be deduced.

Compare the techniques of microlensing and transit photometry for finding exoplanets.

Gravitational lensing is when the gravitational pull of a massive object causes light from distant sources to bend or distort. Microlensing is a form of gravitational lensing, whereby a star in the foreground acts as a lens and bends the light from a background source. When an exoplanet is present, if a lucky alignment occurs, astronomers observe an insightful blip in the brightening curve caused by the foreground lens star. The microlensing method is best at discovering heavier exoplanets and planets far from Earth with orbits far from their stars, but is not as good for finding multiple exoplanets or detecting the same one again.

The transit photometry method involves detecting light from stars and looking for a dimming, which may indicate that an exoplanet is transiting across the star (blocking some of the star’s light as it passes along its orbit). The observed dimming is inversely proportional to the scale of the star’s radius, so this method works better for smaller stars like red dwarfs. Additionally, unlike the microlensing method, this method is better for exoplanets in close orbits. It can be used to calculate the diameters of exoplanets and is a good method for space telescopes like Kepler, but has its limitations: it cannot detect exoplanets that do not cross stars, or accurately estimate exoplanet mass.

SUMMARY QUESTIONS - PHINEHAS

Why do you think red dwarfs are such an interesting area of research?

If our eyes were suited to longer wavelengths near the infrared part of the spectrum, the sky we see would be filled with Red Dwarfs. Due to this and other reasons, these stars (despite being the most abundant in the galaxy) are widely overlooked making it a very interesting area of research. Red Dwarfs are much dimmer compared to stars like our sun which makes it easier to find planets that may surround them called exoplanets and perhaps Earth like planets within the habitable zone. Techniques such as ‘microlensing’ and ‘transit photometry’ have made it much easier to find exoplanets and as a result it has been shown that about 40% of Red Dwarfs host ‘super earth’ planets potentially containing liquid water and liquid oceans. Other reasons why red dwarfs are intriguing is because of their importance. They are used to calculate the age of star clusters as they are so long-lived and they make it easier to study planetary formation as well as ‘the nature of planetary systems’ that orbit it

 Discuss the different ways in which astronomers use spectral lines.

The reasons astronomers know so much about objects and places so far away form Earth is due to the information they get from light. This tells us how bright they are, how they are moving, their composition and this case things that relate to unravelling mysteries about Red Dwarfs. Light is electromagnetic radiation and the different forms of light make up the electromagnetic spectrum. Therefore, information that displayed in the form of wavelength from stars like red dwarfs, is said to be spectra. Astronomers can study the shapes of the lines in such spectra which can provide information about densities, motions and magnetic fields to understand the ‘architecture’ of systems such as ‘Gl 876’. Atoms or ions emit light waves that are specific to it know as emission lines on a spectrum. These changes in the wavelength of the spectral lines can be used to measure motions of stars. This is done by measuring the changes in the wavelength of the emission lines. The Doppler effect further aids in measuring such motions because the wavelength appears shorter when the source approaches, and longer when the source moves away. Therefore, the intensity and changes in the spectra can be used to identify many things about stars. However, there are still types of motion that cannot be detected by spectral lines such as ‘proper motion’ discussed in the article.

Compare the techniques of microlensing and transit photometry for finding exoplanets

The transit method has been the most successful method of discovering exoplanets thus far. It relies on the portion of light that is blocked out when exoplanets pass in front of their host star and measurements are taken. These measurements allow for the creation of a light curve and models to the light curve can extract various data such as orbital motions and atmospheric composition. How much a star dims during a transit is directly related to the relative sizes of the star and the planet. When a small planet transits a large star there is only a slight dimming whereas a large planet while have a larger effect. The main advantage of this method is that it is very sensitive and allows for measurements of physical properties otherwise not measurable. It can also be used effectively with another method for discovering exoplanets: radial- velocity. The main drawback of the method, is that a transit needs to occur which lasts only a tiny fraction of the total orbital period. And not all planets transit their star.

Microlensing unlike the transit method and the radial velocity method does not rely on detecting variations in light from a star but the effect of gravity on light. Predicted by Albert Einstein in his general theory of relativity, it involves understanding that objects with a large mass can bend light around them. The light that is bent can act as a lens and can be enlarged. Usually there will be a smooth light curve as a result however, if a bump appears there is a planet orbiting the star acting as a lens. Unlike the transit method, microlensing is capable of finding the most distant and smallest of planets and even planets that are not orbiting any stars. It is also similar to transit photometry in the sense it targets numerous thousands of planets simultaneously. However, microlensing is dependent on random events that sometimes are extremely rare some planets detected by microlensing may never be observed again. Planets detected thousands and thousands of light years away are based on approximations and there could be a large uncertainty on how far away they actually are; even a few thousand years.

INTRODUCTION

M is the symbol use for a solar mass (the mass of the Sun).

(P1, C1&C2) Red dwarf

A type of star that emits in the infrared part of the spectrum. They are generally smaller than the Sun in terms of mass and radius (they have masses between 0.1 and 0.5 solar masses) and are significantly less luminous (as luminosity scales as M4). But they’re very numerous, and 70% of the stars in the Milky Way are red dwarfs.

(P1, C1) Radiative diffusion

The name given to the process of energy transfer from a star’s core to the surface.

(P1, C1) Hydrostatic equilibrium

The situation in which the forces on the star are balanced. The inwards pull of gravity on the star (from itself) is balanced by the pressure gradient caused by the nuclear reactions (which is trying to expand the star).

(P1, C1) Luminosity

The power output (amount of energy transfer per second) from the surface of a star.

(P1, C1) Luminosity scaling with mass.

Luminosity scales as the fourth power of a star’s mass.

(P1, C2) Black body

An idealised body which can absorb (and emit) radiation at all wavelengths. Contrast this to an element which can only absorb radiation

(P1, C2) Exoplanet

Any planet which orbits a star that is not our own. To be a planet, an object must: orbit a star, be in hydrostatic equilibrium, dominate its orbit.

 INTRODUCTION SUMMARY QUESTIONS

Why are red dwarfs significantly fainter than the Sun?

Because whilst their masses are between 10% and 50% of the Sun, the luminosity scales as M4. As 0.14=0.0001 and 0.54=0.0625, the luminosity of red dwarfs ranges between 0.01% and 6.25% that of the Sun.

Why have astronomers focussed their attention on sun-like stars when looking for exoplanets rather than red dwarfs?

 

What information can you extract from the graph shown in Figure 1 (take note of the unusual x-scale).

Red dwarfs have masses less than 0.6 M. There are significantly more red dwarfs than sun-like stars. As you look at lower masses, we see more and more stars. The distribution is not perfect but samples stars in a generally small region of the universe.

 EARLY DISCOVERIES

(P2, C1) Barnard’s star

The fourth nearest star to the Sun, at a distance of about 6light-years away. It’s red dwarf with a mass of 0.144 M.

(P2, C1) Light year

A unit of distance. It is equivalent to the distance that light can travel, in a vacuum, in a period of one earth year. 1 light year = 9.4607 × 1012 km (nearly 6 million million miles)

(P2, C1) Proper motion (HINT: really think about the diagram at this link)

The stars in the sky are not stationary, despite constellations looking fixed. The proper motion is the angular rate that an astronomical object appears to move across the sky from our point of view. It depends on how far away it is and the component of the object’s velocity that is perpendicular to the line joining the object to us.

(P2, C1) Van de Kamp’s wobbles

The apparent wobbles in the motion of Barnard’s star made Van de Kamp think it was being pulled by its orbiting planets. Further observations showed that the wobbles were erroneous, due to systematic errors in the measurement of position.

This image may be useful for some of the ideas in P2, C2

Taken from Wikipedia https://en.wikipedia.org/wiki/Proper_motion

(P2, C2) Plane of the sky

A plane that is tangent to the celestial sphere and perpendicular to your line of sight.

(P2, C2) Doppler shift

The apparent shift in wavelength of any wave when the emitter and receiver are in motion relative to one another. If they are moving away, we have red shift (the wavelength increases: light shifts towards the redder end of the spectrum, sound becomes lower in pitch). If they are moving towards, we have blue shift (wavelength shortens: light shifts towards the blue end of the spectrum, sound becomes higher pitch).

(P2, C2) Spectral lines

Lines in the electromagnetic spectrum that correspond to specific wavelengths of light. Each atom has a specific ‘fingerprint’ of spectral lines due to the energy levels that the electrons can move between. As the energy levels are discrete, only certain wavelengths of light can be absorbed (and emitted) by each atom or molecule. We can use emission/absorption spectra to identify atoms.

(P2, C2) Does Barnard’s star have planets orbiting?

Observations of its motion suggest (both direct observations of its motion in the plane of the sky and of its radial motion compared to us via Doppler measurements) suggest that it is not wobbling due to the gravitational pull of planets.

(P2, C2) Gl 876

A red dwarf with a mass of 0.3M that is around 15 lightyears from us. It is the first red dwarf discovered to have planets orbiting it via Doppler measurements.

(P2, C2) Astronomical unit

The distance between Earth and the Sun. It’s a useful unit for comparing the orbits of planets around their stars. 1au = 150 million km.

(P2, C2) Gl 876b and Gl 876c

Two planets that orbit the red dwarf Gl 876, found from the Doppler shifting of spectral lines on Gl 876 showing its motion relative was being altered by gravitational effects. Gl 876b is a gas giant with twice the mass of Jupiter. Gl 876c is another gas giant has a mass of 0.7 Jupiter masses. Their orbital periods are both less than 100 days.

(P3, C1) Protoplanetary disc

A dense, rotating disc of dust and gas around a star that might eventually form the planets of the star’s solar system.

(P3, C1) Super earth

Exoplanets with masses greater than Earth but less than that of Uranus and Neptune. The term does not imply a rocky makeup and many so-called super-Earths are in fact gaseous – the terms gas dwarf and mini Neptune are also used.

(P3, C1) Observational bias

When our measurement technique itself is biased towards finding certain objects. For example, our eye has an observational bias towards stars that emit visible light.

 EARLY DISCOVERIES SUMMARY QUESTIONS

Why do astronomers need to use two different techniques to measure the velocity of an astronomical object through the universe?

As the distances and speeds are so disparate to our everyday experiences, measuring velocities is not simple. In the ‘plane of the sky’, we can measure the velocity of an astronomical object as we see it move amongst other objects in the sky. We can use the technique of parallax to measure how distances change over time in the plane of the sky. But, if an object is moving away from or towards us (or has a component in this direction), we don’t detect this in the movement in the plane of the sky. Here, we use Doppler measurements to see how spectral lines are shifted in the light coming from the object.

How do astronomers use spectral lines?

They compare the spectral lines from an astronomical object to the spectral lines from atoms on earth. As stars contain Hydrogen and Helium, the spectral lines from a star should look the same as lab samples of hydrogen and helium. Any shifts are due to the relative motion of the star and us.

How are planets detected around red dwarfs?

Through precise Doppler measurements of the velocity of the star to see the wobbles that occur due the gravitational pull of the planets on the star.

Why are gas giants rare to find orbiting red dwarfs?

Because red dwarfs tend to have low mass protoplanetary discs. Gas giants tend to form around dense, rocky cores that accumulate gas over time. If no such rocky cores exist, then gas giants can’t form and this seems to be the case, typically, for planetary systems surrounding red giants.

 MICROLENSING

(P3, C1) Curving of spacetime

Massive objects (any object with mass) curves space and time around itself. For most objects, this gives no noticeable effect, but for larger objects like stars and galaxies, they can dent spacetime in such a way as to even affect the path of light. We looked at this somewhat in Week 2 when we considered LIGO. This video gives a great visualisation of the curving of spacetime.

(P3, C2) Microlensing (this GIF might help. It’s also helpful to think of it in terms of brightness, I find).

A form of gravitational lensing in which the light from a background source is bent by the gravitational field of a foreground lens. We therefore receive more light than we would have done, either by seeing multiple images (that may be significantly distorted) or, if the alignment is right, an Einstein ring.

(P3, C2) Einstein ring

The effect of gravitational lensing under conditions of perfect alignment where the observer, lens and source are in a perfect line. As the light from the source is bent in all directions by the lens star, we see a ring formed as the image.

 MICROLENSING SUMMARY QUESTIONS

What happens to the brightness of a source star as a lens star moves across our line of sight?

When the lensing is in effect, the brightness increases. The lens object acts to bend more light to our observation. Once the lens star moves out of the way again, so that the light is no longer affected by the perturbation in spacetime, the brightness decreases again to the original level.

How might a planet, if placed in a fortuitous position, alter this lensing by the lens star?

Some of the ‘lensed light’ caused by a lensing star may pass by planets of the lensing star, this creates additional lensing events and an additional sudden increase and decrease in the brightness – see this link.

Why is the microlensing technique biased towards finding heavier planets?

As the degree of additional lensing from the exoplanets will depend on the level of spacetime curvature and hence the mass of the exoplanet. The more massive the planet, the greater the additional lensing effect and the easier it is to detect.

 TRANSITING EXOPLANETS

(P4, C1) M Dwarf

Name given for Earth-sized planets that orbit red dwarf stars.

(P4, C1 & C2) Transit method

By monitoring the brightness of a star, if a planet intersects our line of sight to the star then it will block some of the light, leading to a reduction in brightness (related to the size of the planet). The dips in brightness will come at regular intervals as the orbit of the planet remains stable.

(P4, C1) GJ 1214 and GJ 1214b

GJ 1214 is a red dwarf star 39 light-years from Earth. GJ 1214b is a super-Earth planet that orbits GJ 1214,

(P4, C2) Habitable zone

The region around a star where water can exist in its liquid state. The region is not too close to the star so as to boil the water, and not too far away from the star so that the water freezes.

 TRANSITING EXOPLANETS SUMMARY QUESTIONS

Why is an earth sized planet easier to detect if orbiting a red dwarf?

A red dwarf is a smaller sized star. The dip in light is proportional to so if the star is half as big, then the dip in light will be four times larger.

Why is it easier to detect planets in the habitable zone around red dwarfs than sun-like stars?

 

 

 

It is easier to detect planets that are close to stars using the transit method. The habitable zone of red dwarf stars is very close to the star, whereas the habitable zone around sun-like stars is much further away. This allows the transit method to easily detect planets that are more likely to harbour liquid water.

(You’ll need to read part of the next section to answer this). What are the downsides of using ground-based telescopes to find exoplanets with the transit method?

The atmosphere of the Earth varies in its transparency, so our measurements of brightness are hampered, meaning that ground-based techniques only see larger dips in brightness from stars. Additionally, when the Sun is in our line of sight to the star, the measurements are obviously useless.

KEPLER’S RED DWARF CENSUS

(P4, C2) Kepler space telescope

A space telescope used for detecting Earth-sized planets via the transit method.

(P4, C2) CCD

Charge coupled devices are high quality image sensors

 KEPLER’S RED DWARF CENSUS SUMMARY QUESTIONS

What advantages does Kepler have over ground-based methods?

It collects more photons and is therefore more precise. It can operate for 24hours a day, unhindered by weather conditions.

Why can Kepler make more accurate measurements of the transits of planets around brighter stars?

As there are more photons from the brighter stars. The more photons there are, the more accurately you can determine any changes that occur.

What difficulty do red dwarfs present when hunting for planets?

As red dwarfs are smaller and dimmer, it is hard to accurately determine their radius. The radius of the star is needed to accurately determine the radius of any orbiting planets.

 STELLAR SPECTRA

(P5, C1) Absorption lines

An atom can absorb photons that have an energy equal to any of the differences in energy of electron energy levels. This gives each atom or molecule its own absorption signature as electrons jump between very different energy levels. If we sent an entire spectrum of light through a gas (e.g. hydrogen), only the photons corresponding to the discrete energy level jumps would be absorbed and the rest would pass through unencumbered. If we detect this spectrum afterwards, we will see dark lines – the absorption lines – at energies corresponding to the absorption events.

(P5, C2) Parallax measurements

Measurements of the apparent movement of an astronomical object relative to ‘background stars’ when the earth is at opposite sides of its orbit around the Sun allow us to estimate the distance to the object.

(P5, C2) Metallicity

The abundance of elements heavier than hydrogen or helium in an object. These elements aren’t necessarily metallic at all, but metal is astronomer short-hand for “elements heavier than H or He”.

STELLAR SPECTRA SUMMARY QUESTIONS

Why are the properties of red dwarfs not as well understood as sun-like stars?

We have studied the Sun extensively so can make easy comparisons between the Sun and similar stars. The absorption lines of red dwarfs are not well understood and without a comparison to the Sun to rely on

How do astronomers deduce the mass of a red-dwarf?

There is a correlation between the luminosity of red dwarfs in the infrared part of the spectrum and their mass.

As the luminosity is the total power output of the star, but we can only measure the received power on earth, we need to know how far away the red dwarf is to calculate its luminosity. You can imagine this similarly with a light bulb and a light meter. The light bulb has a fixed power output (its ‘luminosity’) but, the further away we are from the bulb, the dimmer it appears to us – this is because the power is spread out over a larger and larger sphere as we get further away.

 A MINIATURE

(P6, C1) Kepler 42

A small, red dwarf star that has three small planets orbiting it, all with small orbits and short orbital periods.

 A MINIATURE SUMMARY QUESTIONS

What does Figure 5 show?

The size distribution of planets around red dwarf systems, whose orbital periods are less than 150 days. On average, red dwarf systems have 2 planets. The planets are more likely to be smaller (between 0.5 and 1.5 times the earth’s radius). And there are fewer and fewer planets as we look for larger planets. Out of more than 100 similar red dwarf systems, only one contained a planet as large as Jupiter.

What do studies of red dwarf systems suggest about Earth-like planets?

The data suggests that, on average, there should be one Earth-sized planet in the habitable zone for every two red dwarf systems.

 FROM HUNTERS TO GATHERERS SUMMARY QUESTIONS

Why will the next generation of planet hunting instrumentation likely be tailored towards the infrared portion of the electromagnetic spectrum?

As red dwarf systems seem to host a large proportion of earth-sized planets in their habitable zones (0.5 such planets per red dwarf on average). As red dwarfs are more luminous in the near-IR, new instrumentation will likely focus on this.

How will astronomers look for life on planets they think have potential?

They’ll look at absorption spectra from the red dwarf to see the superposed atmospheric features of the planet’s atmosphere as it transits the star. These features will be compared to samples of elements and compounds on Earth to understand which elements/compounds are present in the planet’s atmosphere.

GENERAL INFORMATION

Remember, reading a paper isn't like reading a piece of fiction or a newspaper article. Don't get frustrated if it doesn't immediately make sense - you might need to do a little research of your own to understand some of the ideas. This article gives you an idea of how scientists read differently.

Each question refers to a specific part of the paper e.g. Page 2, Column 3 is written as (P2, C3).

Next week, we'll publish solutions to the questions and the best submitted summaries from students across the country.

NEXT WEEK

We're going to be looking at how one cunning scientist used physics to get out of a traffic offence in the USA. This is more of a theoretical paper and involves some small aspects of the mathematical techniques of integration and differentiation. We will have a separate activity if you're in Y11 and don't feel comfortable with these new concepts but you should hopefully be able to understand it anyway (especially given the work I've seen from Y11 students so far).